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We highlight some problems in this paper. They lead to question the model itself. [ABSTRACT FROM AUTHOR]

Abstract: The immersed boundary method has evolved into one of the most useful computational methods in studying fluid structure interaction. On the other hand, the immersed boundary method is also known to suffer from a severe timestep stability restriction when using an explicit time discretization. In this paper, we propose several efficient semi-implicit schemes to remove this stiffness from the immersed boundary method for the two-dimensional Stokes flow. First, we obtain a novel unconditionally stable semi-implicit discretization for the immersed boundary problem. Using this unconditionally stable discretization as a building block, we derive several efficient semi-implicit schemes for the immersed boundary problem by applying the small scale decomposition to this unconditionally stable discretization. Our stability analysis and extensive numerical experiments show that our semi-implicit schemes offer much better stability property than the explicit scheme. Unlike other implicit or semi-implicit schemes proposed in the literature, our semi-implicit schemes can be solved explicitly in the spectral space. Thus the computational cost of our semi-implicit schemes is comparable to that of an explicit scheme, but with a much better stability property. [Copyright &y& Elsevier]

In this paper, a family of four-stage singly diagonally implicit Runge-Kutta methods are proposed to solve first-/second-order time-dependent problems, exhibiting the following numerical properties: fourth-order accuracy in time, unconditional stability, controllable numerical dissipation, and adaptive time step selection. The BN-stability condition is employed as a constraint to optimize parameters in the Butcher table, having significant benefits, and hence is recommended for nonlinear dynamics problems in contrast to existing methods. Numerical examples involving both first- and second-order linear/nonlinear dynamics problems validate the proposed method, and numerical results reveal that the proposed methods are free from the order reduction phenomenon when applied to nonlinear dynamics problems. The performance of adaptive time-stepping using the embedded scheme is further illustrated by the phase-field modeling problem. Additionally, the advantages and disadvantages of three-stage third-order accurate algebraically stable methods are discussed. The proposed high-order time integration can be readily integrated into high-order spatial discretization methods, such as the high-order spectral element method employed in this paper, to obtain high-order discretization in space and time dimensions. • Three/four-stage SDIRK methods with controllable numerical dissipation. • Algebraically stable for time-dependent nonlinear simulations. • Embedded formulation with accurate error estimation. • Applications to both first- and second-order time-dependent problems. [ABSTRACT FROM AUTHOR]

• A novel positivity-preserving algorithm for implicit NS solver. • A residual correction to compute the correction factor. • A flux correction to enforce the positivity of the solution conservatively. • Positivity-preserving combined with implicit iterations. • Numerical experiments to verify the positivity-preserving capability. This paper presents a general positivity-preserving algorithm for implicit high-order finite volume schemes that solve compressible Euler and Navier-Stokes equations to ensure the positivity of density and internal energy (or pressure). Previous positivity-preserving algorithms are mainly based on the slope limiting or flux limiting technique, which rely on the existence of low-order positivity-preserving schemes. This dependency poses serious restrictions on extending these algorithms to temporally implicit schemes since it is difficult to know if a low-order implicit scheme is positivity-preserving. In the present paper, a new positivity-preserving algorithm is proposed in terms of the flux correction technique. And the factors of the flux correction are determined by a residual correction procedure. For a finite volume scheme that is capable of achieving a converged solution, we show that the correction factors are in the order of unity with additional high-order terms corresponding to the spatial and temporal rates of convergence. Therefore, the proposed positivity-preserving algorithm is accuracy-reserving and asymptotically consistent. The notable advantage of this method is that it does not rely on the existence of low-order positivity-preserving baseline schemes. Therefore, it can be applied to the implicit schemes solving Euler and especially Navier-Stokes equations. In the present paper, the proposed technique is applied to an implicit dual time-stepping finite volume scheme with temporal second-order and spatial high-order accuracy. The present positivity-preserving algorithm is implemented in an iterative manner to ensure that the dual time-stepping iteration will converge to the positivity-preserving solution. Another similar correction technique is also proposed to ensure that the solution remains positivity-preserving at each sub-iteration. Numerical results demonstrate that the proposed algorithm preserves positive density and internal energy in all test cases and significantly improves the robustness of the numerical schemes. [ABSTRACT FROM AUTHOR]

