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In this paper, a compact and high order ADER (Arbitrary high order using DERivatives) scheme using the simple HWENO method (ADER-SHWENO) is proposed for hyperbolic conservation laws. The newly-developed method employs the Lax-Wendroff procedure to convert time derivatives to spatial derivatives, which provides the time evolution of the variables at the cell interfaces. This information is required for the simple HWENO reconstructions, which take advantages of the simple WENO and the classic HWENO. Compared with the original Runge-Kutta HWENO method (RK-HWENO), the new method has two advantages. Firstly, RK-HWENO method must solve the additional equations for reconstructions, which is avoided for the new method. Secondly, the SHWENO reconstruction is performed once with one stencil and is different from the classic HWENO methods, in which both the function and its derivative values are reconstructed with two different stencils, respectively. Thus the new method is more efficient than the RK-HWENO method. Moreover, the new method is more compact than the existing ADER-WENO method. Besides, the new method makes the best use of the information in the ADER method. Thus, the time evolution of the cell averages of the derivatives is simpler than that developed in the work (Li et al. (2021) [17]). Numerical tests indicate that the new method can achieve high order for smooth solutions both in space and time, keep non-oscillatory at discontinuities. • A compact fifth-order ADER scheme is proposed for hyperbolic conservation laws. • A compact reconstruction technique is designed by better utilizing the information in time evolution. • The computational stencil is more compact than the original ADERWENO scheme. • It is more efficient than the original multi-stage RK-HWENO scheme. [ABSTRACT FROM AUTHOR]

Latitude on the choice of initialisation is a shared feature between one-step extended state-space and multi-step methods. The paper focuses on lattice Boltzmann schemes, which can be interpreted as examples of both previous categories of numerical schemes. We propose a modified equation analysis of the initialisation schemes for lattice Boltzmann methods, determined by the choice of initial data. These modified equations provide guidelines to devise and analyze the initialisation in terms of order of consistency with respect to the target Cauchy problem and time smoothness of the numerical solution. In detail, the larger the number of matched terms between modified equations for initialisation and bulk methods, the smoother the obtained numerical solution. This is particularly manifest for numerical dissipation. Starting from the constraints to achieve time smoothness, which can quickly become prohibitive for they have to take the parasitic modes into consideration, we explain how the distinct lack of observability for certain lattice Boltzmann schemes—seen as dynamical systems on a commutative ring—can yield rather simple conditions and be easily studied as far as their initialisation is concerned. This comes from the reduced number of initialisation schemes at the fully discrete level. These theoretical results are successfully assessed on several lattice Boltzmann methods. • We study the initialization of general lattice Boltzmann methods introducing an ad hoc modified equation analysis. • We find the constraints to obtain consistent initialization schemes, preserving second-order for the overall method. • We finely describe initial boundary layers due to dissipation mismatches between bulk and initialization schemes. • We introduce the observability of a lattice Boltzmann scheme, characterizing those with easily-mastered initializations. • We test the introduced analytical tools and their effectiveness through several—very conclusive—numerical experiments. [ABSTRACT FROM AUTHOR]

An artificial viscosity-based shock capturing scheme is extended to the context of high-order triangular Spectral Difference method for solving gas dynamics problems featuring discontinuities and shock waves. The equations are regularised thanks to an artificial diffusivity method combined with a shock sensor based on dilatation and vorticity fields valid for any polynomial order of approximation. The methodological novelty of this paper is an L 2 projection based iterative Restriction Prolongation filtering operation introduced for high-order triangular elements. This filtering operation applied to the artificial bulk viscosity and thermal conductivity stabilises the simulations and improves the quality of the shock capturing approach. The methodology is validated on various canonical relevant compressible test cases. Numerical results demonstrate the performance and the flexibility of the triangular high-order Spectral Difference method for solving simultaneously compressible flows and shock waves. [ABSTRACT FROM AUTHOR]

This paper presents a robust numerical solution to the electromagnetic scattering problem involving multiple multi-layered cavities in both transverse magnetic and electric polarizations. A transparent boundary condition is introduced at the open aperture of the cavity to transform the problem from an unbounded domain into that of bounded cavities. By employing Fourier series expansion of the solution, we reduce the original boundary value problem to a two-point boundary value problem, represented as an ordinary differential equation for the Fourier coefficients. The analytical derivation of the connection formula for the solution enables us to construct a small-scale system that includes solely the Fourier coefficients on the aperture, streamlining the solving process. Furthermore, we propose accurate numerical quadrature formulas designed to efficiently handle the weakly singular integrals that arise in the transparent boundary conditions. To demonstrate the effectiveness and versatility of our proposed method, a series of numerical experiments are conducted. • Innovative Transparent Boundary Conditions: A novel transparent boundary condition is designed to enhance the accuracy and efficiency of simulations. • Enhanced Numerical Quadratures: Highly efficient and accurate numerical quadrature techniques are developed to ensure reliable computations. • Comprehensive Numerical Examples: An array of carefully chosen numerical examples are presented to demonstrate the method's effectiveness and versatility. [ABSTRACT FROM AUTHOR]

Efficient numerical schemes for solving the Poisson equation with jump conditions are of great interest for a variety of problems, including the modeling of electroporation phenomena and filamentary discharges. In this paper, we propose a modification to a finite volume scheme, namely the discrete dual finite volume method, in order to account for jump conditions with surface charges, i.e. with a source term. Our numerical tests demonstrate second-order convergence even with highly distorted meshes. We then apply the proposed method to model electroporation phenomena in biological cells by proposing a model that considers the thickness of the cell membrane as a separate domain, which differs from the literature. We show the advantages of the proposed method in this context through numerical experiments. • The discrete dual finite volume scheme is extended to solve the Poisson equation with jump conditions and surface charges. • The method is shown to exhibit a second-order convergence through canonical numerical tests. • The method is applied to the electroporation phenomena, where accurate modeling of the potential at the membrane is obtained. • Numerical experiments on the stationary and non-stationary case are provided. [ABSTRACT FROM AUTHOR]

