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

In this paper, we propose a unified framework for studying the Cahn–Hilliard equation with two distinct types of dynamic boundary conditions, namely, the Allen–Cahn and Cahn–Hilliard types. Using this unified framework, we develop a linear, second-order, and energy-stable scheme based on the multiple scalar auxiliary variables (MSAV) approach. We design efficient and decoupling algorithms for solving the corresponding linear system in which the unknown variables are intricately coupled both in the bulk and at the boundary. Several numerical experiments are shown to validate the proposed scheme, and to investigate the effect of different dynamical boundary conditions on the dynamics of phase evolution under different scenarios. [ABSTRACT FROM AUTHOR]

The extended Fisher–Kolmogorov (EFK) equation has been used to describe some phenomena in physical, material and biological systems. In this paper, we propose a full-rank splitting scheme and a rank-adaptive splitting approach for this equation. We first use a finite difference method to approximate the space derivatives. Then, the resulting semi-discrete system is split into two stiff linear parts and a nonstiff nonlinear part. This leads to our full-rank splitting scheme. The convergence of the proposed scheme is proved rigorously. Based on the frame of the full-rank splitting scheme, we design a rank-adaptive splitting approach for obtaining a low-rank solution of the EFK equation. Numerical examples show that our methods are robust and accurate. They can also preserve the energy dissipation. • The EFK equation is split into three subproblems, then a full-rank splitting scheme is established. The convergence of this scheme is analyzed. • A rank-adaptive low-rank approach is proposed for the EFK equation. To the best of our knowledge, this is new in the literature for the equation. • Numerical examples show that our methods are robust and accurate. They can also preserve energy dissipation. [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]

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]

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]

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 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]

Accurate and efficient numerical simulations of soil water movement, as described by the highly nonlinear Richards' equation, often require local refinement near recharge or sink/source terms. In this paper, we present a novel numerical scheme for solving Richards' equation. Our approach is based on the vertex-centered finite volume method (VCFVM) and can be easily adapted to locally refined meshes. The proposed scheme offers some key features, including the definition of all unknowns over vertices of the primary mesh, expression of flux crossing dual edges as combinations of hydraulic heads at the vertices of the primary cell, and the capability to handle nonmatching meshes in the presence of local mesh refinement. For performance evaluation, soil water content and soil water potential simulated by the proposed scheme are benchmarked against results produced from HYDRUS (a widely used soil water numerical model) and the observed values in four test cases, including a convergence test case, a synthetic case, a laboratory experiment case and a field experiment case. The comparison results demonstrate the effectiveness and applicability of our scheme across a wide range of soil parameters and boundary conditions. [ABSTRACT FROM AUTHOR]

We propose an easy-to-implement iterative method for resolving the implicit (or semi-implicit) schemes arising in solving reaction-diffusion (RD) type equations. We formulate the nonlinear time implicit scheme as a min-max saddle point problem and then apply the primal-dual hybrid gradient (PDHG) method. Suitable precondition matrices are applied to the PDHG method to accelerate the convergence of algorithms under different circumstances. Furthermore, our method is applicable to various discrete numerical schemes with high flexibility. From various numerical examples tested in this paper, the proposed method converges properly and can efficiently produce numerical solutions with sufficient accuracy. [ABSTRACT FROM AUTHOR]

Stochastic Klein–Gordon–Schrödinger (KGS) equations are important mathematical models and describe the interaction between scalar nucleons and neutral scalar mesons in the stochastic environment. In this paper, we propose novel structure-preserving schemes to numerically solve stochastic KGS equations with additive noise, which preserve averaged charge evolution law, averaged energy evolution law, symplecticity, and multi-symplecticity. By applying central difference, sine pseudo-spectral method, or finite element method in space and modifying finite difference in time, we present some charge and energy preserved fully-discrete schemes for the original system. In addition, combining the symplectic Runge-Kutta method in time and finite difference in space, we propose a class of multi-symplectic discretizations preserving the geometric structure of the stochastic KGS equation. Finally, numerical experiments confirm theoretical findings. [ABSTRACT FROM AUTHOR]