A spatial second-order scheme for the nonlinear radiative transfer equations is introduced in this paper. The discretization scheme is based on the filtered spherical harmonics (F P N) method for the angular variable and the unified gas kinetic scheme (UGKS) framework for the spatial and temporal variables respectively. In order to keep the scheme positive and second-order accuracy, firstly, we use the implicit Monte Carlo (IMC) linearization method [7] in the construction of the UGKS numerical boundary fluxes. This is an essential point in the construction. Then, by carefully analyzing the constructed second-order fluxes involved in the macro-micro decomposition, which is induced by the F P N angular discretization, we establish the sufficient conditions that guarantee the positivity of the radiative energy density and material temperature. Finally, we employ linear scaling limiters for the angular variable in the P N reconstruction and for the spatial variable in the piecewise linear slopes reconstruction respectively, which are shown to be realizable and reasonable to enforce the sufficient conditions holding. Thus, the desired scheme, called the P P F P N -based UGKS, is obtained. Furthermore, we can show that in the regime ϵ ≪ 1 and the regime ϵ = O (1) , the second-order fluxes can be simplified. And, a simplified spatial second-order scheme, called the P P F P N -based SUGKS, is thus presented, which possesses all the properties of the non-simplified one. Inheriting the merit of UGKS, the proposed schemes are asymptotic preserving. By employing the F P N method for the angular variable, the proposed schemes are almost free of ray effects. Moreover, the above-mentioned way of imposing the positivity would not destroy both AP and second-order accuracy properties. To our best knowledge, this is the first time that spatial second-order, positive, asymptotic preserving and almost free of ray effects schemes are constructed for the nonlinear radiative transfer equations without operator splitting. Therefore, this paper improves our previous work on the first-order scheme [42] which could not be directly extended to high order, while keeping the solution positive. Various numerical experiments are included to validate the properties of the proposed schemes. • A spatial second-order FPN scheme with both AP and PP properties is developed for nonlinear radiative transfer equations. • The scheme is almost free of ray effects, and meanwhile can reduce the Gibbs phenomena in the PN approximation. • The IMC linearization method is used in the construction of the UGKS numerical fluxes to make the solution positive. • A simplified scheme with all properties of the non-simplified one is proposed in regimes ϵ ≪ 1 and ϵ = O (1) to reduce the computational costs. • Numerical experiments have validated the spatial second-order accuracy, AP, PP and almost ray effects free properties. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a hybrid LBM-FVM solver for two-phase fluid flow simulations in which interface dynamics is modeled by a conservative phase-field equation. Integrating fluid equations over time is achieved through a velocity-based lattice Boltzmann solver which is improved by a central-moment multiple-relaxation-time collision model to reach higher accuracy. For interface evolution, we propose a finite-volume-based numerical treatment for the integration of the phase-field equation: we show that the second-order isotropic centered stencils for diffusive and separation fluxes combined with the WENO-5 stencils for advective fluxes achieve similar and sometimes even higher accuracy than the state-of-the-art double-distribution-function LBM methods as well as the DUGKS-based method, while requiring less computations and a smaller amount of memory. Benchmark tests (such as the 2D diagonal translation of a circular interface), along with quantitative evaluations on more complex tests (such as the rising bubble and Rayleigh-Taylor instability simulations) allowing comparisons with prior numerical methods and/or experimental data, are presented to validate the advantage of our hybrid solver. Moreover, 3D simulations (including a dam break simulation) are also compared to the time-lapse photography of physical experiments in order to allow for more qualitative evaluations. • This paper proposes a new hybrid LBM-FVM solver to simulate two-phase flows which reduces memory consumption and improves computational accuracy and efficiency. • The momentum equation is solved by a set of lattice Boltzmann equations with a velocity-based high-order CM-MRT model, while the phase-field equation is solved by a WENO-based finite-volume approach. • Our solver is validated through benchmark tests, comparisons, and validation examples, both quantitatively and qualitatively. • Our massively-parallel implementation on GPU offers efficient simulation of two-phase flows for a low memory footprint. [ABSTRACT FROM AUTHOR]