This paper contains the foundation for a new Particle-In-Cell model for gas discharges, based on Îto diffusion and Kinetic Monte Carlo (KMC). In the new model the electrons are described with a microscopic drift-diffusion model rather than a macroscopic one. We discuss the connection of the Îto-KMC model to the equations of fluctuating hydrodynamics and the advection-diffusion-reaction equation which is conventionally used for simulating streamer discharges. The new model is coupled to a particle description of photoionization, providing a non-kinetic all-particle method with several attractive properties, such as: 1) Taking the same input as a fluid model, e.g. mobility coefficients, diffusion coefficients, and reaction rates. 2) Guaranteed non-negative densities. 3) Intrinsic support for reactive and diffusive fluctuations. 4) Exceptional stability properties. The model is implemented as a particle-mesh model on cut-cell grids with Cartesian adaptive mesh refinement. Positive streamer discharges in atmospheric air are considered as the primary application example, and we demonstrate that we can self-consistently simulate large discharge trees. • New, non-kinetic Particle-In-Cell method for electric discharges. • Self-consistent and stochastic evolution of streamer discharge trees. • Compatible with adaptive mesh refinement (AMR) and parallel computations. [ABSTRACT FROM AUTHOR]

The so-called Faster is Slower (FIS) effect is observed in some particular real-life or experimental situations. In the context of an evacuation process, it expresses that increasing the speed (or, more generally, the competitiveness) of individuals may induce a reduction of the flow through the exit door. We propose here a parameter-free model to reproduce and investigate this effect (more precisely its backward "Slower is Faster" equivalent). In spite of its non-smooth character, which makes it difficult to analyze, this granular approach is based on very basic ingredients in terms of behavior. In its native, purely asocial version, individuals are represented by hard-discs, each of which has a desired velocity, and the actual velocity is built as the projection of this field on the set of admissible velocities (which respect the non-overlapping constraints). We implement the slower effect by introducing here an extra step to account for the fact that individuals refrain from pushing, and therefore tend to reduce their desired velocity accounting for the velocities of people upfront. The present paper has two objectives: establish the relevance of this model by showing that it satisfactorily reproduces various empirical effects in highly crowded evacuations with various levels of competitiveness, and explore how it can be implemented to recover and explain the FIS effect. In this spirit, we confront this Inhibition-Based (IB) model to experimental data, focusing on the Faster is Slower effect. We show in particular that this approach makes it possible to accurately recover the effect of competitiveness upon power-law distributions of time lapses which have been experimentally observed. We also study the effect of mixed behaviors, by introducing a two-population model using both approaches. We investigate in particular the effect upon evacuation efficiency of the ratio between competitive agents and non-competitive ones. In a similar context, we investigate the role of an obstacle placed upstream the exit upon evacuation efficiency. • Numerical investigation of the Faster is Slower effect based on a parameter free model. • Integration of an inhibition tendency of polite pedestrians. • Numerical evidence of the fluidizing role of an obstacle upstream the exit during evacuation. • Numerical recovering of an experimental power law for time lapses. [ABSTRACT FROM AUTHOR]

In many Bayesian inverse problems the goal is to recover a spatially varying random field. Such problems are often computationally challenging especially when the forward model is governed by complex partial differential equations (PDEs). The challenge is particularly severe when the spatial domain is large and the unknown random field needs to be represented by a high-dimensional parameter. In this paper, we present a domain-decomposed method to attack the dimensionality issue and the method decomposes the spatial domain and the parameter domain simultaneously. On each subdomain, a local Karhunen-Loève (KL) expansion is constructed, and a local inversion problem is solved independently in a parallel manner, and more importantly, in a lower-dimensional space. After local posterior samples are generated through conducting Markov chain Monte Carlo (MCMC) simulations on subdomains, a novel projection procedure is developed to effectively reconstruct the global field. In addition, the domain decomposition interface conditions are dealt with an adaptive Gaussian process-based fitting strategy. Numerical examples are provided to demonstrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

Parareal is a well-known parallel-in-time algorithm that combines a coarse and fine propagator within a parallel iteration. It allows for large-scale parallelism that leads to significantly reduced computational time compared to serial time-stepping methods. However, like many parallel-in-time methods it can fail to converge when applied to non-diffusive equations such as hyperbolic systems or dispersive nonlinear wave equations. This paper explores the use of exponential integrators within the Parareal iteration. Exponential integrators are particularly interesting candidates for Parareal because of their ability to resolve fast-moving waves, even at the large stepsizes used by coarse propagators. This work begins with an introduction to exponential Parareal integrators followed by several motivating numerical experiments involving the nonlinear Schrödinger equation. These experiments are then analyzed using linear analysis that approximates the stability and convergence properties of the exponential Parareal iteration on nonlinear problems. The paper concludes with two additional numerical experiments involving the dispersive Kadomtsev-Petviashvili equation and the hyperbolic Vlasov-Poisson equation. These experiments demonstrate that exponential Parareal methods offer improved time-to-solution compared to serial exponential integrators when solving certain non-diffusive equations. • Exponential Parareal notably reduces time-to-solution for non-diffusive equations. • Linear analysis accurately predicts Parareal performance on nonlinear problems. • Repartitioning is essential for stabilizing exponential integrators within Parareal. [ABSTRACT FROM AUTHOR]

