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Splitting method is a powerful method to handle application problems by splitting physics, scales, domain, and so on. Many splitting algorithms have been designed for efficient temporal discretization. In this paper, our goal is to use temporal splitting concepts in designing machine learning algorithms and, at the same time, help splitting algorithms by incorporating data and speeding them up. We propose a machine learning assisted splitting scheme which improves the efficiency of the scheme meanwhile preserves the accuracy. We consider a recently introduced multiscale splitting algorithms, where the multiscale problem is solved on a coarse grid. To approximate the dynamics, only a few degrees of freedom are solved implicitly, while others explicitly. This splitting concept allows identifying degrees of freedom that need implicit treatment. In this paper, we use this splitting concept in machine learning and propose several strategies. First, the implicit part of the solution can be learned as it is more difficult to solve, while the explicit part can be computed. This provides a speed-up and data incorporation for splitting approaches. Secondly, one can design a hybrid neural network architecture because handling explicit parts requires much fewer communications among neurons and can be done efficiently. Thirdly, one can solve the coarse grid component via PDEs or other approximation methods and construct simpler neural networks for the explicit part of the solutions. We discuss these options and implement one of them by interpreting it as a machine translation task. This interpretation of the splitting scheme successfully enables us using the Transformer since it can perform model reduction for multiple time series and learn the connection between them. We also find that the splitting scheme is a great platform to predict the coarse solution with insufficient information of the target model: the target problem is partially given and we need to solve it through a known problem which approximates the target. Our machine learning model can incorporate and encode the given information from two different problems and then solve the target problems. We conduct four numerical examples and the results show that our method is stable and accurate. • We have designed a machine learning strategy that is based on stable splitting for multiscale problems. • Our approach uses implicit and explicit discretization combined with the Transformer for multiscale problems to solution prediction. • Numerical results are presented that show the proposed methods provide good prediction and accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the numerical algorithms to capture the blow-up time for a class of convection-diffusion equations with blow-up solutions. The numerical methods for such equations may not be straightforward to construct due to the lack of stability. Moreover, the blow-up time is more difficult to capture since we cannot distinguish whether the blow-up is physical or is due to the instability of the numerical methods. In this paper, we consider a class of convection-diffusion equations with positive blow-up solutions and the blow-up is due to the formation of δ -singularities. We use the positivity-preserving technique to enforce the L 1 -stability and the L 2 -norm of the numerical approximations to detect the blow-up phenomenon. We propose two ways to define the numerical blow-up time and prove their convergence to the exact one. As an application, we extend this method to calculate when the shock appears for scalar hyperbolic conservation laws. Three model problems will be discussed and tested to confirm the convergence numerically. Finally, the method can also be used to test whether an equation has a blow-up solution in finite time. • Construct a sequence of numerical blow-up time and prove its convergence. • Capture the blow-up of the derivative of the primitive function. • Test whether the blow-up appears within the target time interval. • The idea works for general convection-diffusion equations with blow-up solutions. [ABSTRACT FROM AUTHOR]

• Analytical formulas are derived for force and moment coefficients with a manufactured solution. • The formulas allow a quick and rigorous verification of force computation algorithms. • Error cancellations lead to irregular non-monotonic error convergence in total coefficients. • Pressure and viscous contributions should be verified separately. In this paper, we derive analytical formulas for the drag, lift, and moment coefficients of a circular cylinder exposed to a fictitious flow defined by analytical functions in two and three dimensions, and demonstrate that these formulas serve as a useful tool for quickly verifying the implementation of force and moment computation algorithms in computational fluid dynamics solvers. Focusing on a typical second-order unstructured-grid solver, we will show that the lift and drag coefficients consist of second-order accurate pressure and first-order accurate viscous force coefficients, and thus it is not always possible to uniquely determine their orders of accuracy although all will be first-order accurate on fine enough grids. For the circular cylinder geometry, the pitching moment coefficient depends only on the viscous stresses and is therefore first-order accurate. In three dimensions, rolling and yaw moments have both pressure and viscous contributions, and therefore they are asymptotically first-order accurate. Viscous contributions to the force and moment coefficients are first-order accurate even for regular grids because the gradient stencil is not symmetric typically at a boundary and any linearly-exact gradient algorithm can be first-order accurate at best; hence, the velocity gradients used to compute the viscous stresses at a wall are first-order accurate. For verification, therefore, it is necessary to verify the pressure and viscous contributions separately; it can be performed rigorously with the analytical formulas presented in this paper. [ABSTRACT FROM AUTHOR]

This is the first of two companion papers that introduce a new class of positivity-preserving, entropy stable schemes for the 1-D compressible Navier-Stokes equations. In this paper, we develop and analyze a novel first–order finite volume scheme that is discretized on Legendre-Gauss-Lobatto grids used for high-order spectral collocation methods. The scheme is constructed by regularizing the Navier-Stokes equations by adding artificial dissipation in the form of the Brenner-Navier-Stokes diffusion operator. The key distinctive property of the proposed scheme is that it is proven to satisfy the discrete entropy inequality and to guarantee the pointwise positivity of density, temperature, pressure, and internal energy for 1–D compressible viscous flows, while using high–order spectral collocation summation-by-parts operators for discretizing the Navier–Stokes viscous terms. To eliminate time step stiffness caused by the high-order approximation of the viscous terms, we develop two new limiters that bound the magnitude of velocity and temperature gradients and preserve the entropy stability and positivity properties of the baseline scheme. Numerical results demonstrating positivity-preserving and discontinuity-capturing capabilities of the new scheme are presented for viscous and inviscid flows with nearly vacuum regions and very strong shocks and contact discontinuities. • Novel first-order artificial dissipation based on the Brenner diffusion operator. • New 1st-order positivity-preserving L 2 stable scheme for 1D Navier-Stokes equations. • New positivity-preserving entropy stable velocity and temperature limiters. [ABSTRACT FROM AUTHOR]

This is the second of two companion papers, in which we develop and analyze a new class of positivity-preserving, entropy stable spectral collocation schemes of arbitrary order of accuracy for the 1-D compressible Navier-Stokes equations. The key distinctive property of the proposed methodology is that it is proven to guarantee the pointwise positivity of thermodynamic variables for compressible viscous flows. The new schemes are constructed by combining a positivity-violating entropy stable method of arbitrary order of accuracy and a novel first-order positivity-preserving entropy stable method developed in the companion paper [1]. This first-order scheme is discretized on the same Legendre-Gauss-Lobatto points used for the high-order counterpart. As a result, no interpolation or restriction is required between high- and low-order elements which are coupled by using the simultaneous approximation term (SAT) penalty method. To provide positivity preservation and excellent discontinuity-capturing properties, the Navier-Stokes equations are regularized by adding artificial dissipation in the form of the Brenner-Navier-Stokes diffusion operator. The high- and low-order schemes are combined by using a limiting procedure, so that the resultant scheme provides conservation, guarantees pointwise positivity of thermodynamic variables, preserves the design–order of accuracy of the corresponding high-order baseline spectral collocation method for smooth solutions, and satisfies the discrete entropy inequality, thus facilitating a rigorous L 2 -stability proof for the symmetric form of the discretized Navier-Stokes equations. The proposed methodology is very general and can be extended to multiple dimensions and other summation-by-parts SAT-type schemes. • A new physics-based artificial viscosity built as a function of entropy residual. • A novel high-order entropy stable pointwise positivity-preserving flux limiter. • New high-order positivity-preserving L 2 stable schemes for Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

Perfectly Matched Layer (PML) technique is an effective tool to reduce the unbounded wave propagation problem to a bounded domain problem. In this paper, we develop and analyze a fourth-order in space finite difference scheme for solving the Ziolkowski's PML model. Though fourth-order in space schemes have been widely used in solving Maxwell's equations, few papers provide a rigorous convergence analysis. One of the major contributions here is to fill this gap by proving the fourth-order pointwise convergence of the scheme. Numerical results are presented to justify our analysis, and demonstrate the effectiveness of this PML model in absorbing outgoing waves. • The very few papers devoted to convergence analysis for 4th-order spatial FDTD scheme for PML model. • No rigorous error estimate has been proved for 4th-order FDTD scheme, to our best knowledge. • The proof of stability analysis and convergence estimate are original. • Numerical results demonstrate the effectiveness of this PML in absorbing outgoing waves. [ABSTRACT FROM AUTHOR]