In this paper, we develop a fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme for simulating the multicomponent, chemically reacting, compressible Euler equations with complex thermodynamics. The proposed formulation is an extension of the fully conservative, high-order numerical method previously developed by Johnson and Kercher (2020) [14] that maintains pressure equilibrium between adjacent elements. In this first part of our two-part paper, we focus on the one-dimensional case. Our methodology is rooted in the minimum entropy principle satisfied by entropy solutions to the multicomponent, compressible Euler equations, which was proved by Gouasmi et al. (2020) [16] for nonreacting flows. We first show that the minimum entropy principle holds in the reacting case as well. Next, we introduce the ingredients, including a simple linear-scaling limiter, required for the discrete solution to have nonnegative species concentrations, positive density, positive pressure, and bounded entropy. We also discuss how to retain the aforementioned ability to preserve pressure equilibrium between elements. Operator splitting is employed to handle stiff chemical reactions. To guarantee discrete satisfaction of the minimum entropy principle in the reaction step, we develop an entropy-stable discontinuous Galerkin method based on diagonal-norm summation-by-parts operators for solving ordinary differential equations. The developed formulation is used to compute canonical one-dimensional test cases, namely thermal-bubble advection, advection of a low-density Gaussian wave, multicomponent shock-tube flow, and a moving hydrogen-oxygen detonation wave with detailed chemistry. We demonstrate that the formulation can achieve optimal high-order convergence in smooth flows. Furthermore, we find that the enforcement of an entropy bound can considerably reduce the large-scale nonlinear instabilities that emerge when only the positivity property is enforced, to an even greater extent than in the monocomponent, calorically perfect case. Finally, mass, total energy, and atomic elements are shown to be discretely conserved. [ABSTRACT FROM AUTHOR]

In this paper we present a scalable algorithm for fault-tolerant quantum computers for solving the transport equation in two and three spatial dimensions for variable grid sizes and discrete velocities, where the object walls are aligned with the Cartesian grid, the relative difference of velocities in each dimension is bounded by 1 and the total simulated time is dependent on the discrete velocities chosen. We provide detailed descriptions and complexity analyses of all steps of our quantum transport method (QTM) and present numerical results for 2D flows generated in Qiskit as a proof of concept. Our QTM is based on a novel streaming approach which leads to a reduction in the amount of CNOT gates required in comparison to state-of-the-art quantum streaming methods. As a second highlight of this paper we present a novel object encoding method, that reduces the complexity of the amount of CNOT gates required to encode walls, which now becomes independent of the size of the wall. Finally we present a novel quantum encoding of the particles' discrete velocities that enables a linear speed-up in the costs of reflecting the velocity of a particle, which now becomes independent of the amount of velocities encoded. Our main contribution consists of a detailed description of a fail-safe implementation of a quantum algorithm for the reflection step of the transport equation that can be readily implemented on a physical quantum computer. This fail-safe implementation allows for a variety of initial conditions and particle velocities and leads to physically correct particle flow behavior around the walls, edges and corners of obstacles. Combining these results we present a novel and fail-safe quantum algorithm for the transport equation that can be used for a multitude of flow configurations and leads to physically correct behavior. We finally show that our approach only requires O (n w n g 2 + d n t v n v max 2) CNOT gates, which is quadratic in the amount of qubits necessary to encode the grid and the amount of qubits necessary to encode the discrete velocities in a single spatial dimension. This complexity result makes our approach superior to state-of-the-art approaches known in the literature. • Quantum algorithm for the collisionless Boltzmann equation. • Efficient quantum primitive for streaming and reflection. • Fail-safe specular reflection operation. • Detailed complexity analysis in terms of natively implementable two-qubit gates. • Quantum computational fluid method focused on near-term implementability. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the acceleration of numerical simulations. We focus on a hypersonic planetary reentry problem whose simulation involves coupling fluid dynamics and chemical reactions. Simulating chemical reactions takes most of the computational time but, on the other hand, cannot be avoided to obtain accurate predictions. We face a trade-off between cost-efficiency and accuracy: the numerical scheme has to be sufficiently efficient to be used in an operational context but accurate enough to predict the phenomenon faithfully. To tackle this trade-off, we design a hybrid numerical scheme coupling a traditional fluid dynamic solver with a neural network approximating the chemical reactions. We rely on their power in terms of accuracy and dimension reduction when applied in a big data context and on their efficiency stemming from their matrix-vector structure to achieve important acceleration factors (×10 to ×18.6). This paper aims to explain how we design such cost-effective hybrid numerical schemes in practice. Above all, we describe methodologies to ensure accuracy guarantees, allowing us to go beyond traditional surrogate modeling and to use these schemes as references. • Deep Learning-based hybridization speeds up numerical schemes of atmospheric reentry while maintaining high accuracy. • Initializing a scheme with a hybrid code's prediction reduces the convergence time and keeps the exact same guarantees. • Uncertainty analysis provides statistical guarantees concerning approximation errors when using hybridization code. • Neural network approximation error is statistically lower than many other sources of error inherent to numerical simulations. [ABSTRACT FROM AUTHOR]