Robust iterative methods for solving large sparse systems of linear algebraic equations often suffer from the problem of optimizing the corresponding tuning parameters. To improve the performance of the problem of interest, specific parameter tuning is required, which in practice can be a time-consuming and tedious task. This paper proposes an optimization algorithm for tuning the numerical method parameters. The algorithm combines the evolution strategy with the pre-trained neural network used to filter the individuals when constructing the new generation. The proposed coupling of two optimization approaches allows to integrate the adaptivity properties of the evolution strategy with a priori knowledge realized by the neural network. The use of the neural network as a preliminary filter allows for significant weakening of the prediction accuracy requirements and reusing the pre-trained network with a wide range of linear systems. The detailed algorithm efficiency evaluation is performed for a set of model linear systems, including the ones from the SuiteSparse Matrix Collection and the systems from the turbulent flow simulations. The obtained results show that the pre-trained neural network can be effectively reused to optimize parameters for various linear systems, and a significant speedup in the calculations can be achieved at the cost of about 100 trial solves. The hybrid evolution strategy decreases the calculation time by more than 6 times for the black box matrices from the SuiteSparse Matrix Collection and by a factor of 1.4–2 for the sequence of linear systems when modeling turbulent flows. This results in a speedup of up to 1.8 times for the turbulent flow simulations performed in the paper. • A hybrid evolution strategy for optimizing the parameters of linear solvers is proposed. • The pre-trained neural network used as a pre-filter allows for improving the quality of optimization. • The pre-trained neural networks can be reused to optimize a variety of linear systems across different compute platforms. • Optimizing the parameters of linear solvers allows for the acceleration of incompressible turbulent flow simulations. [ABSTRACT FROM AUTHOR]

We present a new exactly divergence-free and well-balanced hybrid finite volume/finite element scheme for the numerical solution of the incompressible viscous and resistive magnetohydrodynamics (MHD) equations on staggered unstructured mixed-element meshes in two and three space dimensions. The equations are split into several subsystems, each of which is then discretized with a particular scheme that allows to preserve some fundamental structural features of the underlying governing PDE system also at the discrete level. The pressure is defined on the vertices of the primary mesh, while the velocity field and the normal components of the magnetic field are defined on an edge-based/face-based dual mesh in two and three space dimensions, respectively. This allows to account for the divergence-free conditions of the velocity field and of the magnetic field in a rather natural manner. The non-linear convective and the viscous terms in the momentum equation are solved at the aid of an explicit finite volume scheme, while the magnetic field is evolved in an exactly divergence-free manner via an explicit finite volume method based on a discrete form of the Stokes law in the edges/faces of each primary element. The latter method is stabilized by the proper choice of the numerical resistivity in the computation of the electric field in the vertices/edges of the 2D/3D elements. To achieve higher order of accuracy, a piecewise linear polynomial is reconstructed for the magnetic field, which is guaranteed to be exactly divergence-free via a constrained L 2 projection. Finally, the pressure subsystem is solved implicitly at the aid of a classical continuous finite element method in the vertices of the primary mesh and making use of the staggered arrangement of the velocity, which is typical for incompressible Navier-Stokes solvers. In order to maintain non-trivial stationary equilibrium solutions of the governing PDE system exactly, which are assumed to be known a priori , each step of the new algorithm takes the known equilibrium solution explicitly into account so that the method becomes exactly well-balanced. We show numerous test cases in two and three space dimensions in order to validate our new method carefully against known exact and numerical reference solutions. In particular, this paper includes a very thorough study of the lid-driven MHD cavity problem in the presence of different magnetic fields and the obtained numerical solutions are provided as free supplementary electronic material to allow other research groups to reproduce our results and to compare with our data. We finally present long-time simulations of Soloviev equilibrium solutions in several simplified 3D tokamak configurations, showing that the new well-balanced scheme introduced in this paper is able to maintain stationary equilibria exactly over very long integration times even on very coarse unstructured meshes that, in general, do not need to be aligned with the magnetic field. • Semi-implicit FV/FE method for incompressible viscous and resistive MHD equations. • Well-balanced and exactly divergence-free on general unstructured mixed-element grids. • Constrained L2 projection for an exactly divergence-free reconstruction. • Thorough study of the lid-driven MHD cavity problem (reference solution is provided). • Stable long-time simulation of Grad-Shafranov equilibria in 3D tokamak geometries. [ABSTRACT FROM AUTHOR]

This paper proposes a novel deep learning approach for learning operators in semigroup, with applications to modeling unknown autonomous dynamical systems using time series data collected at varied time lags. It is a sequel to the previous flow map learning (FML) works [Qin et al. (2019) [29] ], [Wu and Xiu (2020) [30] ], and [Chen et al. (2022) [31] ], which focused on learning single evolution operator with a fixed time step. This paper aims to learn a family of evolution operators with variable time steps, which constitute a semigroup for an autonomous system. The semigroup property is very crucial and links the system's evolutionary behaviors across varying time scales, but it was not considered in the previous works. We propose for the first time a framework of embedding the semigroup property into the data-driven learning process, through a novel neural network architecture and new loss functions. The framework is very feasible, can be combined with any suitable neural networks, and is applicable to learning general autonomous ODEs and PDEs. We present the rigorous error estimates and variance analysis to understand the prediction accuracy and robustness of our approach, showing the remarkable advantages of semigroup awareness in our model. Moreover, our approach allows one to arbitrarily choose the time steps for prediction and ensures that the predicted results are well self-matched and consistent. Extensive numerical experiments demonstrate that embedding the semigroup property notably reduces the data dependency of deep learning models and greatly improves the accuracy, robustness, and stability for long-time prediction. • Propose a deep learning framework for learning flow map operators in semigroup. • Learn unknown ODEs and PDEs from time series data collected at varied time lags. • Embed semigroup property into deep learning by novel neural networks and loss functions. • Present error estimates and variance analysis to understand the accuracy and robustness. • Semigroup awareness significantly improves the prediction accuracy and enhances the stability. [ABSTRACT FROM AUTHOR]