We propose a generalized Eulerian-Lagrangian (GEL) discontinuous Galerkin (DG) method. The method is a generalization of the Eulerian-Lagrangian (EL) DG method for transport problems proposed in Cai et al. (2021) [5] , which tracks solution along approximations to characteristics in the DG framework, allowing extra large time stepping size with stability. The newly proposed GEL DG method in this paper is motivated for solving linear hyperbolic systems with variable coefficients, where the velocity field for adjoint problems of the test functions is frozen to constant. In this paper, in a simplified scalar setting, we propose the GEL DG methodology by freezing the velocity field of adjoint problems, and by formulating the semi-discrete scheme over the space-time region partitioned by linear lines approximating characteristics. The fully-discrete schemes are obtained by method-of-lines Runge-Kutta methods. We further design flux limiters for the schemes to satisfy the discrete geometric conservation law (DGCL) and maximum principle preserving (MPP) properties. Numerical results on 1D and 2D linear transport problems are presented to demonstrate great properties of the GEL DG method. These include the high order spatial and temporal accuracy, stability with extra large time stepping size, and satisfaction of DGCL and MPP properties. • We develop a generalized Eulerian Lagrangian (EL) discontinuous Galerkin (DG) method. • We establish connection of the generalized EL DG method with the EL DG method. • Numerical results on linear transport problems. [ABSTRACT FROM AUTHOR]

This paper is focused on the optimization approach to the solution of inverse problems. We introduce a stochastic dynamical system in which the parameter-to-data map is embedded, with the goal of employing techniques from nonlinear Kalman filtering to estimate the parameter given the data. The extended Kalman filter (which we refer to as ExKI in the context of inverse problems) can be effective for some inverse problems approached this way, but is impractical when the forward map is not readily differentiable and is given as a black box, and also for high dimensional parameter spaces because of the need to propagate large covariance matrices. Application of ensemble Kalman filters, for example use of the ensemble Kalman inversion (EKI) algorithm, has emerged as a useful tool which overcomes both of these issues: it is derivative free and works with a low-rank covariance approximation formed from the ensemble. In this paper, we work with the ExKI, EKI, and a variant on EKI which we term unscented Kalman inversion (UKI). The paper contains two main contributions. Firstly, we identify a novel stochastic dynamical system in which the parameter-to-data map is embedded. We present theory in the linear case to show exponential convergence of the mean of the filtering distribution to the solution of a regularized least squares problem. This is in contrast to previous work in which the EKI has been employed where the dynamical system used leads to algebraic convergence to an unregularized problem. Secondly, we show that the application of the UKI to this novel stochastic dynamical system yields improved inversion results, in comparison with the application of EKI to the same novel stochastic dynamical system. The numerical experiments include proof-of-concept linear examples and various applied nonlinear inverse problems: learning of permeability parameters in subsurface flow; learning the damage field from structure deformation; learning the Navier-Stokes initial condition from solution data at positive times; learning subgrid-scale parameters in a general circulation model (GCM) from time-averaged statistics. • A filtering-based method for inverse problems with novel stochastic dynamical systems. • Derive extended, ensemble, and unscented Kalman inversions for inverse problems. • The method induces a form of Tikhonov regularization to overcome ill-posedness. • Prove that the method features exponential convergence for linear inverse problems. • Test on 8 inverse problems, including calibrating a 3D climate model. [ABSTRACT FROM AUTHOR]

• Construct implicit pressure explicit concentration methods. • Add a correction step to gain second-order accuracy in time. • The method saves 90% of the CPU time. • The full algorithm preserves the physical bounds of the concentration. In this paper, we construct bound-preserving interior penalty discontinuous Galerkin (IPDG) methods with a second-order implicit pressure explicit concentration (SIPEC) time marching for the coupled system of two-component compressible miscible displacements. The SIPEC method is a crucial innovation based on the traditional second-order strong-stability-preserving Runge-Kutta (SSP-RK2) method. The main idea is to treat the pressure equation implicitly and the concentration equation explicitly. However, this treatment would result in a first-order accurate scheme. Therefore, in all previous works, only the combination of forward and backward Euler time integrations was considered. In this paper, we propose a correction stage to compensate for the second-order accuracy in each time step. There are two main difficulties in constructing a second-order scheme. Firstly, in the concentration equation, correction of the diffusion term will cause anti-diffusion, leading to malfunction of the bound-preserving technique. We can deal with the velocity in the diffusion term explicitly to avoid correction of the diffusion term. Secondly, we need to ensure that the bound-preserving technique for the convection and source terms can be applied when the correction stage has been established. In fact, in the correction stage, the new approximation to the concentration can be chosen as the numerical solution in the previous stage, so the numerical cell averages are positivity-preserving. Moreover, we use the same correction technique for the pressure, so that the consistent flux pairs would guarantee the preservation of the upper bound 1 of the concentration. Numerical experiments will be given to demonstrate that the proposed scheme can reduce the computational cost significantly compared with explicit schemes if the diffusion coefficient D is small in the concentration equation. Moreover, the proposed method also yields much larger cfl number compared with first-order implicit pressure explicit concentration schemes. Moreover, the effectiveness of the bound-preserving technique will also be presented. [ABSTRACT FROM AUTHOR]

• We have designed a generalized multiscale model for heterogeneous Cosserat media without scale separation. • We have designed multiscale spaces with and without coupling of processes and investigated them. • Numerical results are presented that show the proposed methods provide a good accuracy with a few degrees of freedom. In this paper, we develop a multiscale computational approach for heterogeneous Cosserat media without scale separation and high contrast. Cosserat medium has been used in many applications to describe materials whose particles possess rotational degrees of freedom. Many previous findings investigate Cosserat medium in the presence of heterogeneities. Approaches that include homogenization and numerical homogenization have been developed for problems with scale separation, such as periodicity. In these approaches, macroscopic models are derived that show the resulting equations can be Cauchy medium or Cosserat medium depending on the Cosserat intrinsic length scale. In our paper, we consider computational macroscopic models for Cosserat medium without scale separation. We use Generalized Multiscale Finite Element Method (GMsFEM) to derive a macroscopic model on a coarse computational grid, that does not resolve the small scales and the contrast. In this approach, multiple macroscopic basis functions are used to describe the small scales. The resulting macroscopic model is similar to multicontinua model as it contains multiple macroscopic parameters at each macroscopic point (or grid). We present numerical results for different heterogeneous media types. Our numerical results show that using a few multiscale basis functions (i.e., macroscopic parameters), we can achieve a good accuracy. [ABSTRACT FROM AUTHOR]

• Identification of conditions that should be ensured by a Particle Shifting Technique. • Theoretical study of existing PSTs linked to the targeted list of conditions. • Proposition of an enhanced PST ensuring the targeted conditions. • Validation on 2D and 3D test cases involving free-surface and solid boundaries. This paper addresses the Particle Shifting Technique (PST) in the SPH schemes. Improving the accuracy of SPH schemes leads to particle clustering along the flow streamlines which turns to be detrimental for the simulations. PSTs aim at avoiding this adverse effect by slightly disordering the particles, allowing to retrieve a regular particle distribution within the kernel interpolation support. The gain in accuracy is such that this technique is now commonly adopted by the SPH practitioners, however the conditions that should be respected by a PST are not clearly discussed in the literature. In this paper, such conditions are exposed and their fulfillment by the main existing PSTs of the literature is analyzed. None of these existing PSTs fully satisfying these conditions, a novel PST is introduced. The proposed PST is validated for three different SPH schemes on 2D and 3D test cases, in presence of free-surface and solid boundaries. [ABSTRACT FROM AUTHOR]

• The pressure term follows directly from a first-principles form of the pressure force. • It involves integration over the entire water column lying on a grid cell. • If the free surface is level, the integrand is zero, so the pressure term is zero. • This representation of pressure forcing is therefore well-balanced. • This work is one step towards handling bottom topography in multi-layer ocean models. This paper begins a development of methods for addressing variable bottom topography in discontinuous Galerkin numerical methods for multi-layer, variable-density models of ocean circulation. For numerical models of ocean circulation, it is a widespread practice to split the fast (external) and slow (internal) dynamics into separate subsystems that are solved by different techniques. The fast dynamics are modeled by a vertically-integrated system that is very similar to the shallow water equations for a hydrostatic fluid of constant density. As a first step, the present paper focuses on variable bottom topography for the shallow water equations; extensions to the multi-layer case will be reported elsewhere. A central point of this work, for both the shallow water and the multi-layer cases, is the representation of the pressure forcing in the momentum equations. For the shallow water case, the present work does not use the standard representation of the pressure forcing that is widely used for the shallow water system. Instead, it begins with a more basic form of the momentum equations for a fluid flow, and it proceeds directly to a weak Galerkin form via integration over a suitable fluid region. The resulting formulation of the momentum equations is automatically well-balanced, subject to an assumption that the algorithms used to compute quantities at cell edges reproduce the continuous values in the case where all functions involved are continuous. In addition, this formulation has a structure that is analogous to that of the momentum equations in the individual layers of a multi-layer fluid, and this facilitates consistency between the two subsystems that are used to model such a fluid. The numerical computations described here include tests of well-balancing and tests of wave propagation over variable topography. [ABSTRACT FROM AUTHOR]