We propose an unconditionally energy-stable, orthonormality-preserving, component-wise splitting iterative scheme for the Kohn-Sham gradient flow based model in the electronic structure calculation. We first study the scheme discretized in time but still continuous in space. The component-wise splitting iterative scheme changes one wave function at a time, similar to the Gauss-Seidel iteration for solving a linear equation system. At the time step n , the orthogonality of the wave function being updated to other wave functions is preserved by projecting the gradient of the Kohn-Sham energy onto the subspace orthogonal to all other wave functions known at the current time, while the normalization of this wave function is preserved by projecting the gradient of the Kohn-Sham energy onto the subspace orthogonal to this wave function at t n + 1 / 2. The unconditional energy stability is nontrivial, and it comes from a subtle treatment of the two-electron integral as well as a consistent treatment of the two projections. Rigorous mathematical derivations are presented to show our proposed scheme indeed satisfies the desired properties. We then study the fully-discretized scheme, where the space is further approximated by a conforming finite element subspace. For the fully-discretized scheme, not only the preservation of orthogonality and normalization (together we called orthonormalization) can be quickly shown using the same idea as for the semi-discretized scheme, but also the highlight property of the scheme, i.e., the unconditional energy stability can be rigorously proven. The scheme allows us to use large time step sizes and deal with small systems involving only a single wave function during each iteration step. Several numerical experiments are performed to verify the theoretical analysis, where the number of iterations is indeed greatly reduced as compared to similar examples solved by the Kohn-Sham gradient flow based model in the literature. • This paper proposes a novel and efficient numerical scheme for the Kohn-Sham gradient flow based model. • The scheme is an unconditionally energy-stable, orthonormality-preserving, component-wise splitting iterative scheme. • The scheme does not modify the original energy, allows large time step sizes, and solves small systems at each time step. • The rigorous proof is presented in the paper. • Several numerical examples are illustrated to verify the theoretical analysis. [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]

An algorithm is presented to calculate the field components of the magnetic field [ B x (x , y , z) , B y (x , y , z) , B z (x , y , z) ] at a point (x , y , z) in space, from the knowledge of the components [ B x (x , y = 0 , z) , B y (x , y = 0 , z) , B z (x , y = 0 , z) ] on a "reference plane", which is normal to the y -axis at y = 0. The algorithm, which is a general one and is not restricted to fields with mid-plane symmetry is based on the Maclaurin series expansion of the magnetic field components at any point in space in terms of the distance (y) of the point from the reference plane. The coefficients of the Maclaurin series expansion are expressed in terms of the on-plane field components and their partial derivatives with respect to spatial coordinates (x , z). The field components are usually generated from magnetic field measurements on a rectangular grid on the plane. This algorithm was employed in 1986 in the RAYTRACE computer code to help calculate the optical properties of magnets and of the Alternating Gradient Synchrotron (AGS) at the Brookhaven National Laboratory (BNL). A general mathematical formulation of this algorithm based on the differential algebraic method was presented by Makino in 2011. This paper presents the step by step derivation of the algorithm and provides the necessary formulas to be introduced by the reader in any computer code which requires the field components generated by magnetic devices. In addition provides an example of the use of the algorithm and its limitations as applied to a Halbach type magnet with or without median plane symmetry. • The field of accelerator physics relays on magnetic field measurements of the devices used in beam lines or accelerators. • Such magnetic field measurements are limited in space for any particular magnet therefor algorithms like the one presented in this paper are important to calculate the values of the magnetic fields to a larger region of space than the space covered by the measured magnetic fields. [ABSTRACT FROM AUTHOR]

In the present paper, a multi-step reconstruction procedure is proposed for high order finite volume schemes on unstructured grids using compact stencil. The procedure is a recursive algorithm that can eventually provide sufficient relations for high order reconstruction in a multi-step procedure. Two key elements of this procedure are the partial inversion technique and the continuation technique. The partial inversion can be used not only to obtain lower order reconstruction based on existing reconstruction relations, but also to regularize the existing reconstruction relations to provide new relations for higher order reconstructions. The continuation technique is to extend the regularized relations on the face-neighboring cells to current cell as additional reconstruction relations. This multi-step procedure is operationally compact since in each step only the relations defined on a compact stencil are used. In the present paper, the third and fourth order finite volume schemes based on two-step quadratic and three-step cubic reconstructions are studied. [ABSTRACT FROM AUTHOR]