Liouville's equation describes light propagation through an optical system. It governs the evolution of an energy distribution, the basic luminance, on phase space. The basic luminance is discontinuous across optical interfaces, as is described by non-local boundary conditions at each optical interface. Curved optical interfaces manifest themselves as moving boundaries on phase space. A common situation is that an optical system is described by a piecewise constant refractive index field. Away from optical interfaces, the characteristics of Liouville's equation reduce to straight lines. This property is exploited in the novel solver developed in this paper by employing a semi-Lagrangian discontinuous Galerkin (SLDG) scheme away from optical interfaces. Close to optical interfaces we apply an ADER discontinuous Galerkin (ADER-DG) method on a moving mesh. The ADER-DG method is a fully discrete explicit scheme which must obey a CFL condition that restricts the stepsize, whereas the SLDG scheme can be CFL-free. A moving mesh is used to align optical interfaces with the mesh. Very small elements cannot always be avoided, even when applying mesh refinement. Local time stepping is introduced in the solver to ensure these very small elements only have a local impact on the stepsize. By construction we allow elements of SLDG type to run at a stepsize independent of these small elements. The proposed SLDG scheme uses the exact evolution of the solution, as is described by the characteristics, to update the numerical solution. We impose the condition that no characteristic emanating from an SLDG element can cross an optical interface. In the novel hybrid SLDG and ADER-DG solver this condition is used to naturally divide the spatio-temporal domain into different regions, describing where the SLDG scheme and where the ADER-DG scheme need to be used. An intermediate element is introduced to efficiently couple an SLDG region with an ADER-DG region. Numerical experiments validate that the hybrid solver obeys energy conservation up to machine precision and numerical convergence results show the expected order of convergence. The performance of the hybrid solver is compared to a pure ADER-DG scheme with global time stepping to show the efficiency of the hybrid solver. In particular, a speed-up of 1.6 to 10, in favour of the hybrid solver, for computation times up to 4 minutes was seen in an example. Finally, the hybrid solver and the pure ADER-DG scheme are compared to quasi-Monte Carlo ray tracing. In the examples considered, amongst the three methods the hybrid solver is shown to converge the fastest to high accuracies. • Hybrid semi-Lagrangian (SL) discontinuous Galerkin (DG) and ADER-DG. • Local time stepping leads to high efficiency. • Conservative coupling between SLDG and ADER-DG elements. • High order energy-preserving method for geometrical optics. [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]

Abstract In this paper, numerical schemes are introduced for the efficient resolution of anisotropic equations including high order differential operators. The model problem investigated in this paper, though simplified, is representative of the difficulties encountered in the modeling of Tokamak plasmas. The occurrence of high order differential operators introduces specific difficulties for the design of effective numerical methods. On the one hand, regular discretizations of the problem provide matrices characterized by a condition number that blows up with increasing anisotropy strength. On the other hand, matrices issued from Asymptotic-Preserving methods preserve a condition number bounded with respect to the anisotropy strength, nonetheless it scales very poorly as the mesh is refined. Both alternatives reveal to be inoperative in this specific framework to address the targeted values of anisotropy on refined meshes. We therefore introduce two successful methods offering the advantages of each approach: a condition number unrelated to the anisotropy strength and scaling as favorably as standard discretizations with the mesh refinement. Highlights • Numerical methods for anisotropic equations arising in tokamak turbulence plasma simulation. • Improved numerical efficiency for large anisotropy strengths and refined meshes compared to available numerical methods. • System matrix condition number uniformly bounded with respect to the anisotropy strength. • System matrix condition number scaling similarly to isotropic problems with respect to the mesh refinements. [ABSTRACT FROM AUTHOR]

Abstract Solving the interface positioning equation is the key procedure of the PLIC-VoF methods. Most of previous research only focused on the planar constant calculation and paid less attention to the relationship between the planar constant and the approximate interface orientation. The latter issue is important especially for the second order iteration based PLIC-VoF method, such as the MoF and LVIRA method. In these methods, the most accurate interface orientation is calculated through an iterative procedure, so the interface positioning equation has to be solved multiple times for the given volume fraction with different interface orientations. In this situation, if the incremental relation between the planar constant and the interface orientation is known, a predicted planar constant can be estimated. In this paper, we deduce the analytical partial derivatives of the planar constant with respect to the interface orientation and use them to predict the planar constant. A predicted-Newton's method is proposed to solve the interface positioning equation which takes the predicted planar constant as the initial guess. A great deal of numerical tests are also presented in this paper to verify the robustness of the new scheme. The efficiency of the proposed predicted-Newton's method is compared with the commonly used secant/bisection method by Ahn and Shashkov, and the numerical results indicate that the new method can reduce the iteration steps by 60 % ∼ 66 % in solving the interface positioning equation and reduce the CPU time by 32 % ∼ 39 % when implemented in the MoF method. [ABSTRACT FROM AUTHOR]