Inspired by our previous work on a quadratic approximation manifold [1] , we propose in this paper a computationally tractable approach for combining a projection-based reduced-order model (PROM) and an artificial neural network (ANN) to mitigate the Kolmogorov barrier to reducibility of parametric and/or highly nonlinear, high-dimensional, physics-based models. The main objective of our PROM-ANN concept is to reduce the dimensionality of the online approximation of the solution beyond what is achievable using affine and quadratic approximation manifolds, while maintaining accuracy. In contrast to previous approaches that exploited one form or another of an ANN, the training of the ANN part of our PROM-ANN does not involve data whose dimension scales with that of the high-dimensional model; and the resulting PROM-ANN can be efficiently hyperreduced using any well-established hyperreduction method. Hence, unlike many other ANN-based model order reduction approaches, the PROM-ANN concept we propose in this paper should be practical for large-scale and industry-relevant computational problems. We demonstrate the computational tractability of its offline stage and the superior wall clock time performance of its online stage for a large-scale, parametric, two-dimensional, model problem that is representative of shock-dominated unsteady flow problems. • New concept of computationally tractable arbitrarily nonlinear approximation manifold using an artificial neural network. • Resulting projection-based reduced-order model is hyperreducible by standard hyperreduction methods. • Wall clock time performance demonstrated for a parametric transport problem that exhibits the Kolmogorov n -width issue. • Up to seventy-fold acceleration of the online performance of a traditional, nonlinear, projection-based reduced-order model. [ABSTRACT FROM AUTHOR]

Large-scale cosmological simulations are an indispensable tool for modern cosmology. To enable model-space exploration, fast and accurate predictions are critical. In this paper, we show that the performance of such simulations can be further improved with time-stepping schemes that use input from cosmological perturbation theory. Specifically, we introduce a class of time-stepping schemes derived by matching the particle trajectories in a single leapfrog/Verlet drift-kick-drift step to those predicted by Lagrangian perturbation theory (LPT). As a corollary, these schemes exactly yield the analytic Zel'dovich solution in 1D in the pre-shell-crossing regime (i.e. before particle trajectories cross). One representative of this class is the popular ' FastPM ' scheme by Feng et al. 2016 [1] , which we take as our baseline. We then construct more powerful LPT-inspired integrators and show that they outperform FastPM and standard integrators in fast simulations in two and three dimensions with O (1 − 100) timesteps, requiring fewer steps to accurately reproduce the power spectrum and bispectrum of the density field. Furthermore, we demonstrate analytically and numerically that, for any integrator, convergence is limited in the post-shell-crossing regime (to order 3 2 for planar-wave collapse), owing to the lacking regularity of the acceleration field, which makes the use of high-order integrators in this regime futile. Also, we study the impact of the timestep spacing and of a decaying mode present in the initial conditions. Importantly, we find that symplecticity of the integrator plays a minor role for fast approximate simulations with a small number of timesteps. • We construct new efficient time integrators for cosmological simulations. • For their construction, we use results from Lagrangian perturbation theory. • Our integrators produce the exact solution in 1D before particle trajectories cross. • The crossing of particle trajectories limits the convergence order of integrators. • Numerical experiments demonstrate the effectiveness of our integrators. [ABSTRACT FROM AUTHOR]

Learning unknown stochastic differential equations (SDEs) from observed data is a significant and challenging task with applications in various fields. Current approaches often use neural networks to represent drift and diffusion functions, and construct likelihood-based loss by approximating the transition density to train these networks. However, these methods often rely on one-step stochastic numerical schemes, necessitating data with sufficiently high time resolution. In this paper, we introduce novel approximations to the transition density of the parameterized SDE: a Gaussian density approximation inspired by the random perturbation theory of dynamical systems, and its extension, the dynamical Gaussian mixture approximation (DynGMA). Benefiting from the robust density approximation, our method exhibits superior accuracy compared to baseline methods in learning the fully unknown drift and diffusion functions and computing the invariant distribution from trajectory data. And it is capable of handling trajectory data with low time resolution and variable, even uncontrollable, time step sizes, such as data generated from Gillespie's stochastic simulations. We then conduct several experiments across various scenarios to verify the advantages and robustness of the proposed method. • Novel density approximations are proposed for learning SDE via maximum likelihood estimation. • The proposed density approximation allows for multiple time steps, effectively reducing discretization errors. • The proposed method exhibits superior accuracy in learning the governing functions and computing the invariant distribution. • The proposed method is capable of handling trajectory data with low time resolution and variable time step sizes. [ABSTRACT FROM AUTHOR]

This paper presents a new methodology allowing the discretisation of phase specific transport equations within the Volume-of-Fluid interface capturing method framework. The method uses a sharp interface algorithm to compute the transport of species concentration. The interface reconstruction and advection is provided by the geometric advection scheme isoAdvector [1] implemented in the OpenFOAM® library. When discretising the transport equation for species, lack of consistency with the free surface advection scheme can lead to numerical errors, causing conservation or boundedness issues. This work addresses the issue of consistency in convective transport of species and is divided in two parts. First, a new interpolation procedure is used to compute face values from cell-centered values. Then, the diffusive operator of the transport equation is corrected. Finally, a set of test cases are presented to validate the transport equation's consistency with the free surface advection. Species transport across the interface is not part of the scope of this article, however, this methodology can further be used to study mass transfer at the gas-liquid interface using additional mass source terms that are not discussed here. • A new methodology to discretize a transport equation for species concentration has been described. • This methodology is consistent with the geometric Volume-of-Fluid method isoAdvector which is available in OpenFOAM. • The developments made in this work avoid artificial mass transfer between the phases. • Diffusive fluxes were also corrected to avoid non-physical fluxes using the species concentration inside the phase. [ABSTRACT FROM AUTHOR]