• Highlight of this paper The highlight of the paper is in the following aspects. • The dynamic and adaptive way of evolving low rank basis for the Vlasov equation. • The hierarchical Tucker tensor techniques for high dimensional Vlasov equation. • The curse of dimensionality is being alleviated for a grid based high order method. • The low rank evolution of the flow map. We propose a low-rank tensor approach to approximate linear transport and nonlinear Vlasov solutions and their associated flow maps. The approach takes advantage of the fact that the differential operators in the Vlasov equation are tensor friendly, based on which we propose a novel way to dynamically and adaptively build up low-rank solution basis by adding new basis functions from discretization of the PDE, and removing basis from an SVD-type truncation procedure. For the discretization, we adopt a high order finite difference spatial discretization and a second order strong stability preserving multi-step time discretization. We apply the same procedure to evolve the dynamics of the flow map in a low-rank fashion, which proves to be advantageous when the flow map enjoys the low rank structure, while the solution suffers from high rank or displays filamentation structures. Hierarchical Tucker decomposition is adopted for high dimensional problems. An extensive set of linear and nonlinear Vlasov test examples are performed to show the high order spatial and temporal convergence of the algorithm with mesh refinement up to SVD-type truncation, the significant computational savings of the proposed low-rank approach especially for high dimensional problems, the improved performance of the flow map approach for solutions with filamentations. [ABSTRACT FROM AUTHOR]

In this paper, we mainly concern with the order reduction for the unknown solution coefficient vectors about the classical continuous space-time finite element (CCSTFE) method of the two-dimensional (2D) non-stationary incompressible Navier-Stokes equations. Therefore, we first build the CCSTFE scheme for the 2D non-stationary Navier-Stokes equations with respect to stream–vorticity functions, give the existence and stability together with convergence to the CCSTFE solutions, and rewrite the CCSTFE scheme as a matrix form. And then, we use a proper orthogonal decomposition (POD) technique to build a reduced-order extrapolated continuous space-time finite element (ROECSTFE) scheme in the matrix form including very few unknowns but possessing high enough accuracy, employ matrix analysis to demonstrate the existence and stability together with convergence of the ROECSTFE solutions such that the theoretical analysis becomes very concise and convenient, and summarize the flowchart for solving the ROECSTFE scheme. Finally, we take advantage of the numerical simulations for the back-step flow and the flow around airfoil problem to verify the correctness of theoretical results. • The continuous space-time finite element (CSTFE) method for unsteady Navier-Stokes equations is proposed for the first time. • The reduced-order extrapolated CSTFE scheme is established in this paper for the first time. • Using matrix tool to argue the existence, stability, convergence of the numerical solutions leads to very laconic theoretic analysis. • The back-step flow and the flow around airfoil problem are simulated. [ABSTRACT FROM AUTHOR]

• Intrusive generalised Polynomial Chaos (gPC). • More efficient than non-intrusive gPC. • Monte Carlo (MC) resolution of the gPC based reduced model. • Efficient uncertainty propagation with a combined MC/gPC Chaos resolution. • Analytic uncertain benchmarks from well-known benchmarks of the literature. In this paper, we are interested in taking into account uncertainties for k eff computations in neutronics. More generally, the material of this paper can be applied to propagate uncertainties in eigenvalue/eigenvector computations for the linear Boltzmann equation. In [1,2] , an intrusive MC solver for the gPC based reduced model of the instationary linear Boltzmann equation has been put forward. The MC-gPC solver presents interesting characteristics (mainly a better efficiency than non-intrusive strategies and spectral convergence): our aim is to recover these characteristics in an eigenvalue/eigenvector estimation context. This is done in practice at the price of few well identified modifications of an existing Monte Carlo implementation. [ABSTRACT FROM AUTHOR]

• This paper addresses one unresolved issue for the SAV method. • It introduces a relaxation step for calculating the auxiliary variables. • The relaxation strategy improves the accuracy and consistency of the SAV method. The scalar auxiliary variable (SAV) method was introduced by Shen et al. in [36] and has been broadly used to solve thermodynamically consistent PDE problems. By utilizing scalar auxiliary variables, the original PDE problems are reformulated into equivalent PDE problems. The advantages of the SAV approach, such as linearity, unconditionally energy stability, and easy-to-implement, are prevalent. However, there is still an open issue unresolved, i.e., the numerical schemes resulting from the SAV method preserve a "modified" energy law according to the auxiliary variables instead of the original variables. Truncation errors are introduced during numerical calculations so that the numerical solutions of the auxiliary variables are no longer equivalent to their original continuous definitions. In other words, even though the SAV scheme satisfies a modified energy law, it does not necessarily satisfy the energy law of the original PDE models. This paper presents one essential relaxation technique to overcome this issue, which we named the relaxed-SAV (RSAV) method. Our RSAV method penalizes the numerical errors of the auxiliary variables by a relaxation technique. In general, the RSAV method keeps all the advantages of the baseline SAV method and improves its accuracy and consistency noticeably. Several examples have been presented to demonstrate the effectiveness of the RSAV approach. [ABSTRACT FROM AUTHOR]

In the framework of fusion energy research, divertor design and model calibration based on plasma edge codes currently rely either on manual iterative tuning of parameters, or alternatively on parameter scans. This, combined with the complex and computationally expensive nature of plasma edge codes, makes these procedures extremely cumbersome. Gradient-based optimization methods can significantly reduce this effort, but require an efficient strategy for sensitivity calculation. Algorithmic Differentiation (AD) offers an efficient and accurate solution, allowing semi-automatic sensitivity computation in complex, continuously developed codes. The adjoint AD mode is especially attractive, as its cost is independent of the number of input parameters. In this paper, adjoint AD sensitivity calculation is deployed for the first time in plasma edge codes, applying the TAPENADE AD tool to SOLPS-ITER. Adjoint AD results are verified to be machine precision accurate compared to tangent AD mode, and up to 10−9 compared to finite differences. Scalings of AD computational efforts prove the advantages of adjoint compared to tangent AD, while memory requirements rapidly increase for adjoint, showing the need for an improved checkpointing strategy. With the new tool, a wealth of information becomes accessible to the modeler. An adjoint sensitivity analysis for a COMPASS density scan identifies the input parameters with largest impact on the solution, and 2D sensitivity maps show their spatial dependence. • Algorithmic Differentiation allows adjoint sensitivity computation in complex codes. • Algorithmic Differentiation is applied to the plasma edge code SOLPS-ITER. • Adjoint sensitivities are obtained for the first time in plasma edge codes. • The computational cost of the sensitivities agrees with the expected performance. • A large amount of new sensitivity information is made available for the community. [ABSTRACT FROM AUTHOR]

Jump penalty stabilization techniques for under-resolved turbulence have been recently proposed for continuous and discontinuous high order Galerkin schemes [1–3]. The stabilization relies on the gradient or solution discontinuity at element interfaces to incorporate localised numerical diffusion in the numerical scheme. This diffusion acts as an implicit subgrid model and stabilizes under-resolved turbulent simulations. This paper investigates the effect of jump penalty stabilization methods (penalising gradient or solution) for stabilization and improvement of high-order discontinuous Galerkin schemes in turbulent regime. We analyze these schemes using an eigensolution analysis, a 1D non-linear Burgers equation (mimicking a turbulent cascade) and 3D turbulent Navier-Stokes simulations (Taylor-Green Vortex and Kelvin-Helmholtz Instability problems). We show that the two jump penalty stabilization techniques can stabilize under-resolved simulations thanks to the improved dispersion-dissipation characteristics (when compared to non-penalized schemes) and provide accurate results for turbulent flows. The numerical results indicate that the proposed jump penalty methods stabilize under-resolved simulations and improve the simulations, when compared to the original unpenalized scheme and to classic explicit subgrid models (Smagorinsky and Vreman). • Stabilisation of DG for under-resolved turbulence through penalising gradient or solution jumps at element interfaces. • Implementation for 1D advection-diffusion and 3D Navier-Stokes equations. • Eigensolution analysis for dispersion-dissipation behaviour for various polynomial orders and Riemann fluxes. • Validation on Burgers and Taylor-Green vortex problems and comparison with classic explicit subgrid models. [ABSTRACT FROM AUTHOR]