This paper deals with the implementation of high order semi-Lagrangian particle methods to handle high dimensional Vlasov–Poisson systems. It is based on recent developments in the numerical analysis of particle methods and the paper focuses on specific algorithmic features to handle large dimensions. The methods are tested with uniform particle distributions in particular against a recent multi-resolution wavelet based method on a 4D plasma instability case and a 6D gravitational case. Conservation properties, accuracy and computational costs are monitored. The excellent accuracy/cost trade-off shown by the method opens new perspective for accurate simulations of high dimensional kinetic equations by particle methods. [ABSTRACT FROM AUTHOR]

In this study, we focus on the mathematical model of hyperthermia treatment as one the most constructive and effective procedures. Considering the sophisticated nature of involving phenomena in bioheat transfer inside a living tissue, several models with different levels of simplifications have been proposed. One of the general forms of the bioheat transfer equation which is introduced and studied in this paper for the first time, is the 2D-transient, dual phase lag (DPL), variable-order fractional energy equation. For finding the numerical solution of this general case, we propose an efficient semi-discrete method based on the two-dimensional Legendre wavelets (2D LWs). Precisely, the variable-order fractional derivatives of the model are discretized in the first stage, and then the response of the model is expanded by the 2D LWs. Consequently, the main problem is transformed into an equivalent system of algebraic equations, which can be simply tackled. The stability of the proposed method is examined theoretically and experimentally. Also, the procedure is described for one example to examine the computational efficiency of method. The experimental results show the stability and spectral accuracy of the proposed method. According to the achieved results, increasing the fractional order from 0.1 to 1.0, leads to increment of maximum tissue temperatures by about 29% near the center of the targeted region. [ABSTRACT FROM AUTHOR]

In this paper, the Monte Carlo Markov chain method for solving the modified anomalous fractional sub-diffusion equation is studied. Most of the previous methods are low in temporal and spatial accuracy order. Based on the idea of Monte Carlo Markov chain method and compact finite difference schemes, a probability model for solving the modified anomalous fractional sub-diffusion equation is established. Numerical examples are given to show the feasibility of the proposed scheme. Compared with the compact finite difference method, the present method is truly meshless and is easy to be implemented with high temporal and spatial accuracy order. And it is also applied to solve partial differential equation in irregular domains. • The Monte Carlo Markov chains method is developed to solve the modified anomalous fractional sub-diffusion equation. • The present method is truly meshless and is easy to be implemented with high temporal and spatial accuracy order. • The convergence orders are analyzed and discussed by calculating the numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we develop the fully discrete local discontinuous Galerkin (LDG) methods coupled with the implicit-explicit (IMEX) multistep time marching for solving the nonlinear Cahn-Hilliard equation. To be more specific, we rewrite the Cahn-Hilliard equation in a novel form by adding and subtracting a "linear" term. Then we discretize the spatial derivatives with the LDG methods and the temporal derivative with the IMEX multistep method. Finally, a series of numerical experiments are given to verify the accuracy, efficiency and validness of proposed method. [ABSTRACT FROM AUTHOR]

This paper shows how the combination of Strang splitting and the exact integration of SPH pair-wise viscous interactions, enables highly viscous flows such as lava or magma, to be integrated efficiently, even when the typical time scale of the dissipation terms is much less than the time scale arising from other constraints such as the Courant condition. We first apply the algorithm to the simulation of a planar Couette flow in two dimensions and find it is stable up to the highest viscosity coefficient used which, in SI units is μ = 10 5 which is equivalent to 108 times the viscosity of water. The accuracy is excellent up to μ = 800 but for larger values of μ the errors are larger until the flow is close to the final state. The second application is to the two dimensional flow of a viscous fluid under gravity over a rigid surface and with a free upper surface. The agreement with the similarity solution of Huppert (1982) [3] is very satisfactory even when the time step is ∼600 times the stable time step for an explicit integration. • Simple algorithm • No iteration required • High accuracy [ABSTRACT FROM AUTHOR]