In a previous paper, [1] , a new formulation for isothermal gas flow in a single pipeline including gravity and friction effects has been introduced and solved by finite element methods. A performance analysis has been done by using some tests both manufactured and experimental. The goal of the present work is to extend the ideas to model gas flow in a transportation network. As a first step, in this paper the network only includes pipes connected at nodes which can be emission or consumption points, or simply structural connections between pipes. In addition to the mass and momentum conservation equations inside the pipes that are partial differential equations, we need to write the mass conservation equations at the nodes. The latter appear to be a set of constraints (one per node) whose respective Lagrange multipliers are the (continuous) pressures at the nodes. Thus, the whole model is a Partial Differential-Algebraic Equation (PDAE). For numerical solution, an implicit time discretization is proposed first and then a weak formulation of the resulting semi-discrete problem is introduced. After doing a finite element discretization we are led to solve a nonlinear system of numerical equations per time step what is done by Newton's method. In order to reduce the computing time, the algorithm can be parallelized in such a way that, at each time step, every pipe can be solved separately in one processor (or core). Moreover, we show that the proposed scheme exactly conserves the mass at the nodes and in the whole network, and a new simple and compact way of computing the network line-pack over the time is given. Finally, the introduced methodology is validated by applying the computer program to a triangular network, which is a standard test in references on the subject, and to a real network for which we have got measurements that enable us to create the boundary conditions for the model and also to compare the numerical results with experimental data. • Numerical solution of Euler-like models with friction and variable topography. • An equivalent formulation with one nonlinear PDE of second order in time and space. • An implicit time discretization combined with a finite element method is used. • Accuracy has been tested with experimental data and numerical results from papers. [ABSTRACT FROM AUTHOR]

• Introduce a new third-order conservative sign-preserving and steady-state-preserving RK method. • The ODE solver is A (π 4) -stable. • The technique can be applied to stiff detonation with large time step. In this paper, we develop third-order conservative sign-preserving and steady-state-preserving time integrations and seek their applications in multispecies and multireaction chemical reactive flows. In this problem, the density and pressure are nonnegative, and the mass fraction for the i th species, denoted as z i , 1 ≤ i ≤ M , should be between 0 and 1, where M is the total number of species. There are four main difficulties in constructing high-order bound-preserving techniques for multispecies and multireaction detonations. First of all, most of the bound-preserving techniques available are based on Euler forward time integration. Therefore, for problems with stiff source, the time step will be significantly limited. Secondly, the mass fraction does not satisfy a maximum-principle and hence it is not easy to preserve the upper bound 1. Thirdly, in most of the previous works for gaseous denotation, the algorithm relies on second-order Strang splitting methods where the flux and stiff source terms can be solved separately, and the extension to high-order time discretization seems to be complicated. Finally, most of the previous ODE solvers for stiff problems cannot preserve the total mass and the positivity of the numerical approximations at the same time. In this paper, we will construct third-order conservative sign-preserving Rugne-Kutta and multistep methods to overcome all these difficulties. The time integrations do not depend on the Strang splitting, i.e. we do not split the flux and the stiff source terms. Moreover, the time discretization can handle the stiff source with large time step and preserves the steady-state. Numerical experiments will be given to demonstrate the good performance of the bound-preserving technique and the stability of the scheme for problems with stiff source terms. [ABSTRACT FROM AUTHOR]

This paper presents a topology optimization method in rarefied gas flow problems to obtain the optimal structure of a flow channel as a configuration of gas and solid domains. In this paper, the kinetic equation, the governing equation of rarefied gas flows, is extended over the entire design domain including solid domains assuming the solid as an imaginary gas for implicitly handling the gas-solid interfaces in the optimization process. Based on the extended equation, a 2D flow channel design problem is formulated, and the design sensitivity is obtained based on the Lagrange multiplier method and adjoint variable method. Both the rarefied gas flow and the adjoint flow are computed by a deterministic method based on a finite discretization of the molecular velocity space, rather than the DSMC method. The validity and effectiveness of our proposed method are confirmed through several numerical examples. • We constructed a topology optimization method for rarefied gas flow problems. • The BGK equation was extended over the domain composed of gas and solid. • The sensitivity analysis was performed based on the adjoint variable method. • Numerical examples showed the validity and usefulness of our proposed method. [ABSTRACT FROM AUTHOR]