The opacity of FRTE depends on not only the material temperature but also the frequency, whose values may vary several orders of magnitude for different frequencies. The gray radiation diffusion and frequency-dependent diffusion equations are two simplified models that can approximate the solution to FRTE in the thick opacity regime. The frequency discretization for the two limit models highly affects the numerical accuracy. However, classical frequency discretization for FRTE considers only the absorbing coefficient. In this paper, we propose a new decomposed multi-group method for frequency discretization that is not only AP in both gray radiation diffusion and frequency-dependent diffusion limits, but also the frequency discretization of the limiting models can be tuned. Based on the decomposed multi-group method, a full AP scheme in frequency, time, and space is proposed. Several numerical examples are used to verify the performance of the proposed scheme. • The frequency discretization can be consistent with the required multi-group diffusion model. • Accuracy is guaranteed in both optically thin and thick regimes with large spatial meshes. • The required temporary step is independent of the light speed. • Nonlinearity appears only in the system with macroscopic quantities. • The microscopic quantities can be updated linearly. [ABSTRACT FROM AUTHOR]

Recently, there has been a growing interest in efficient numerical algorithms based on tensor networks and low-rank techniques to approximate high-dimensional functions and solutions to high-dimensional PDEs. In this paper, we propose a new tensor rank reduction method based on coordinate transformations that can greatly increase the efficiency of high-dimensional tensor approximation algorithms. The idea is simple: given a multivariate function, determine a coordinate transformation so that the function in the new coordinate system has smaller tensor rank. We restrict our analysis to linear coordinate transformations, which gives rise to a new class of functions that we refer to as tensor ridge functions. Leveraging Riemannian gradient descent on matrix manifolds we develop an algorithm that determines a quasi-optimal linear coordinate transformation for tensor rank reduction. The results we present for rank reduction via linear coordinate transformations open the possibility for generalizations to larger classes of nonlinear transformations. • Riemannian optimization of tensor ridge functions. • Tensor rank reduction via coordinate flows. • PDEs in curvilinear coordinate systems. • Rank-adaptive tensor methods. [ABSTRACT FROM AUTHOR]

Summation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP operators was that polynomials can accurately approximate the solution, and SBP operators should thus be exact for them. However, polynomials do not provide the best approximation for some problems, with other approximation spaces being more appropriate. We recently addressed this issue and developed a theory for one-dimensional SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the theory of FSBP operators to multiple dimensions. We focus on their existence, connection to quadratures, construction, and mimetic properties. A more exhaustive numerical demonstration of multi-dimensional FSBP (MFSBP) operators and their application will be provided in future works. Similar to the one-dimensional case, we demonstrate that most of the established results for polynomial-based multi-dimensional SBP (MSBP) operators carry over to the more general class of MFSBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently done. This can increase the accuracy of the numerical solutions and/or provide stability to the methods. • Multi-dimensional summation-by-parts operators for general (non-polynomial) function spaces. • Existence proof that connects them to quadratures. • Construction procedure for general multi-dimensional domains/elements/blocks. • Proof of discrete conservation and energy-boundedness. • Numerical comparison demonstrating that non-polynomial function spaces can be of advantage. [ABSTRACT FROM AUTHOR]

The discrete-equations method (DEM) [1] provides a universal approach to solve multi-phase-flow equations as it combines the solutions of pairwise Riemann problems. Although very robust, the original DEM with piecewise-constant volume fractions suffers from strong diffusion preventing accurate interface capturing. High-order interface reconstruction, however, introduces a restrictive time-step limit. This paper presents RDEMIC, a robust extension of DEM for accurate interface capturing on Cartesian meshes. By a modified partitioning of the Riemann solutions and a specific combination of fluxes and non-conservative terms, the time-step restriction is effectively prevented, which is critical for making the method practically applicable. Moreover, the accuracy of interface and shock-wave propagation is maintained. RDEMIC is not limited to two-phase flow but defined for an arbitrary number of phases. The method is combined with a THINC scheme [2] to reconstruct volume fractions. The reconstruction is enhanced by a positivity-preserving averaging procedure, which is consistent with the underlying multi-stage Runge–Kutta scheme of the flow solver. The resulting scheme consisting of RDEMIC and the positivity-preserving THINC reconstruction is very robust and captures the interface with high accuracy. We demonstrate its performance for various cases of shock-interface interactions, which show very good agreement with reference results from literature. • Robust high-resolution numerical model and algorithm for compressible multi-phase flow with arbitrary number of phases. • No further time-step restrictions implied by high-order interface reconstruction. • Accurate capturing and advection of multi-phase interfaces with positivity preservation. • No numerical artifacts for wave propagation across interfaces. [ABSTRACT FROM AUTHOR]

This paper introduces a novel method for the computation of accurate normal fields from volume fractions on unstructured polyhedral meshes. For surfaces of considerable curvature variation, the method is second-order accurate with respect to symmetric volume differences and first-order accurate with respect to normal field angular deviation. For this purpose, an averaged normal is locally computed, i.e. in each mesh cell, by fitting a plane in a least squares sense with appropriate weights to the volume fraction data of neighbouring cells while explicitly accounting for volume conservation in the cell at hand. The resulting minimization problem is solved approximately by means of a Newton -type method. Moreover, employing the Reynolds transport theorem allows for both assessing the regularity of the derivatives of the least squares error functional with respect to the free parameters and to exactly compute these derivatives at small additional costs. Since the divergence theorem implies that the volume fraction can be written as a sum of face-based quantities, our method considerably simplifies the numerical procedure for applications in three spatial dimensions while demonstrating an inherent ability to robustly deal with unstructured meshes. We discuss the theoretical foundations, regularity and appropriate error measures, along with the details of the numerical algorithm. Finally, numerical results for convex and non-convex hypersurfaces embedded in cuboidal and tetrahedral meshes are presented together with a convergence study considering two different error measures. Moreover, our findings are complemented by profound insights into the minimization procedure. • Efficient second-order accurate computation of normal fields solely from volume fractions. • Applicable to unstructured grids with arbitrary polyhedral cells in three spatial dimensions. • Face-based normal reconstruction with intrinsic exact volume conservation. • Dynamic stencil extension and bulk cell weighting for removal of outliers. • Second-order convergence in symmetric volume difference for moderate curvature variations. [ABSTRACT FROM AUTHOR]

Partial discharges and the electrical treeing represent one of key aspects for the ageing of the insulating parts of electrical components. To predict the end of life of dielectric systems several different modelling approaches have been developed. One of the most promising is the modelling using first principles and a set of partial differential equations. In this field, one of the toughest challenges is to cope with the very different scales of the treeing (each branch is a few micrometers wide) and of electrical components (raging from a few centimetres to a few meters). In this paper we propose an approach, based on the finite element and finite volume methods, which uses two completely independent grids to describe internal defects and electrical components. Some interpolation algorithms have been developed to couple the solution on the two grids. We have also tested the accuracy of our approach by comparing it with previously published methods that use only one grid. • Partial discharges simulation using plasma physics for challenging geometries. • Efficient treatment of tiny internal defects in electric equipments. • Grid decoupling between a gaseous defect and a dielectric bulk. [ABSTRACT FROM AUTHOR]

Aircraft aerodynamic design often involves evaluating flow conditions that span low subsonic to transonic or even supersonic Mach numbers. Compressible flow solvers are a natural choice for such design problems, but these solvers suffer from reduced accuracy and efficiency at low Mach numbers. In addition, simulations with supersonic conditions can be challenging to converge because of large gradients in the flow field. This paper presents three contributions to address these issues in the context of an approximate Newton–Krylov solver for the Reynolds-averaged Navier–Stokes equations. First, we propose a method for scaling the artificial dissipation terms in the Jameson–Schmidt–Turkel scheme to improve its accuracy at low Mach numbers while retaining the simplicity of the original scalar dissipation formulation. Second, we show that characteristic time-stepping combined with an approximate Newton method can accelerate convergence for low Mach number flows by reducing the stiffness in the linear system for each Newton iteration. Third, we introduce a dissipation-based continuation method for flows with shocks that improves robustness and accelerates convergence without sacrificing accuracy. These methods can make compressible flow solvers more accurate and efficient across low and high Mach number regimes. • A simple change to a scalar dissipation scheme improves accuracy at low Mach numbers. • Characteristic time-stepping reduces linear system stiffness for Newton-type solvers. • Scaling artificial dissipation is needed to achieve speedups at low Mach numbers. • Dissipation-based continuation improves robustness for flows with shocks. [ABSTRACT FROM AUTHOR]

In this paper, we propose a cusp-capturing physics-informed neural network (PINN) to solve discontinuous-coefficient elliptic interface problems whose solution is continuous but has discontinuous first derivatives on the interface. To find such a solution using neural network representation, we introduce a cusp-enforced level set function as an additional feature input to the network to retain the inherent solution properties; that is, capturing the solution cusps (where the derivatives are discontinuous) sharply. In addition, the proposed neural network has the advantage of being mesh-free, so it can easily handle problems in irregular domains. We train the network using the physics-informed framework in which the loss function comprises the residual of the differential equation together with certain interface and boundary conditions. We conduct a series of numerical experiments to demonstrate the effectiveness of the cusp-capturing technique and the accuracy of the present network model. Numerical results show that even using a one-hidden-layer (shallow) network with a moderate number of neurons and sufficient training data points, the present network model can achieve prediction accuracy comparable with traditional methods. Besides, if the solution is discontinuous across the interface, we can simply incorporate an additional supervised learning task for solution jump approximation into the present network without much difficulty. • Cusp-capturing physics informed neural network for elliptic interface problems. • Our network can present continuous solutions that inherently have discontinuous first derivatives on interfaces. • A mesh-free approach for solving PDEs with interfaces is presented. • Our network can achieve high prediction accuracy with large contrast discontinuous coefficients. [ABSTRACT FROM AUTHOR]