The lacking of analytic solutions of diverse partial differential equations (PDEs) gives birth to a series of computational techniques for numerical solutions. Although numerous latest advances are accomplished in developing neural operators, a kind of neural-network-based PDE solver, these solvers become less accurate and explainable while learning long-term behaviors of non-linear PDE families. In this paper, we propose the Koopman neural operator (KNO), a new neural operator, to overcome these challenges. With the same objective of learning an infinite-dimensional mapping between Banach spaces that serves as the solution operator of the target PDE family, our approach differs from existing models by formulating a non-linear dynamic system of equation solution. By approximating the Koopman operator, an infinite-dimensional operator governing all possible observations of the dynamic system, to act on the flow mapping of the dynamic system, we can equivalently learn the solution of a non-linear PDE family by solving simple linear prediction problems. We validate the KNO in mesh-independent, long-term, and zero-shot predictions on five representative PDEs (e.g., the Navier-Stokes equation and the Rayleigh-Bénard convection) and three real dynamic systems (e.g., global water vapor patterns and western boundary currents). In these experiments, the KNO exhibits notable advantages compared with previous state-of-the-art models, suggesting the potential of the KNO in supporting diverse science and engineering applications (e.g., PDE solving, turbulence modeling, and precipitation forecasting). • We propose a new neural operator model for partial differential equation (PDE) solving and realize an effective linear prediction of non-linear dynamics via Koopman operator. • Our approach achieves robust and accurate modeling of the long-term non-linear dynamics of PDE families or real systems. • Our model may serve as a basic unit for constructing new frameworks of PDE solving and dynamic system modeling. [ABSTRACT FROM AUTHOR]

Small adhesive particles with non-spherical shapes are common in natural and industrial processes. However, it is challenging to accurately simulate the dynamic behaviors of non-spherical particulate systems, not only due to the complex local geometry at the contact region but also to the non-linear coupling between the van der Waals adhesion and the elastic deformation. In this paper, we develop a discrete element method (DEM) framework for small non-spherical particles. Particularly, we present a simple algorithm for solving the implicit equations of the dimensionless generalized JKR model, to accurately resolve the local surface geometry at the contact region and the strong coupling between the elastic deformation and the surface adhesion. New explicit expressions are established for the pull-off force, overlap and minor-to-major ratio of the contact ellipse, to naturally capture the pull-off point upon collision. Then the actual contact parameters including the contact radius, curvatures and overlap are used to modify the interparticle interactions, including the normal elastic-adhesive-dissipative force and sliding-rolling-twisting resistances. Finally, the framework is quantitatively validated by four random packing problems, including adhesive spheres, ellipsoids, spherocylinders and dimers, and the representative non-adhesive ellipsoids. Both the microscopic and macroscopic properties of packings agree well with a large number of classic theories, experiments and simulations, showing the satisfactory accuracy and effectiveness. The newly developed adhesive DEM framework has the full capability to dynamically simulate the collision-related behaviors of complex shaped particles in the presence of van der Waals adhesion and frictional forces. • A simple algorithm is presented for solving the implicit equations of the generalized JKR model. • The non-linear coupling between surface adhesion and elastic deformation is accurately resolved. • New criterion is established to naturally capture the pull-off point upon collision. • Microscopic and macroscopic properties of simulated packing problems agree well with reported results. [ABSTRACT FROM AUTHOR]

Establishing multi-component phase field models on dynamic surfaces, and exploring the similarities and differences in phase field evolution between evolving and static surfaces, are interesting studies in multi-phase flow problems. This paper focuses on mathematical modeling and numerical simulation of the N-component Cahn-Hilliard model on evolving surfaces. In the modeling process, the evolution of the energy functional, componential mass conservation, and point-wise mass conservation (hyperplane link condition) are considered. On an evolving surface, the energy dissipation law does not hold because the surface velocity can be viewed as an external force in the system. Meanwhile, due to the velocity, the surface area may change, the componential mass conservation property and the hyperplane link condition can not be satisfied simultaneously. From the point of view of preserving these two physical properties of mass conservation, three types of N-component Cahn-Hilliard model are established on the evolving surface: componential mass conservation, point-wise mass conservation, and both componential and point-wise mass conservation. For the numerical simulation, the evolving surface finite element method is considered for the space-time discretization of the proposed model. In order to obtain a linear, decoupled, high-accurate, and stable scheme for long time numerical simulations, the stabilized semi-implicit method is integrated into the framework of the evolving surface finite element method. Through several numerical examples, the rationality of the model and the efficiency of the numerical method are shown. Additionally, three- and four-component phase separation phenomena are shown to investigate the N-component Cahn-Hilliard dynamic on various evolving surfaces. • Mathematical modeling of the N-component Cahn-Hilliard model on evolving surfaces. • The componential and point-wise mass conservation and their discrete preservations on evolving surfaces. • The stabilized semi-implicit method is integrated into the framework of the evolving surface finite element method. • Investigation on the similarities and differences in phase field evolution between evolving and static surfaces. [ABSTRACT FROM AUTHOR]

Variables interpolation is one of the key concepts of MUSCL schemes. Originally developed for one-dimensional frameworks, many improvements have been made over the last decades to extend these methods to general unstructured multi-dimensional meshes. It results that scalar interpolation in a finite volume cell-centered framework is already quite well understood. It is known that linear reconstructions have to be limited in order to prevent non-physical oscillations of the solutions while ensuring a spatial second-order accuracy for smooth solutions. This is done thanks to a limiting function which allows the reconstruction to satisfy a monotonicity property, which then ensure the stability of the scheme. Nevertheless, some difficulties arise when we try to extend this process to vectorial variables. Generally, vectorial reconstructions are done componentwise, but this process reveals to be frame-dependent and leads to a loss of precision due to false detection of extrema. In this paper, we present a new method dealing with vectorial reconstructions in a multislope MUSCL context. • The cell-centered finite volume method on general unstructured meshes is considered. • Alternative formulations of the scalar multislope MUSCL method are introduced. • The flaws of componentwise vectorial reconstructions are illustrated. • A new vectorial multislope MUSCL method is introduced from the scalar procedure. • Numerical results show its accuracy, frame-independence, efficiency and robustness. [ABSTRACT FROM AUTHOR]