Computational electrodynamics (CED), the numerical solution of Maxwell's equations, plays an incredibly important role in several problems in science and engineering. High accuracy solutions are desired, and the discontinuous Galerkin (DG) method is one of the better ways of delivering high accuracy in numerical CED. Maxwell's equations have a pair of involution constraints for which mimetic schemes that globally satisfy the constraints at a discrete level are highly desirable. Balsara and Käppeli (2019) presented a von Neumann stability analysis of globally constraint-preserving DG schemes for CED up to fourth order. That paper was focused on developing the theory and documenting the superior dissipation and dispersion of DGTD schemes in media with constant permittivity and permeability. In this paper we present working DGTD schemes for CED that go up to fifth order of accuracy and analyze their performance when permittivity and permeability vary strongly in space. Our DGTD schemes achieve constraint preservation by collocating the electric displacement and magnetic induction as well as their higher order modes in the faces of the mesh. Our first finding is that at fourth and higher orders of accuracy, one has to evolve some zone-centered modes in addition to the face-centered modes. It is well-known that the limiting step in DG schemes causes a reduction of the optimal accuracy of the scheme; though the schemes still retain their formal order of accuracy with WENO-type limiters. In this paper, we document simulations where permittivity and permeability vary by almost an order of magnitude without requiring any limiting of the DG scheme. This very favorable second finding ensures that DGTD schemes retain optimal accuracy even in the presence of large spatial variations in permittivity and permeability. We also study the conservation of electromagnetic energy in these problems. Our third finding shows that the electromagnetic energy is conserved very well even when permittivity and permeability vary strongly in space; as long as the conductivity is zero. • Globally constraint-preserving FR/DG schemes have been designed for CED up to fifth order. • At fourth and higher orders, one has to evolve a few zone-centered modes in addition to the facial modes. • Limiters are not needed even when material properties vary by an order of magnitude. • The high order schemes show excellent conservation of electromagnetic energy even on coarse meshes. [ABSTRACT FROM AUTHOR]

In this paper we propose a novel set of first-order hyperbolic equations that can model dispersive non-hydrostatic free surface flows. The governing PDE system is obtained via a hyperbolic approximation of the family of non-hydrostatic free-surface flow models recently derived by Sainte-Marie et al. in [1]. Our new hyperbolic reformulation is based on an augmented system in which the divergence constraint of the velocity is coupled with the other conservation laws via an evolution equation for the depth-averaged non-hydrostatic pressure, similar to the hyperbolic divergence cleaning applied in generalized Lagrangian multiplier methods (GLM) for magnetohydrodynamics (MHD). We suggest a formulation in which the divergence errors of the velocity field are transported with a large but finite wave speed that is directly related to the maximal eigenvalue of the governing PDE. We then use arbitrary high order accurate (ADER) discontinuous Galerkin (DG) finite element schemes with an a posteriori subcell finite volume limiter in order to solve the proposed PDE system numerically. The final scheme is highly accurate in smooth regions of the flow and very robust and positive preserving for emerging topographies and wet-dry fronts. It is well-balanced making use of a path-conservative formulation of HLL-type Riemann solvers based on the straight line segment path. Furthermore, the proposed ADER-DG scheme with a posteriori subcell finite volume limiter adapts very well to modern GPU architectures, resulting in a very accurate, robust and computationally efficient computational method for non-hydrostatic free surface flows. The new model proposed in this paper has been applied to idealized academic benchmarks such as the propagation of solitary waves, as well as to more challenging physical situations that involve wave runup on a shore including wave breaking in both one and two space dimensions. In all cases the achieved agreement with analytical solutions or experimental data is very good, thus showing the validity of both, the proposed mathematical model and the numerical solution algorithm. • New family of hyperbolic reformulations of depth-averaged models for nonlinear dispersive water waves. • Governing PDE system fulfills an extra energy conservation law (convex extension). • Discretization with arbitrary high order accurate discontinuous Galerkin finite element schemes. • The new approach is computationally efficient and allows large explicit time steps. • Straightforward and highly efficient GPU implementation. [ABSTRACT FROM AUTHOR]

• Deep learning accelerates vapor liquid equilibrium prediction with verified accuracy. • Comprehensive comparison between deep learning and various flash calculation methods. • Optimized deep learning model for estimating VLE properties. • Understandable introduction of deep learning and flash calculation for general audience. • Remarks on further applications of deep learning in classical engineering problems. In the past two decades, researchers have made remarkable progress in accelerating flash calculation, which is very useful in a variety of engineering processes. In this paper, general phase splitting problem statements and flash calculation procedures using the Successive Substitution Method are reviewed, while the main shortages are pointed out. Two acceleration methods, Newton's method and the Sparse Grids Method are presented afterwards as a comparison with the deep learning model proposed in this paper. A detailed introduction from artificial neural networks to deep learning methods is provided here with the authors' own remarks. Factors in the deep learning model are investigated to show their effect on the final result. A selected model based on that has been used in a flash calculation predictor with comparison with other methods mentioned above. It is shown that results from the optimized deep learning model meet the experimental data well with the shortest CPU time. More comparison with experimental data has been conducted to show the robustness of our model. [ABSTRACT FROM AUTHOR]