Kernels obtained from the heat equation arise in several modeling contexts, like some double porosity models, or viscous flows in networks of thin tubes. These kernels are weakly singular at initial time. An accurate approximation must therefore take this singularity into account. In this paper we obtain an asymptotic expansion for small times, which we use to build a numerical scheme for approximating the kernels. Convergence of the scheme and relevance of a correction through the asymptotics are proven both analytically and numerically. Finally, we show that this approximation applies to the model on the graph studied by the authors in Canon et al. [4] (2021). • Small time asymptotic expansion of weakly singular kernels • Convergent numerical approximation of a diffusion equation with incompatible initial and boundary data • Application to simplified models for viscous fluids in networks in relation to blood vessels flows and microfluidics [ABSTRACT FROM AUTHOR]

The hypersonic flow is in a thermochemical nonequilibrium state due to the high-temperature caused by the strong shock compression. In a thermochemical nonequilibrium flow, the distribution of molecular internal energy levels strongly deviates from the equilibrium distribution (i.e., the Boltzmann distribution). It is intractable to directly obtain the microscopic nonequilibrium distribution from existed experimental measurements usually described by macroscopic field variables such as temperature or velocity. Motivated by the idea of deep multi-scale multi-physics neural network (DeepMMNet) proposed in [1] , we develop in this paper a data assimilation framework called DeepStSNet to accurately reconstruct the quantum state-resolved thermochemical nonequilibrium flowfield by using sparse experimental measurements of vibrational temperature and pre-trained deep neural operator networks (DeepONets). In particular, we first construct several DeepONets to express the coupled dynamics between field variables in the thermochemical nonequilibrium flow and to approximate the state-to-state (StS) approach, which traces the variation of each vibrational level of molecule accurately. These proposed DeepONets are then trained by using the numerical simulation data, and would later be served as building blocks for the DeepStSNet. We demonstrate the effectiveness and accuracy of DeepONets with different test cases showing that the density and energy of vibrational groups as well as the temperature and velocity fields are predicted with high accuracy. We then extend the architectures of DeepMMNet by considering a simplified thermochemical nonequilibrium model, i.e., the 2T model, showing that the entire thermochemical nonequilibrium flowfield is well predicted by using scattered measurements of full or even partial field variables. We next consider a more accurate and complex thermochemical nonequilibrium model, i.e., the StS-CGM model, and develop a DeepStSNet for this model. In this case, we employ the coarse-grained method, which divides the vibrational levels into groups (vibrational bins), to alleviate the computational cost for the StS approach in order to achieve a fast but reliable prediction with DeepStSNet. We test the present DeepStSNet framework with sparse numerical simulation data showing that the predictions are in excellent agreement with the reference data for test cases. We further employ the DeepStSNet to assimilate a few experimental measurements of vibrational temperature obtained from the shock tube experiment, and the detailed non-Boltzmann vibrational distribution of molecule oxygen is reconstructed by using the sparse experimental data for the first time. Moreover, by considering the inevitable uncertainty in the experimental data, an average strategy in the predicting procedure is proposed to obtain the most probable predicted fields. The present DeepStSNet is general and robust, and can be applied to build a bridge from sparse measurements of macroscopic field variables to a microscopic quantum state-resolved flowfield. This kind of reconstruction is beneficial for exploiting the experimental measurements and uncovering the hidden physicochemical processes in hypersonic flows. • Two DeepMMNets for the 2T model are developed to predict all flowfields with scarce data. • A DeepStSNet is developed to reconstruct the quantum state-resolved thermochemical nonequilibrium flowfield with scarce data. • The microscopic vibrational distribution function of O 2 is reconstructed for the first time with scarce experimental data. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel methodology for automatic adaptive weighting of Bayesian Physics-Informed Neural Networks (BPINNs), and we demonstrate that this makes it possible to robustly address multi-objective and multiscale problems. BPINNs are a popular framework for data assimilation, combining the constraints of Uncertainty Quantification (UQ) and Partial Differential Equation (PDE). The relative weights of the BPINN target distribution terms are directly related to the inherent uncertainty in the respective learning tasks. Yet, they are usually manually set a-priori, that can lead to pathological behavior, stability concerns, and to conflicts between tasks which are obstacles that have deterred the use of BPINNs for inverse problems with multiscale dynamics. The present weighting strategy automatically tunes the weights by considering the multitask nature of target posterior distribution. We show that this remedies the failure modes of BPINNs and provides efficient exploration of the optimal Pareto front. This leads to better convergence and stability of BPINN training while reducing sampling bias. The determined weights moreover carry information about task uncertainties, reflecting noise levels in the data and adequacy of the PDE model. We demonstrate this in numerical experiments in Sobolev training, and compare them to analytically ε -optimal baseline, and in a multiscale Lotka-Volterra inverse problem. We eventually apply this framework to an inpainting task and an inverse problem, involving latent field recovery for incompressible flow in complex geometries. • Adaptive weighting enables robust uncertainty quantification in multitask Bayesian inference. • It avoids Bayesian PINNs training failure and provides unbiased uncertainty estimates. • It avoids manual noise parameter tuning and allows solving multi-scale problems. [ABSTRACT FROM AUTHOR]

In this paper, we propose an efficient quadratic interpolation formula utilizing solution gradients computed and stored at nodes and demonstrate its application to a third-order cell-centered finite-volume discretization on tetrahedral grids. The proposed quadratic formula is constructed based on an efficient formula of computing a projected derivative. It is efficient in that it completely eliminates the need to compute and store second derivatives of solution variables or any other quantities, which are typically required in upgrading a second-order cell-centered unstructured-grid finite-volume discretization to third-order accuracy. Moreover, a high-order flux quadrature formula, as required for third-order accuracy, can also be simplified by utilizing the efficient projected-derivative formula, resulting in a numerical flux at a face centroid plus a curvature correction not involving second derivatives of the flux. Similarly, a source term can be integrated over a cell to high-order in the form of the source term evaluated at the cell centroid plus a curvature correction, again, not requiring second derivatives of the source term. The discretization is defined as an approximation to an integral form of a conservation law but the numerical solution is defined as a point value at a cell center, leading to another feature that there is no need to compute and store geometric moments for a quadratic polynomial to preserve a cell average. Third-order accuracy and improved second-order accuracy are demonstrated and investigated for simple but illustrative test cases in three dimensions. • Efficient quadratic interpolation scheme with no second derivatives at all. • Third-order cell-centered finite-volume discretization on tetrahedral grids. • Third-order accuracy achieved with tetrahedra with planar boundary faces. • Only a single numerical flux (plus a correction term) per face. • Conservative discretization with point-valued solutions at centroids. [ABSTRACT FROM AUTHOR]