The numerical approximation of the shallow-water equations, which support geophysical wave propagation, namely the fast external Poincaré and Kelvin waves and the slow large-scale planetary Rossby waves, is a delicate and difficult problem. Indeed, the coupling between the momentum and the continuity equations may lead to the presence of erratic or spurious solutions for these waves, e.g. the spurious pressure and inertial modes. In addition, the presence of spurious branches and the emergence of spectral gaps in the dispersion relation at specific wavenumbers may lead to anomalous dissipation/dispersion in the representation of both fast and slow waves. The aim of the present study is to propose a class of possible discretization schemes, via the discontinuous Galerkin method, that is not affected by the above-mentioned problems. A Fourier/stability analysis of the 2D shallow-water model is performed by analysing the finite volume method and the linear discontinuous P 1 D G and non conforming P 1 N C Galerkin discretizations. These are stabilized by means of a family of numerical fluxes via the Polynomial Viscosity Matrix approach. A long time stability result is proven for all schemes and fluxes examined here. Further, a super-convergent result is demonstrated for the P 1 N C discrete frequencies compared to the P 1 D G ones, except for the slow mode in the Roe flux case. Indeed, we show that the Roe flux yields spurious frequencies and sub-optimal rates of convergence for the slow mode, for both the P 1 D G and P 1 N C methods. Finally, numerical solutions of linear and non-linear test problems confirm the theoretical results and reveal the computational cost efficiency of the P 1 N C approach. • A Fourier analysis of the 2D shallow-water model is performed for the linear DG and NC Galerkin discretizations. • A long time stability result is proven for all schemes and fluxes examined in this paper. • A super-convergent result is demonstrated for the NC frequencies compared to the DG ones. • We show that the Roe flux yields spurious frequencies in O(h) for the Rossby frequency for both the DG and NC methods. • The CFL number of the NC method is 5/3 as large as that of the DG method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new set of midpoint-based high-order discretization schemes for computing straight and mixed nonlinear second derivative terms that appear in the compressible Navier-Stokes equations. Firstly, we detail a set of conventional fourth and sixth-order baseline schemes that utilize central midpoint derivatives for the calculation of second derivatives terms. To enhance the spectral properties of the baseline schemes, an optimization procedure is proposed that adjusts the order and truncation error of the midpoint derivative approximation while still constraining the same overall stencil width and scheme order. A new filter penalty term is introduced into the midpoint derivative calculation to help achieve high wavenumber accuracy and high-frequency damping in the mixed derivative discretization. Fourier analysis performed on the both straight and mixed second derivative terms show high spectral efficiency and minimal numerical viscosity with no odd-even decoupling effect. Numerical validation of the resulting optimized schemes is performed through various benchmark test cases assessing their theoretical order of accuracy and solution resolution. The results highlight that the present optimized schemes efficiently utilize the inherent viscosity of the governing equations to achieve improved simulation stability - a feature attributed to their superior spectral resolution in the high wavenumber range. The method is also tested and applied to non-uniform structured meshes in curvilinear coordinates, employing a supersonic impinging jet test case. [ABSTRACT FROM AUTHOR]

In this paper we develop a non-diffusive neural network (NDNN) algorithm for accurately computing weak solutions to hyperbolic conservation laws. The principle is to construct these weak solutions by computing smooth local solutions in subdomains bounded by discontinuity lines (DLs), the latter defined from the Rankine-Hugoniot jump conditions. The proposed approach allows to efficiently consider an arbitrary number of entropic shock waves, shock wave generation, as well as wave interactions. Some numerical experiments are presented to illustrate the strengths and properties of the algorithms. • Conservation laws. • Neural network solver. • Accurate discontinuity tracking. • Shock wave generation. • Wave interaction. [ABSTRACT FROM AUTHOR]

A full approximation scheme (FAS) nonlinear multigrid solver for two-phase flow and transport problems driven by wells with multiple perforations is developed. It is an extension to our previous work on FAS solvers for diffusion and transport problems. The solver is applicable to discrete problems defined on unstructured grids as the coarsening algorithm is aggregation-based and algebraic. To construct coarse basis that can better capture the radial flow near wells, coarse grids in which perforated well cells are not near the coarse-element interface are desired. This is achieved by an aggregation algorithm proposed in this paper that makes use of the location of well cells in the cell-connectivity graph. Numerical examples in which the FAS solver is compared against Newton's method on benchmark problems are given. In particular, for a refined version of the SAIGUP model, the FAS solver is at least 35% faster than Newton's method for time steps with a CFL number greater than 10. [ABSTRACT FROM AUTHOR]

We propose high-resolution numerical schemes for two-phase compressible flow simulations to reproduce dynamically created gas/vapor-liquid interfaces with phase change. In the Godunov-type finite volume framework, suppressing numerical dissipation errors in numerical schemes is crucial for capturing the discontinuous solutions. The MUSCL scheme has second-order accuracy for smooth solutions and non-oscillatory behavior near discontinuous solutions. However, the MUSCL scheme introduces excessive numerical dissipation and diffuses the discontinuities nonphysically, leading to difficulties in distinguishing gas/vapor and liquid phases, and thus causing a blur in the interfaces during the multi-phase flow simulations. The hybrid-type boundary variation diminishing (BVD) scheme in this paper combines the MUSCL scheme and the THINC scheme to reduce the numerical dissipation errors near the discontinuities. The MUSCL-THINC-BVD scheme applies the MUSCL scheme for smooth solutions and the THINC scheme for discontinuous solutions, resulting in the successful capture of the discontinuities including the dynamically created gas/vapor-liquid interfaces. The Adaptive THINC-BVD scheme, which switches two types of THINC schemes with different values of gradient parameter, also captures the discontinuities clearly. The numerical results of the benchmark tests show that the proposed BVD schemes can lucidly reproduce the vapor-liquid interfaces newly created during the dynamical process of phase change. • Hybrid BVD schemes are presented to compute gas/vapor interfaces in 6-equation model for multi-phase flows. • Interfaces newly generated between liquid and vapor during the evaporation process are well-reproduced for the first time. • Basic benchmark validations are conducted for two-phase flows associated with phase changes. • The numerical methods presented are well-suited for simulating multi-phase flows involving phase changes. [ABSTRACT FROM AUTHOR]