In this paper, we consider Maxwell's equations in linear dispersive media described by a single-pole Lorentz model for electronic polarization. We study two classes of commonly used spatial discretizations: finite difference methods (FD) with arbitrary even order accuracy in space and high spatial order discontinuous Galerkin (DG) finite element methods. Both types of spatial discretizations are coupled with second order semi-implicit leap-frog and implicit trapezoidal temporal schemes. By performing detailed dispersion analysis for the semi-discrete and fully discrete schemes, we obtain rigorous quantification of the dispersion error for Lorentz dispersive dielectrics. In particular, comparisons of dispersion error can be made taking into account the model parameters, and mesh sizes in the design of the two types of schemes. This work is a continuation of our previous research on energy-stable numerical schemes for nonlinear dispersive optical media [6,7]. The results for the numerical dispersion analysis of the reduced linear model, considered in the present paper, can guide us in the optimal choice of discretization parameters for the more complicated and nonlinear models. The numerical dispersion analysis of the fully discrete FD and DG schemes, for the dispersive Maxwell model considered in this paper, clearly indicate the dependence of the numerical dispersion errors on spatial and temporal discretizations, their order of accuracy, mesh discretization parameters and model parameters. The results obtained here cannot be arrived at by considering discretizations of Maxwell's equations in free space. In particular, our results contrast the advantages and disadvantages of using high order FD or DG schemes and leap-frog or trapezoidal time integrators over different frequency ranges using a variety of measures of numerical dispersion errors. Finally, we highlight the limitations of the second order accurate temporal discretizations considered. • Maxwell's equations. • Lorentz model. • Numerical dispersion. • Finite differences. • Discontinuous Galerkin finite elements. [ABSTRACT FROM AUTHOR]

In this paper we propose a fluctuation splitting finite volume scheme for a non-conservative modeling of shear shallow water flow (SSWF). This model was originally proposed by Teshukov (2007) in [14] and was extended to include modeling of friction by Gavrilyuk et al. (2018) in [7]. The directional splitting scheme proposed by Gavrilyuk et al. (2018) in [7] is tricky to apply on unstructured grids. Our scheme is based on the physical splitting in which we separate the characteristic waves of the model to form two different hyperbolic sub-systems. The fluctuations associated with each sub-systems are computed by developing Riemann solvers for these sub-systems in a local coordinate system. These fluctuations enable us to develop a Godunov-type scheme that can be easily applied on mixed/unstructured grids. While the equation of energy conservation is solved along with the SSWF model in [7] , in this paper we solve only SSWF model equations. We develop a cell-centered finite volume code to validate the proposed scheme with the help of some numerical tests. As expected, the scheme shows first order convergence. The numerical simulation of 1D roll waves shows a good agreement with the experimental results. The numerical simulations of 2D roll waves show similar transverse wave structures as observed in [7]. • A finite volume scheme is proposed for non-conservative shear shallow water equations. • A preferable 'family of paths' is used to design approximate Riemann solvers. • These Riemann solvers are used in to design a fluctuation based Godunov-type scheme. • The proposed scheme is validated by performing some numerical tests. • The scheme can be a useful tool to study the shear shallow water equations. [ABSTRACT FROM AUTHOR]

In this paper, we propose an extended model for the numerical simulation of evaporating droplets within the framework of interface capturing or interface tracking methods. Most existing works make several limiting assumptions that need to be overcome for a more accurate description of the evaporation of droplets. In particular, the variations of several physical variables with local temperature and mass fraction fields must be accounted for in order to perform more realistic computations. While taking into account the variations of some of these physical properties, as viscosity, seems rather obvious, variations of other variables, as density and surface tension, involve additional source terms in the fundamental equations for which a suitable discretization must be developed. The paper presents a numerical strategy to account for such an extended model along with several original test-cases allowing to demonstrate both the accuracy of the proposed numerical schemes and the strong interest in developing such an extended model for the simulation of droplet evaporation. In particular, the impact of thermo-capillary convection will be highlighted on the vapor film thickness between a superheated wall and a static Leidenfrost droplet levitating above this wall. • An extended model is proposed for the direct numerical simulation of evaporating droplets. • A suitable numerical framework is developed to solve this extended model. • Novel benchmarks are proposed to validate the extended model and the proposed numerical approach. • A significant impact of the density variations on the Nusselt number is reported for a moving and evaporating droplet. • A strong effect of the Marangoni convection is also reported for Leidenfrost droplets. [ABSTRACT FROM AUTHOR]