The volume penalization (VP) or the Brinkman penalization (BP) method is a diffuse interface method for simulating multiphase fluid-structure interaction (FSI) problems in ocean engineering and/or phase change problems in thermal sciences and engineering. The method relies on a penalty factor (which is inversely related to body's permeability κ) that must be large to enforce rigid body velocity in the solid domain. When the penalty factor is large, the discrete system of equations becomes stiff and difficult to solve numerically. In this paper, we propose a projection method-based preconditioning strategy for solving volume penalized (VP) incompressible and low-Mach Navier-Stokes equations. The projection preconditioner enables the monolithic solution of the coupled velocity-pressure system in both single phase (uniform density and viscosity) and multiphase (variable density and viscosity) flow settings. In this approach, the penalty force is treated implicitly, which is allowed to take arbitrary large values without affecting the solver's convergence rate or causing numerical stiffness. It is made possible by including the penalty term in the pressure Poisson equation (PPE), which was not included in previous works that solved VP incompressible Navier-Stokes equations using the projection method. We show how and where the Brinkman penalty term enters the PPE by re-deriving the projection algorithm for the VP method. Solver scalability under grid refinement is demonstrated, i.e., convergence is achieved with the same number of iterations regardless of the problem size. A manufactured solution in a single phase setting is used to determine the spatial accuracy of the penalized solution. Various values of body's permeability κ are considered. Second-order pointwise accuracy is achieved for both velocity and pressure solutions for reasonably small values of κ. Error saturation occurs when κ is extremely small, but the convergence rate of the solver does not degrade. The solver converges faster as κ decreases, contrary to prior experience. Two multiphase fluid-structure interaction (FSI) problems from the ocean engineering literature are also simulated to evaluate the solver's robustness and performance (in terms of its number of iterations). The proposed solver also allows us to investigate the effect of κ on the motion of the contact line over the surface of the immersed body. It also allows us to investigate the dynamics of the free surface of a solidifying metal. • A projection preconditioner for volume penalized multiphase incompressible/low Mach Navier-Stokes equations is proposed. • The new projection method considers the Brinkman penalty term in both pressure Poisson and velocity update equations. • In contrast to prior experience, the solver convergence rate improves with decrease in body's permeability. • Second-order pointwise accuracy is achieved for both velocity and pressure solutions for reasonably small permeability values. • Challenging multiphase fluid-structure interaction and phase change problems are simulated using the proposed preconditioner. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse acoustic obstacle problem for sound-soft star-shaped obstacles in two dimensions wherein the boundary of the obstacle is determined from measurements of the scattered field at a collection of receivers outside the object. One of the standard approaches for solving this problem is to reformulate it as an optimization problem: finding the boundary of the domain that minimizes the L 2 distance between computed values of the scattered field and the given measurement data. The optimization problem is computationally challenging since the local set of convexity shrinks with increasing frequency and results in an increasing number of local minima in the vicinity of the true solution. In many practical experimental settings, low frequency measurements are unavailable due to limitations of the experimental setup or the sensors used for measurement. Thus, obtaining a good initial guess for the optimization problem plays a vital role in this environment. We present a neural network warm-start approach for solving the inverse scattering problem, where an initial guess for the optimization problem is obtained using a trained neural network. We demonstrate the effectiveness of our method with several numerical examples. For high frequency problems, this approach outperforms traditional iterative methods such as Gauss-Newton initialized without any prior (i.e., initialized using a unit circle), or initialized using the solution of a direct method such as the linear sampling method. The algorithm remains robust to noise in the scattered field measurements and also converges to the true solution for limited aperture data. However, the number of training samples required to train the neural network scales exponentially in frequency and the complexity of the obstacles considered. We conclude with a discussion of this phenomenon and potential directions for future research. • Propose a neural network warm-start approach for the inverse scattering problem with high frequency. • Demonstrate its outperformance compared with the Gauss-Newton method and linear sampling method. • Demonstrate its performance with noisy data or partial aperture data. • Investigate scaling of the number of training data with frequency. [ABSTRACT FROM AUTHOR]

This article proposes an efficient numerical method for solving nonlinear partial differential equations (PDEs) based on sparse Gaussian processes (SGPs). Gaussian processes (GPs) have been extensively studied for solving PDEs by formulating the problem of finding a reproducing kernel Hilbert space (RKHS) to approximate a PDE solution. The approximated solution lies in the span of base functions generated by evaluating derivatives of different orders of kernels at sample points. However, the RKHS specified by GPs can result in an expensive computational burden due to the cubic computation order of the matrix inverse. We conjecture that a solution exists on a "condensed" subspace that can achieve similar approximation performance, and we propose a SGP-based method to reformulate the optimization problem in the "condensed" subspace. This significantly reduces the computation burden while retaining desirable accuracy. The paper rigorously formulates this problem and provides error analysis and numerical experiments to demonstrate the effectiveness of this method. The numerical experiments show that the SGP method uses fewer than half the uniform samples as inducing points and achieves comparable accuracy to the GP method using the same number of uniform samples, resulting in a significant reduction in computational cost. Our contributions include formulating the nonlinear PDE problem as an optimization problem on a "condensed" subspace of RKHS using SGP, as well as providing an existence proof and rigorous error analysis. Furthermore, our method can be viewed as an extension of the GP method to account for general positive semi-definite kernels. • We propose a new algorithm for solving nonlinear partial differential equations based on sparse Gaussian processes. • Our algorithms base on theories of kernel methods and sparse Gaussian processes. • Our algorithm is meshless and flexible to the shape of domains. • The sparse Gaussian process method reduces computational complexity by using low-rank kernels. [ABSTRACT FROM AUTHOR]

In this paper, the Half-Space Matching (HSM) method, first introduced for scalar problems, is extended to elastodynamics, to solve time-harmonic 2D scattering problems, in locally perturbed infinite anisotropic homogeneous media. The HSM formulation couples a variational formulation around the perturbations with Fourier integral representations of the outgoing solution in four overlapping half-spaces. These integral representations involve outgoing plane waves, selected according to their group velocity, and evanescent waves. Numerically, the HSM method consists in a finite element discretization of the HSM formulation, together with an approximation of the Fourier integrals. Numerical results, validating the method, are presented for different materials, isotropic and anisotropic. Comparisons with the Perfectly Matched Layers (PML) method are performed for several anisotropic materials. These results highlight the robustness of the HSM method compared to the sensitivity of the PML method with respect to its parameters. [ABSTRACT FROM AUTHOR]

In this paper, the multivariate well-conditioned asymptotic waveform evaluation (MWCAWE) algorithm is presented. It is based on a univariate version of the algorithm, previously introduced for fast frequency sweeps. Taking advantage of the robustness of this univariate algorithm, the MWCAWE enables to effectively generate multi-parameter reduced-order models. Additionally, a residue-based contour following approach is introduced for a two-stage generation of a projection basis, here applied on bi-variate acoustic problems where frequency and a material parameter are in focus. The effectiveness of the proposed method is demonstrated on two poro-acoustic problems, highlighting both the good convergence property of the algorithm and the potential of the multi-patch strategy for parametric applications. • A parametric model order reduction is demonstrated on vibro-acoustic problems. • A multivariate moment-matching algorithm is proposed based on the WCAWE algorithm. • A residue-based, contour-following sampling is introduced for multi-interval strategies. • The effectiveness of the method is demonstrated on 3D bivariate FE problems. [ABSTRACT FROM AUTHOR]

The extremely high dimensionality and nonlinearity in the Boltzmann equation bring tremendous difficulties to the study of rarefied gas dynamics. This paper addresses a neural network-based surrogate model that provides a structure-preserving approximation for the fivefold collision integral. The notion originates from the similarity in structure between the BGK-type relaxation model and residual neural network (ResNet) when a particle distribution function is treated as the input to the neural network function. Therefore, we extend the ResNet architecture and construct what we call the relaxation neural network (RelaxNet). Specifically, two feed-forward neural networks with physics-informed connections and activations are introduced as building blocks in RelaxNet, which provide bounded and physically realizable approximations of the equilibrium distribution and velocity-dependent relaxation time respectively. The evaluation of the collision term is thus significantly accelerated due to the fact that the convolution in the fivefold integral is replaced by tensor multiplication in the neural network. We fuse the mechanical advection operator and the RelaxNet-based collision operator into a unified model named the universal Boltzmann equation (UBE). We prove that UBE preserves the key structural properties in a many-particle system, i.e., positivity, conservation, invariance, H-theorem, and correct fluid dynamic limit. These properties promise that RelaxNet is superior to strategies that naively approximate the right-hand side of the Boltzmann equation using a machine learning model. A novel sampling strategy based on closure hierarchies of the moment system of the Boltzmann equation is developed to generate reliable and unbiased sampling for the supervised learning. The construction of the RelaxNet-based UBE and its solution algorithm are demonstrated in detail. Several numerical experiments, where the ground-truth datasets are produced by the Shakhov model, velocity-dependent ν -BGK model, and the full Boltzmann equation, are investigated. The capability of the current approach for simulating non-equilibrium flow physics is validated through satisfactory in- and out-of-distribution performance. The open-source codes to reproduce the numerical results are available under the MIT license [1]. • RelaxNet as a structure-preserving surrogate model for the collision operator in the Boltzmann equation. • Universal Boltzmann equation (UBE) which preserves all the key structural properties of a many-particle system. • Novel sampling strategy based on closure hierarchies of the moment system of the Boltzmann equation. • Solution algorithm to efficiently and accurately solve the UBE model. [ABSTRACT FROM AUTHOR]