The present paper develops two new techniques, namely additive correction multicloud (ACMC) and smoothed restriction multicloud (SRMC), for the efficient solution of systems of equations arising from Radial Basis Function-generated Finite Difference (RBF-FD) meshless discretizations of partial differential equations (PDEs). RBF-FD meshless methods employ arbitrary distributed nodes, without the need to generate a mesh, for the numerical solution of PDEs. The proposed techniques are specifically designed for the RBF-FD data structure and employ simple restriction and interpolation strategies in order to obtain a hierarchy of coarse-level node distributions and the corresponding correction equations. Both techniques are kept as simple as possible in terms of code implementation, which is an important feature of meshless methods. The techniques are verified on 2D and 3D Poisson equations, defined on non-trivial domains , showing very high benefits in terms of both time consumption and work to convergence when comparing the present techniques to the most common solver approaches. These benefits make the RBF-FD approach competitive with standard grid-based approaches when the number of nodes is very high, allowing large size problems to be tackled by the RBF-FD method. • Multilevel principles for convergence acceleration are applied to RBF-FD meshless discretizations. • Two simple multicloud techniques have been developed and successfully applied. • Code implementation is kept as simple as possible. • Convergence work and computing time are considerably reduced (up to 10× and 20×, respectively). • Possibility to face very large size problems with the RBF-FD meshless method. [ABSTRACT FROM AUTHOR]

Liouville's equation describes light propagation through an optical system. It governs the evolution of an energy distribution on phase space. This energy distribution is discontinuous across optical interfaces. Curved optical interfaces manifest themselves as moving boundaries on phase space. In this paper, an ADER discontinuous Galerkin (DG) method on a moving mesh is applied to solve Liouville's equation. In the ADER approach a temporal Taylor series is computed by replacing temporal derivatives with spatial derivatives using the Cauchy-Kovalewski procedure. The result is a fully discrete explicit scheme of arbitrary high order of accuracy. A moving mesh is not sufficient to be able to solve Liouville's equation numerically for the optical systems considered in this article. To that end, we combine the scheme with a new method we refer to as sub-cell interface method. When dealing with optical interfaces in phase space, non-local boundary conditions arise. These are incorporated in the DG method in an energy-preserving manner. Numerical experiments validate energy-preservation up to machine precision and show the high order of accuracy. Furthermore, the DG method is compared to quasi-Monte Carlo ray tracing for two examples showing that the DG method yields better accuracy in the same amount of computational time. • Explicit Taylor expansion on moving mesh for discontinuous Galerkin. • Non-local boundary conditions on phase space. • High order energy-preserving method for geometrical optics. • Outperforms quasi-Monte Carlo ray tracing in a few examples. [ABSTRACT FROM AUTHOR]

Fluid dynamical simulations are often performed using cheap macroscopic models like the Euler equations. For rarefied gases under near-equilibrium conditions, however, macroscopic models are not sufficiently accurate and a simulation using more accurate microscopic models is often expensive. In this paper, we introduce a hierarchical micro-macro acceleration based on moment models that combines the speed of macroscopic models and the accuracy of microscopic models. The hierarchical micro-macro acceleration is based on a flexible four step procedure including a micro step, restriction step, macro step, and matching step. We derive several new micro-macro methods from that and compare to existing methods. In 1D and 2D test cases, the new methods achieve high accuracy and a large speedup. [ABSTRACT FROM AUTHOR]