Understanding the complexity of the nucleation and transition between the crystalline and quasicrystalline is significant because the structural incommensurability is anisotropic and of significance in revealing material properties. This paper reports the two- and three-dimensional nucleation and transition investigation from quasicrystals to crystals using a phase field method. The investigation starts with the Lifshitz–Petrich (LP) model, which is derived from the Landau theory for the exploration of critical nuclei. By the variational derivation, we construct two phase field models with tenth- and eighth-order, which provide the possibility of exploring the transition under stable phase states. In order to dissipate the original Lifshitz–Petrich energy, we apply a Lagrange multiplier method to modify the two models and solve them by the Fourier spectral method. Whereas the nonlinearity leads to expensive computational burden and extra stiffness, the designed algorithm can effectively avoid the numerical oscillations caused by rigidity and keep an O (N log ⁡ N) computational complexity, where N is the mesh grid size. To further demonstrate the robustness and advantages of the proposed method for handling phase-field modeling of crystalline structures, we compare its performance with other methods for constructing unconditionally stable methods. Our method can be directly implemented on a GPU for acceleration and achieves multiple times faster performance compared to CPU-only alternatives. Various numerical tests have been used to validate that our method works well for revealing the transition between different stable states during the nucleation process. [ABSTRACT FROM AUTHOR]

In this paper, the Edge-Based Interface Tracking (EBIT) method is extended to simulate multiphase flows with phase change. As a novel Front-Tracking method, the EBIT method binds interfacial markers to the Eulerian grid to achieve automatic parallelization. To include phase change effects, the energy equation for each phase is solved, with the temperature boundary condition at the interface sharply imposed. When using collocated grids, the cell-centered velocity is approximately projected. This will lead to unphysical oscillations in the presence of phase change, as the velocity will be discontinuous across the interface. It is demonstrated that this issue can be addressed by using the ghost fluid method, in which the ghost velocity is set according to the jump condition, thereby removing the discontinuity. A series of benchmark tests are performed to validate the present method. It is shown that the numerical results agree very well with the theoretical solutions and the experimental results. • The Edge-Based Interface Tracking (EBIT) method has been extended to simulate multiphase flows with phase change. • The ghost fluid method is employed to suppress unphysical oscillations in the approximate projection method. • The present method has been validated by various benchmarks, demonstrating a second-order convergence rate. • The present method is implemented in the free software Basilisk, and all the codes are freely available in the Basilisk sandbox. [ABSTRACT FROM AUTHOR]

Inductive biases play a critical role in enabling Graph Networks (GN) to learn particle and mesh-based physics simulations. In this paper, we propose two generalizable inductive biases that minimize rollout error and energy accumulation. GNs conditioned on the input states and relying on the Mean Squared Error (MSE) loss function implicitly assume Gaussian-distributed output errors. Consequently, GNs may either assign probability densities to infeasible regions in the state space of the deterministic physics problem or fail to assign densities to feasible regions. Instead, we advocate for maximizing the likelihood of the actual target distribution, challenging the underlying assumptions of MSE-based regression models using our proposed conditional normalizing flows (cNF) decoder. We discover that this inductive bias enables GNs to significantly improve their next state prediction accuracy. Existing sequential GNs encode temporal dependencies by autoregressively processing the latent representations of the input data. In our work, we find that inducing the Arrow-of-Time inductive bias through an auto-regressive encoding step before autoregressively processing the resulting latent vectors enables GNs to better minimize rollout error. We critically analyze the impact of existing inductive biases on rollout error and energy accumulation and discover that the choice of biases encoded in a GN, rather than the number of inductive biases, has a substantial impact on forward simulation prediction. • Probabilistic inductive biases are crucial for minimizing rollout error/energy accumulation when learning physics simulations. • Inducing Arrow-of-Time bias via auto-regressive encoding improves Graph Networks' simulation prediction accuracy. • A conditional Normalizing Flows decoder minimizes prediction and energy accumulation errors in Graph Networks. [ABSTRACT FROM AUTHOR]

In this paper, we present a robust formulation to address nonlinear equality-constrained and inequality-constrained multi-objective optimization problems for multi-cell accelerating cavities with TESLA-like geometry under consideration of uncertain parameters. We propose to explore a robust approach that employs the polynomial chaos expansion (PCE) to model the uncertainty. For the uncertainty quantification (UQ), the resulting random-dependent eigenvalue problem is solved using the stochastic collocation method (SCM) combined with PCE. As a result, the shape optimization of the studied multi-cell cavity is defined in terms of statistical moments, which serve as the cost functionals in the multi-objective (MO) steepest descent algorithm. Two versions of this method are developed and implemented. Finally, the optimization results and their implications for the operation of a multi-cell cavity are discussed. • Key component of the particle accelerators, i.e., the shape of an accelerating cavity. • Stochastic Maxwell's eigenvalue problem to quantify uncertainty propagation. • Parametric multi-objective shape optimization under uncertainty using a robust formulation. • Lie derivative and shape derivative for continuous robust shape sensitivity. • Two efficient gradient-based approaches to find TESLA-like designs, which are robust against industrial imperfections. [ABSTRACT FROM AUTHOR]