We compare high-order methods including spectral difference (SD), flux reconstruction (FR), the entropy-stable discontinuous Galerkin spectral element method (ES-DGSEM), modal discontinuous Galerkin methods, and WENO to select the best candidate to simulate strong shock waves characteristic of hypersonic flows. We consider several benchmarks, including the Leblanc and modified shock-density wave interaction problems that require robust stabilization and positivity-preserving properties for a successful flow realization. We also perform simulations of the three-species Sod problem with simplified chemistry with the chemical reaction source terms introduced in the Euler equations. The ES-DGSEM scheme exhibits the highest stability, negligible numerical oscillations, and requires the least computational effort in resolving reactive flow regimes with strong shock waves. Therefore, we extend the ES-DGSEM to hypersonic Euler equations by deriving a new set of two-point entropy conservative fluxes for a five-species gas model. In this paper, hypersonic Euler equations refer to the multi-species Euler equations for which the internal energy and thermodynamic properties are computed using the Rigid-Rotor Harmonic-Oscillator model. Stabilization for capturing strong shock waves occurs by blending high-order entropy conservative fluxes with low-order finite volume fluxes constructed using the HLLC Riemann solver. The hypersonic Euler solver is verified using the non-equilibrium chemistry Sod problem. To this end, we adopt the Mutation++ library to compute the reaction source terms, thermodynamic properties, and transport coefficients. We also investigate the effect of real chemistry versus ideal chemistry, and the results demonstrate that the ideal chemistry assumption fails at high temperatures, hence real chemistry must be employed for accurate predictions. Finally, we consider a viscous hypersonic flow problem to verify the transport coefficients and reaction source terms determined by the Mutation++ library. • Resolving extreme pressure ratio shock waves in near vacuum conditions. • Constructing entropy-stable high order schemes for the hypersonic Euler equations. • Deriving Entropy-conservative fluxes for flows modeled with the RRHO method. • Comparing real chemistry versus ideal chemistry in hypersonic flows. • Hypersonic NS solver based on the entropy-stable Euler scheme and the BR1 approach. [ABSTRACT FROM AUTHOR]

In this paper, we propose a self-adaptive algorithm of physics-informed neural networks (PINNs) for 2D and 3D linear and nonlinear Biot models, including solving the forward and inverse problems. Firstly, we apply the original PINNs algorithm to 2D and 3D linear and nonlinear Biot models. Secondly, we explore the performance of PINNs in solving Biot model when λ → ∞ to show the potential of PINNs in avoiding locking phenomenon for displacement and pressure oscillation compared to the traditional numerical algorithms. Then, we propose a self-adaptive PINNs algorithm to solve the Biot model with high spatial-temporal complexity and apply the proposed algorithm to simulate the brain pressure distribution problem with irregular boundary. And the numerical results show that the physics-informed neural network has good precision in solving nonlinear problems, inverse problems and high-dimensional problems. Finally, we draw conclusions to summarize the main results of this work. [ABSTRACT FROM AUTHOR]

This paper proposes the first Lagrangian free-stream boundary condition for weakly-compressible smoothed particle hydrodynamics (WCSPH), significantly decreasing the domain-size effect for simulating flows around static or moving geometries and increasing computational efficiency. The boundary condition is implemented based on several numerical techniques, as follows. First, a spatio-temporal identification and an ensuring technique precisely identify surface particles without misjudgment. Second, the far-field corrections of density and velocity precisely target surface particles. Third, the improved in- and outlet boundaries guarantee the smooth inflow profile and free-outflow particles without the issue of the truncated kernel. The test cases of flows over a flat plate and around a circular cylinder have validated the accuracy and computational efficiency of the present method. In order to demonstrate its applicability, the method has also been tested with more complex examples, such as a fluid-structure interaction (FSI) problem and a 3-D flow around a sphere. • First Lagrangian free-stream boundary condition for SPH method. • Domain size required for convergence significantly reduced. • Spatio-temporal surface-particle identification and ensuring methods. [ABSTRACT FROM AUTHOR]

In this paper, a new type of hybrid weighted essentially non-oscillatory (WENO) schemes with third-order and fourth-order accuracy is developed to simulate compressible flow problems on unstructured meshes. First, an improved discontinuity sensor is designed on unstructured meshes, which can automatically and accurately identify the troubled cells without manually adjusting the parameters related to the problem. Second, a hybrid unequal-sized WENO (US-WENO) scheme with different accuracy is developed based on the new hybridization strategy and the existing US-WENO scheme to reduce the computational overhead caused by the computation of nonlinear weights and local characteristic decompositions in WENO procedures. Finally, some benchmark examples are employed to verify the performance of the proposed method in terms of numerical dissipation, shock capture ability, and computational efficiency. Numerical experiments show that the proposed hybrid US-WENO schemes can inherit the excellent characteristics of the original US-WENO schemes while improving computational efficiency. • An improved, simple and artificial parameter free discontinuity sensor is presented on triangular meshes. • An efficient hybrid unequal-sized WENO scheme is designed to solve compressible problems on triangular meshes. • These new hybrid WENO schemes are more efficient than the existing unequal-sized WENO scheme. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of unsteady Stokes equations in patient-specific arterial-like domains in 3D. A Stokes-like solver is a necessary component in a more sophisticated nonlinear Navier-Stokes method, for which several multilevel domain decomposition methods have been introduced recently. Because of the complex geometry, the construction and the solve of the coarse problem usually take a large percentage of the total compute time. In this paper, we introduce a parameterized one-dimensional Stokes solver defined along the centerline of the artery and use its stabilized finite element discretization to construct a coarse preconditioner. With suitable 3D-to-1D restriction and 1D-to-3D extension operators on fully unstructured meshes, a two-level additive Schwarz preconditioner can be constructed. Some numerical experiments for flows in realistic arteries are presented to show the efficiency and robustness of the new coarse preconditioner whose computational cost is considerably lower than the existing three-dimensional coarse preconditioners. • A two-level DDM is proposed for unsteady Stokes flows in patient-specific arteries. • The coarse preconditioner is obtained by a 1D Stokes model on the centerline. • Multiscale 1D-3D restriction and extension matrices are introduced by interpolations. • Numerical results show high efficiencies of the proposed method for complex arteries. [ABSTRACT FROM AUTHOR]

The multi-scale nature of gaseous flows poses tremendous difficulties for theoretical and numerical analysis. The Boltzmann equation, while possessing a wider applicability than hydrodynamic equations, requires significantly more computational resources due to the increased degrees of freedom in the model. The success of a hybrid fluid-kinetic flow solver for the study of multi-scale flows relies on accurate prediction of flow regimes. In this paper, we draw on binary classification in machine learning and propose the first neural network classifier to detect near-equilibrium and non-equilibrium flow regimes based on local flow conditions. Compared with classical semi-empirical criteria of continuum breakdown, the current method provides a data-driven alternative where the parameterized implicit function is trained by solutions of the Boltzmann equation. The ground-truth labels are derived rigorously from the deviation of particle distribution functions and the approximations based on the Chapman-Enskog ansatz. Therefore, no tunable parameter is needed in the criterion. Following the entropy closure of the Boltzmann moment system, a data generation strategy is developed to produce training and test sets. Numerical analysis shows its superiority over simulation-based samplings. A hybrid Boltzmann-Navier-Stokes flow solver is built correspondingly with an adaptive partition of local flow regimes. Numerical experiments including the one-dimensional Riemann problem, shear flow layer, and hypersonic flow around a circular cylinder are presented to validate the current scheme for simulating cross-scale and non-equilibrium flow physics. The quantitative comparison with a semi-empirical criterion and benchmark results demonstrates the capability of the current neural classifier to accurately predict continuum breakdown. The code for the data generator, hybrid solver, and neural network implementation is available in the open source repositories [1,2]. • The first neural network to predict equilibrium and non-equilibrium flow regimes is proposed. • A data generation strategy is developed based on the entropy closure theory of the Boltzmann equation. • The label of flow regime is rigorously derived with no tunable parameters. • A hybrid Boltzmann-Navier-Stokes flow solver is built upon the neural classifier. [ABSTRACT FROM AUTHOR]

Physics-informed Neural Network (PINN) is a promising tool that has been applied in a variety of physical phenomena described by partial differential equations (PDE). However, it has been observed that PINNs are difficult to train in certain "stiff" problems, which include various nonlinear hyperbolic PDEs that display shocks in their solutions. Recent studies added a diffusion term to the PDE, and an artificial viscosity (AV) value was manually tuned to allow PINNs to solve these problems. In this paper, we propose three approaches to address this problem, none of which rely on an a priori definition of the artificial viscosity value. The first method learns a global AV value, whereas the other two learn localized AV values around the shocks, by means of a parametrized AV map or a residual-based AV map. We applied the proposed methods to the inviscid Burgers equation and the Buckley-Leverett equation, the latter being a classical problem in Petroleum Engineering. The results show that the proposed methods are able to learn both a small AV value and the accurate shock location and improve the approximation error over a nonadaptive global AV alternative method. • Methods to add adaptive artificial viscosity to train PINNs to solve hyperbolic PDEs with shocks. • Adaptive methods accurately learn both the value and location of the artificial viscosity. • Successfully solved the Inviscid Burgers and Buckley-Leverett equations. • Additional computation cost over the baseline PINN is small. [ABSTRACT FROM AUTHOR]