This paper presents a new cell-centered numerical method for ideal magnetohydrodynamics (MHD) that can be used in Lagrangian or Eulerian discretization. A two-dimensional Riemann solver based on the HLLC-type MHD method is established, which can be viewed as an extension from the HLLC-2D in hydrodynamics. The main feature of the algorithm is to introduce a nodal contact velocity and ensure the compatibility between edge fluxes and the nodal flow intrinsically. It transforms naturally from Lagrangian setting to the Eulerian setting in terms of grid nodal velocity, and gains benefits of the Lagrangian nature of the scheme. In the Lagrangian approach, the finite volume scheme itself can keep the magnetic field divergence-free strictly, while in the Eulerian case, a special constrained transport (CT) algorithm is constructed from the discontinuous fluxes on cell interfaces to ensure solenoidal nature again. Numerical tests are presented to demonstrate the performance of this new solver and compare the difference between the Lagrangian and Eulerian methods. • A genuine 2D Riemann solver for MHD is developed. • The divergence-free constraint is naturally satisfied in the Lagrangian setting. • In the Eulerian case, a constrained transport method is constructed. [ABSTRACT FROM AUTHOR]

The paper is concerned with the numerical solution of the linear steady-state regularized 13-moment equations in two space dimensions. To facilitate the understanding of the specific challenges, the equations are first divided into two subsystems before the full system is approached. The arising problems mainly stem from the complicated saddle-point structure as well as the non-standard nature of the boundary conditions. A continuous interior penalty method is presented and the pronounced advantages of utilizing high order basis functions in this setting are illustrated. To render the presented approach more efficient, a hybridization technique is presented that originates in the discontinuous Galerkin method. [ABSTRACT FROM AUTHOR]

• Uncertainty quantification for the Boltzmann equation. • Variance reduction of standard Monte Carlo sampling methods. • Control variate methods based on reduced models obtained in a multi-scale setting. • Asymptotic behavior of the control variate methods and fluid-dynamic limit. • Error estimates show a strong acceleration of the convergence with respect to standard Monte Carlo. Kinetic equations play a major rule in modeling large systems of interacting particles. Uncertainties may be due to various reasons, like lack of knowledge on the microscopic interaction details or incomplete informations at the boundaries. These uncertainties, however, contribute to the curse of dimensionality and the development of efficient numerical methods is a challenge. In this paper we consider the construction of novel multi-scale methods for such problems which, thanks to a control variate approach, are capable to reduce the variance of standard Monte Carlo techniques. [ABSTRACT FROM AUTHOR]

Abstract Acoustical shock waves can be generated by numerous atmospheric sources, either natural – like thunder and volcanoes – or anthropic – like explosions, sonic boom or buzz saw noise. The prediction of their long-range propagation remains a numerical challenge at 3D because of the large propagation distance to wavelength ratio, and of the high frequency / small wavelength content associated to shocks. In this paper, an original numerical method for propagating acoustical shock waves in three-dimensional heterogeneous media is proposed. Heterogeneities can result from temperature or density gradients and also from atmospheric shear and turbulent flows. The method called FLHOWARD (for FLow and Heterogeneities in a One-Way Approximation of the nonlineaR wave equation in 3D) is based on a one-way solution of a generalized nonlinear wave equation. Even though backscattering is neglected, it does not suffer from the limitations of classical ray theory nor from the angular limitations of the popular parabolic methods. The numerical approach is based on a split-step method, which has the advantage of splitting the original equation into simpler ones associated with specific physical mechanisms: diffraction, flows, heterogeneities, nonlinearities, absorption and relaxation. The method has been developed on parallel architecture for very high demanding 3D configurations using the Single Method Multiple Data paradigm. The method is validated through several test cases. A study of the lateral cut-off of the sonic boom finally illustrates the potentialities of the method for realistic cases. [ABSTRACT FROM AUTHOR]