Projection methods are applicable in many fields. It is a natural and practical approach to devise the invariant-preserving schemes for conservative systems. The idea is to project the solution of any underlying numerical scheme onto the manifold determined by the invariant, and this process will be referred to as the projection integrator. Generally, the projection integrator chooses the gradient of invariant as its projection direction and has the same order as the underlying method. In this paper, we propose a different projection direction to construct a new projection integrator whose order is higher than the underlying method. According to this novel direction, we further summarize high-order projection integrators with superconvergence and rigorously prove the truncation error by utilizing the linear integral method as a central tool. Apart from the invariant-preserving property, symmetry is an important geometric property for reversible differential equations. The design and analysis of another high-order projection integrators with symmetry and superconvergence are also presented in this paper. Numerical experiments are provided to verify our theoretical results and illustrate that our proposed projection integrators have superior behaviors in a long time numerical simulation. [ABSTRACT FROM AUTHOR]

Recently, there has been a series of works on the positivity-preserving discontinuous Galerkin methods for stationary hyperbolic equations, where the notion of mass conservation follows from a straightforward analogy of that of time-dependent problems, i.e. conserving the mass = preserving cell averages during limiting. Based on such a notion, the implementations and theoretical proofs of positivity-preserving limited methods for stationary equations are unnecessarily complicated and constrained. As will be shown in this paper, in some extreme cases, their convergence could even be problematic. In this work, we clarify a more appropriate definition of mass conservation for limiters applied to stationary hyperbolic equations and establish the genuinely conservative high-order positivity-preserving limited discontinuous Galerkin methods based on this definition. The new methods are able to preserve the positivity of solutions of scalar linear equations and scalar nonlinear equations with invariant wind direction, with much simpler implementations and easier proofs for accuracy and the Lax-Wendroff theorem, compared with the existing methods. Two types of positivity-preserving limiters preserving the local mass of stationary equations are developed to accommodate for the new definition of conservation and their accuracy are investigated. We would like to emphasize that a major advantage of the original DG scheme presented in [24] is a sweeping procedure, which allows for the computation of conservative steady-state solutions explicitly, cell by cell, without iterations, even for nonlinear equations as long as the wind direction is fixed. The main contribution of this paper is to introduce a limiting procedure to enforce positivity without changing the conservative property of this original DG scheme. The good performance of the algorithms for stationary hyperbolic equations and their applications in time-dependent problems are demonstrated by ample numerical tests. • A new definition of local conservation is given for stationary hyperbolic systems. • This allows the design of positivity-preserving discontinuous Galerkin (DG) schemes in more general cases than before. • Such high order positivity-preserving DG schemes are more general than before. [ABSTRACT FROM AUTHOR]

This paper primarily concerns the discontinuities capturing problems in nonconservative and nonconvex conservative hyperbolic systems. For the Godunov scheme of nonconservative hyperbolic systems, the numerical dissipation at discontinuous points in the simulation process is analyzed grid point by grid point through a new perspective. The numerical paths implied in the nonconservative variables represent different averaging and dissipating processes from the conservative cases. Unphysical dissipation of nonconservative variables ruins Rankine-Hugoniot relations, contributing to incorrect jumps, wrong propagation speed, and spurious fluctuations in other characteristic fields. For the discontinuities capturing problem of nonconservative hyperbolic systems, a novel numerical path preserving (NPP) method is proposed to modify the original Godunov schemes so that the dissipation of the numerical methods at discontinuities is carried out strictly following the consistent numerical path. Numerical simulations of the nonconservative systems are performed for isothermal and Euler equations. The results indicate that the NPP method can correctly capture the discontinuous structures and verify the correctness of our theoretical analysis. Additionally, the NPP method is extended to nonconvex hyperbolic conservation systems, and the Alfvénic wave (discontinuity) of the one-dimensional ideal magnetohydrodynamic (MHD) equations is simulated. It is found that the nonconvex nature of the flux causes unphysical compound wave structures while simulating the Alfvénic wave with the popular schemes nowadays, e.g., Roe of flux difference splitting method and WENO with flux vector splitting method. Modifying the original Godunov schemes with the NPP method proposed in this paper ensures correct simulation of Alfvénic discontinuity in MHD equations. It validates the effectiveness of the NPP method for nonconvex hyperbolic conservation systems. [ABSTRACT FROM AUTHOR]