An accurate modeling of the optical interactions in metallic nanostructures with subnanometer features requires an accurate description of quantum effects at the scale of billions of atoms. At such scale, first-principle methods are not computationally viable. Quantum hydrodynamic theory (QHT) has emerged as a powerful method that includes nonlocal contributions of the kinetic energy and the spatial dependence of the electron density, and it can predict both plasmon energy and spill-out effects in large metal nanoparticles. In this paper, we introduce a hybridizable discontinuous Galerkin method for solving Maxwell's equations coupled with a QHT model in order to account for quantum effects in three-dimensional metallic nanostructures. The coupled system of Maxwell's equations and QHT model is not only nonlinear but also multi-scale due to the interaction between the micrometer electromagnetic waves and the nanometer cavities of metallic nanostructures. We present extensive numerical experiments to validate the QHT model and demonstrate the capability of the HDG method to provide accurate solutions in the presence of strong nonlinearities and multiple length scales. These results offer a possibility to enhance nonlinear optical effects or to harness quantum mechanical electron tunneling by engineering metallic nanostructures at the quantum level. • An HDG method to simulate the equilibrium equation of quantum hydrodynamic theory. • An HDG method to simulate the first-order linear system of quantum hydrodynamic theory coupled with Maxwell's equations. • 2D simulation of metal nanowire and full 3D simulation of a periodic nanocoax array accounting for quantum effects. • Comparison of the optical response using quantum effects to that of local and nonlocal electron models for both applications. [ABSTRACT FROM AUTHOR]

Solving the chemical master equation is an indispensable tool in understanding the behavior of biological and chemical systems. In particular, it is increasingly recognized that commonly used ODE models are not able to capture the stochastic nature of many cellular processes. Solving the chemical master equation directly, however, suffers from the curse of dimensionality. That is, both memory and computational effort scale exponentially in the number of species. In this paper we propose a dynamical low-rank approach that enables the simulation of large biological networks. The approach is guided by partitioning the network into biological relevant subsets and thus avoids the use of single species basis functions that are known to give inaccurate results for biological systems. We use the proposed method to gain insight into the nature of asynchronous vs. synchronous updating in Boolean models and successfully simulate a 41 species apoptosis model on a standard desktop workstation. • First dynamical low-rank approximation (DLRA) for the Boolean chemical master equation. • Drastic reduction in both memory and computational cost. • Proposal of an alternative to single orbital basis functions based on the structure of the problem. [ABSTRACT FROM AUTHOR]

In this paper, we present a discontinuous Galerkin (DG) method for the simulation of incompressible and immiscible two-phase flow in fractured porous media with capillary pressure. The hybrid-dimensional fracture models are considered in which fractures are treated as (d − 1) -dimensional interfaces in the d -dimensional domain and fluid exchange between fractures and surrounding matrix is taken into account. The approximation schemes combining the DG method with the fully implicit backward Euler time discretization are formulated for models with a single fracture and intersecting fractures network, and the stability and unique solvability are established. Optimal error estimates in the discrete H 1 -norm for saturation and pressure are derived. Numerical experiments in two-dimensional geometry are conducted to verify the accuracy of our theoretical analysis. In addition to the backward Euler formulation, we also implement the algorithm with a kind of conventional IMplicit Pressure Explicit Saturation (IMPES) method. • DG methods for hybrid-dimensional fracture models of two-phase flow are presented. • The stability and unique solvability of the fully discrete solutions are proved. • Optimal order convergence rates of the pressure and saturation are obtained. • Numerical experiments are realized to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

The solution of Poisson-like equations defined on a complex geometry is required for gyrokinetic simulations, which are important for the modelling of plasma turbulence in nuclear fusion devices such as the ITER tokamak. In this paper, we compare three existing solvers finely tuned to solve this problem, in terms of the accuracy of the solution, and their computational efficiency. We also consider practical implementation aspects, including the parallel efficiency of the code, potentially enabling an integration of the solvers in a state-of-the-art first-principle gyrokinetic simulation framework. The first, the Spline FEM solver, uses C 1 polar splines to construct a finite elements method which solves the equation on curvilinear coordinates. The resulting linear system is solved using a conjugate gradient method. The second, the GMGPolar solver, uses a symmetric finite difference method to discretise the differential equation. The resulting linear system is solved using a tailored geometric multigrid scheme, with a combination of zebra circle and radial line smoothers, together with an implicit extrapolation scheme. The third, the Embedded Boundary solver, uses a finite volumes method on Cartesian coordinates with an embedded boundary scheme. The resulting linear system is solved using a multigrid scheme. The Spline FEM solver is shown to be the most accurate. The GMGPolar solver is shown to use the least memory. The Embedded Boundary solver is shown to be the fastest in most cases. All three solvers are shown to be capable of solving the equation on a realistic non-analytical geometry. The Embedded Boundary solver is additionally used to attempt to solve an X-point geometry. • Solver comparison for the gyrokinetic Poisson-like equation on complex geometry. • High-order spline FEM solver based on C1 polar splines and conjugate gradient. • Curvilinear finite difference method using multigrid with low memory footprint. • Embedded boundary scheme with fast multigrid for simple X-point geometry. [ABSTRACT FROM AUTHOR]

While deep learning algorithms demonstrate a great potential in scientific computing, its application to multi-scale problems remains to be a big challenge. This is manifested by the "frequency principle" that neural networks tend to learn low frequency components first. Novel architectures such as multi-scale deep neural network (MscaleDNN) were proposed to alleviate this problem to some extent. In this paper, we construct a subspace decomposition based DNN (dubbed SD2NN) architecture for a class of multi-scale problems by combining MscaleDNN algorithms with traditional numerical analysis ideas. The proposed architecture includes one low frequency normal DNN submodule, and one (or a few) high frequency MscaleDNN submodule(s), which are designed to capture the smooth part and the oscillatory part of the multi-scale solutions simultaneously. We demonstrate that the SD2NN outperforms existing models such as MscaleDNN, through several benchmark multi-scale problems in regular or perforated domains. • We propose SD2NN, a new architecture, combining subspace decomposition from numerical analysis and multiscale DNN solvers. • SD2NN has low/high frequency submodules, simultaneously capturing smooth/oscillatory parts of multi-scale solutions. • The new method outperforms state-of-the-art DNN solvers for 1D-3D multiscale PDEs on regular and perforated domains. [ABSTRACT FROM AUTHOR]

The solution of time dependent differential equations with neural networks has attracted a lot of attention recently. The central idea is to learn the laws that govern the evolution of the solution from data, which might be polluted with random noise. However, in contrast to other machine learning applications, usually a lot is known about the system at hand. For example, for many dynamical systems physical quantities such as energy or (angular) momentum are exactly conserved. Hence, the neural network has to learn these conservation laws from data and they will only be satisfied approximately due to finite training time and random noise. In this paper we present an alternative approach which uses Noether's Theorem to inherently incorporate conservation laws into the architecture of the neural network. We demonstrate that this leads to better predictions for three model systems: the motion of a non-relativistic particle in a three-dimensional Newtonian gravitational potential, the motion of a massive relativistic particle in the Schwarzschild metric and a system of two interacting particles in four dimensions. • Construction of Lagrangian neural networks invariant under continuous symmetries. • Noether's Theorem guarantees exact conservation of corresponding physical quantities. • Numerically demonstrated conservation of angular momentum for three model systems. • Symmetry-enforcing layers improve stability under perturbations of initial condition. [ABSTRACT FROM AUTHOR]

In this paper, a high-order semi-implicit (SI) asymptotic preserving (AP) and divergence-free finite difference weighted essentially nonoscillatory (WENO) scheme is proposed for magnetohydrodynamic (MHD) equations. We consider the sonic Mach number ε ranging from 0 to O (1). High-order accuracy in time is obtained by SI implicit-explicit Runge–Kutta (IMEX-RK) time discretization. High-order accuracy in space is achieved by finite difference WENO schemes with characteristic-wise reconstructions. A constrained transport method is applied to maintain a discrete divergence-free condition. We formally prove that the scheme is AP. Asymptotic accuracy (AA) in the incompressible MHD limit is obtained if the implicit part of the SI IMEX-RK scheme is stiffly accurate. Numerical experiments are provided to validate the AP, AA, and divergence-free properties of our proposed approach. Besides, the scheme can well capture discontinuities such as shocks in an essentially non-oscillatory fashion in the compressible regime, while it is also a good incompressible solver with uniform large-time step conditions in the low sonic Mach limit. • High order AP schemes for compressible ideal MHD equations are developed. • Our schemes are uniformly stable with respect to the sonic Mach number. • Discrete divergence-free magnetic field is achieved via a CT method. • Asymptotic preserving and asymptotically accurate properties are formally proved. • The schemes perform well in both compressible and incompressible regimes. [ABSTRACT FROM AUTHOR]