Abstract The construction of numerical schemes for the Kuramoto model is challenging due to the structural properties of the system which are essential in order to capture the correct physical behavior, like the description of stationary states and phase transitions. Additional difficulties are represented by the high dimensionality of the problem in presence of multiple frequencies. In this paper, we develop numerical methods which are capable to preserve these structural properties of the Kuramoto equation in the presence of diffusion and to solve efficiently the multiple frequencies case. The novel schemes are then used to numerically investigate the phase transitions in the case of identical and nonidentical oscillators. Highlights • We develop deterministic schemes able to capture accurately both continuous and discontinuous phase transitions. • Structure preserving methods allow to compute stationary states of Kuramoto oscillators with general frequencies. • A comparison to spectral methods for the stationary kinetic Kuramoto model and MC approaches is made. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the normalized gradient flow method to compute the ground states of Bose–Einstein condensates (BEC) with higher order interactions (HOI), which is modeled via the modified Gross–Pitaevskii equation (MGPE). Schemes constructed in naive ways suffer from severe stability problems due to the high restrictions on time steps. To build an efficient and stable scheme, we split the HOI term into two parts with each part treated separately. The part corresponding to a repulsive/positive energy is treated semi-implicitly while the one corresponding to an attractive/negative energy is treated fully explicitly. Based on the splitting, we construct the BEFD-splitting and BESP-splitting schemes. A variety of numerical experiments show that the splitting will improve the stability of the schemes significantly. Besides, we will show that the methods can be applied to multidimensional problems and to the computation of the first excited state as well. [ABSTRACT FROM AUTHOR]

This paper introduces two families of A-stable algorithms for the integration of y ′ = F ( y , t ) y : the extended predictor–corrector (EPC) and the exponential–linear (EL) methods. The structure of the algorithm families are described, and the method of derivation of the coefficients presented. The new algorithms are then tested on a simple deterministic problem and a Monte Carlo isotopic evolution problem. The EPC family is shown to be only second order for systems of ODEs. However, the EPC-RK45 algorithm had the highest accuracy on the Monte Carlo test, requiring at least a factor of 2 fewer function evaluations to achieve a given accuracy than a second order predictor–corrector method (center extrapolation / center midpoint method) with regards to Gd-157 concentration. Members of the EL family can be derived to at least fourth order. The EL3 and the EL4 algorithms presented are shown to be third and fourth order respectively on the systems of ODE test. In the Monte Carlo test, these methods did not overtake the accuracy of EPC methods before statistical uncertainty dominated the error. The statistical properties of the algorithms were also analyzed during the Monte Carlo problem. The new methods are shown to yield smaller standard deviations on final quantities as compared to the reference predictor–corrector method, by up to a factor of 1.4. [ABSTRACT FROM AUTHOR]

In this paper we study two multidimensional nonlinear dispersive systems: the Serre-Green-Naghdi (SGN) equations describing dispersive shallow water flows, and the Iordanskii-Kogarko-Wijngaarden (IKW) equations describing fluids containing small compressible gas bubbles. These models are Euler-Lagrange equations for a given Lagrangian and share common mathematical structure, namely the dependence of the pressure on material derivatives of macroscopic variables. We develop a generic dispersive model such that SGN and IKW systems become its special cases if only one specifies the appropriate Lagrangian, and then use the extended Lagragian approach proposed in Favrie and Gavrilyuk (2017) to build its hyperbolic approximation. The new approximate model is unconditionally hyperbolic for both SGN and IKW cases, and accurately describes dispersive phenomena, which allows to impose discontinuous initial data and study dispersive shock waves. We consider the 2-D hyperbolic version of SGN system as an example for numerical simulations and apply a second order implicit-explicit scheme in order to numerically integrate the system. The obtained 1-D and 2-D results are in close agreement with available exact solutions and numerical tests. • Hyperbolization of dispersive systems is efficient for numerical simulations. • HLLC Riemann solver is accurate for the simulations of dispersive shock waves. • The implicit-explicit method requires little mesh points to get the good precision. [ABSTRACT FROM AUTHOR]