Robust and efficient solvers for coupled-adjoint linear systems are crucial to successful aerostructural optimization. Monolithic and partitioned strategies can be applied. The monolithic approach is expected to offer better robustness and efficiency for strong fluid-structure interactions. However, it requires a higher implementation cost and convergence may depend on appropriate scaling and initialization strategies. On the other hand, the modularity of the partitioned method enables a straightforward implementation while its convergence may require relaxation. In addition, a partitioned solver often leads to a higher number of iterations to get the same level of convergence as the monolithic one. The objective of this paper is to accelerate the fluid-structure coupled-adjoint partitioned solver by considering techniques borrowed from approximate invariant subspace recycling strategies adapted to sequences of linear systems with varying right-hand sides. Indeed, in a partitioned framework, the structural source term attached to the fluid block of equations affects the right-hand side with the nice property of quickly converging to a constant value. For the fluid block of equations, we also consider deflation of approximate eigenvectors in conjunction with advanced inner-outer Krylov solvers. To demonstrate the effectiveness of our proposed recycling strategy, we compute the coupled derivatives for two emblematic problems in transonic viscous flow. The first one is an aeroelastic configuration of the ONERA-M6 fixed wing. For this exercise, the fluid grid was coupled to a structural model specifically designed to exhibit a high flexibility. The second one is a wing-body aeroelastic configuration of the Common Research Model. All computations are performed using a fully linearized one-equation Spalart-Allmaras turbulence model. Numerical simulations show up to 39% reduction in matrix-vector products for GCRO-DR and up to 25% for the nested FGCRO-DR solver. • Improve and accelerate partitioned solution of coupled-adjoint linear systems. • Review of approximate invariant subspace recycling theory with suggestions for practical implementation. • Adaptation to the specific case of fluid adjoint systems with a varying structural source term. • Numerical demonstration on a realistic test case (ONERA M6 wing in RANS flow and a Spalart Allmaras turbulence model). [ABSTRACT FROM AUTHOR]

Dynamical system state estimation and parameter calibration problems are ubiquitous across science and engineering. Bayesian approaches to the problem are the gold standard as they allow for the quantification of uncertainties and enable the seamless fusion of different experimental modalities. When the dynamics are discrete and stochastic, one may employ powerful techniques such as Kalman, particle, or variational filters. Practitioners commonly apply these methods to continuous-time, deterministic dynamical systems after discretizing the dynamics and introducing fictitious transition probabilities. However, approaches based on time-discretization suffer from the curse of dimensionality since the number of random variables grows linearly with the number of time-steps. Furthermore, the introduction of fictitious transition probabilities is an unsatisfactory solution because it increases the number of model parameters and may lead to inference bias. To address these drawbacks, the objective of this paper is to develop a scalable Bayesian approach to state and parameter estimation suitable for continuous-time, deterministic dynamical systems. Our methodology builds upon information field theory. Specifically, we parameterize the dynamical system state path using a linear basis. Then, we construct a physics-informed prior probability measure for the coefficients of the linear basis such that functions that satisfy the physics are more likely. This prior allows us to quantify model form errors. We connect the system's response to observations through a probabilistic model of the measurement process. The joint posterior over the system responses and all parameters is given by Bayes' rule. To approximate the intractable posterior, we develop a stochastic variational inference algorithm. In summary, the developed methodology offers a powerful framework for Bayesian estimation in dynamical systems. [ABSTRACT FROM AUTHOR]

The present paper addresses the problem of multiphase incompressible suspension flows where an interface with a gas phase is explicitly taken into account in a sharp manner. We define the suspension phase as a mixture of a liquid and non-colloidal particles and a level-set method is used to track the position of the interface between the suspension and the gas phase. The presence of particles in the liquid phase is accounted for in an Euler-Euler manner: properties of the suspension vary locally with the particle volume fraction which is accurately resolved with the use of high-order schemes. Since two-phase flows are considered, specific attention is given to the interface region in order to limit spurious forces and mass loss. The overall method turns out to be robust and accurately captures complex suspension flows as illustrated by selected results. • New numerical methodology to simulate incompressible solid-liquid suspension with an interface with gas, tracked by a level-set method. • Suspension is modeled in an Euler-Euler manner with advanced rheological properties that depend on the local particle volume fraction. • Specific jump conditions at the suspension/air interface are handled in a sharp manner through an extension of the Ghost Fluid Method. • Particle fraction transport in suspension phase is computed by an algorithm that prevents mass loss while maintaining physical relevance. [ABSTRACT FROM AUTHOR]

In this paper, we present a pre-classification overset grid assembly (OGA) based on the wall distance criterion. In contrast to the conventional OGA, where donor search is preceded by point classification, the proposed methodology first performs point classification based on the wall distance criterion, and then performs donor search on the reduced set of cells. To robustly profile the given grid components, a wall distance evaluation is performed exactly by considering the full geometry of the discretized surface while maintaining efficiency. The efficiency of the algorithm is achieved by redefining the methodology into a bounded form and establishing a conservative correlation between the vertex-based and element-based distances through a rigorous examination of geometric relations. From these observations, wall distance calculation, which requires a tremendous amount of computation, is divided into sequential steps of vertex-based approximate calculation and element-based exact calculation, dramatically reducing the computation load. To better adapt to parallel environments, load balancing is presented, and additionally, some voxelization techniques are developed to efficiently process the required geometric data. Numerous test cases are analyzed to demonstrate the efficiency and robustness of the proposed algorithm in various situations. The proposed algorithm consistently outperforms the conventional OGA, and also proves its robustness in the presence of high proximity objects, which highlights its suitability for large-scale multi-body problems. • Acceleration of overset grid assembly by pre-classifying the points. • Optimization of methodology with bounded distance form. • Conservative relationship between vertex-based and element-based distances. • Octree and volumetric ray casting for efficient data processing. • Robust and efficient method for large-scale multi-body simulation. [ABSTRACT FROM AUTHOR]