Since proposed in Zhang and Shu (2010) [1] , the Zhang–Shu framework has attracted extensive attention and motivated many bound-preserving (BP) high-order discontinuous Galerkin and finite volume schemes for various hyperbolic equations. A key ingredient in the framework is the decomposition of the cell averages of the numerical solution into a convex combination of the solution values at certain quadrature points, which helps to rewrite high-order schemes as convex combinations of formally first-order schemes. The classic convex decomposition originally proposed by Zhang and Shu has been widely used over the past decade. It was verified, only for the 1D quadratic and cubic polynomial spaces, that the classic decomposition is optimal in the sense of achieving the mildest BP CFL condition. Yet, it remained unclear whether the classic decomposition is optimal in multiple dimensions. In this paper, we find that the classic multidimensional decomposition based on the tensor product of Gauss–Lobatto and Gauss quadratures is generally not optimal, and we discover a novel alternative decomposition for the 2D and 3D polynomial spaces of total degree up to 2 and 3, respectively, on Cartesian meshes. Our new decomposition allows a larger BP time step-size than the classic one, and moreover, it is rigorously proved to be optimal to attain the mildest BP CFL condition, yet requires much fewer nodes. The discovery of such an optimal convex decomposition is highly nontrivial yet meaningful, as it may lead to an improvement of high-order BP schemes for a large class of hyperbolic or convection-dominated equations, at the cost of only a slight and local modification to the implementation code. Several numerical examples are provided to further validate the advantages of using our optimal decomposition over the classic one in terms of efficiency. • Make the first effort on questing the optimal convex decomposition for bound-preserving (BP) multidimensional schemes. • Find the classic convex decomposition widely used in the past decade not optimal for multi-dimensional Pk spaces. • Discover the optimal decomposition for P2 and P3, which attains the mildest BP CFL condition yet requires much fewer nodes. • Improve high-order BP schemes for a large class of hyperbolic PDEs with only a slight local modification to the code. • Demonstrate by numerical tests the advantages of the optimal decomposition over the classic one in terms of efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new finite element approach to simulate the time-dependent Ginzburg-Landau equations under the temporal gauge, and design an efficient preconditioner for the Newton iteration of the resulting discrete system. The new approach solves the magnetic potential in H (curl) space by the lowest order of the second kind Nédélec element. This approach offers a simple way to deal with the boundary condition, and leads to a stable and reliable performance when dealing with the superconductor with reentrant corners. The comparison in numerical simulations verifies the efficiency of the proposed preconditioner, which can significantly speed up the simulation in large-scale computations. • Energy stability is analyzed for the new approach under the temporal gauge. • The boundedness and coercivity are analyzed to motivate the design of preconditioner. • The preconditioner is efficient and can significantly speed up large-scale simulations. • Extensive numerical experiments are provided. [ABSTRACT FROM AUTHOR]

In this paper we present a novel particle method for the Vlasov–Poisson equation. Unlike in conventional particle methods, the particles are not interpreted as point charges, but as point values of the distribution function. In between the particles, the distribution function is reconstructed using mesh-free interpolation. Our numerical experiments confirm that this approach results in significantly increased accuracy and noise reduction. At the same time, many benefits of the conventional schemes are preserved. • Long-term accurate, high-order particle method for the Vlasov–Poisson equation. • Stable approach without remeshing. • RKHS-based, grid-free interpolation scheme used. • Method is auto-adaptive. • Domain-decomposition can be used for speed-up and parallelisation. [ABSTRACT FROM AUTHOR]

This paper develops a physics-informed neural network (PINN) combined with a Schwarz waveform relaxation (SWR) method for solving local and nonlocal advection-diffusion-reaction equations. Specifically, we derive the algorithm by constructing subdomain-dependent local solutions by minimizing local loss functions, allowing the decomposition of the training process in different domains in an embarrassingly parallel procedure. Provided the convergence of PINN, the overall proposed algorithm is convergent. By constructing local solutions, one can, in particular, adapt the depth of the deep neural networks, depending on the solution's spectral space and time complexity in each subdomain. One of the main advantages of using NN compared to standard solvers, is that the PINN algorithm introduces some learning in the SWR algorithm allowing for an acceleration of the overall algorithm, especially close to SWR convergence. We present some numerical experiments based on classical and Robin-SWR algorithms to illustrate the performance and comment on the convergence of the proposed method. • Schwarz Waveform Relaxation algorithms combined with Physics-Informed Neural Networks. • Mathematical justification of the proposed algorithms. • Analysis of complexity and gain compared to standard approach. • Numerical experiments. [ABSTRACT FROM AUTHOR]

Based on the numerical method proposed in Hu et al. (2018) [22] for Kohn-Sham equation, further improvement on the efficiency is obtained in this paper by i). designing a numerical method with the strategy of separately handling the nonlinear Hartree potential and exchange-correlation potential, and ii). parallelizing the algorithm in an eigenpairwise approach. The feasibility of two approaches is analyzed in detail, and the new algorithm is described completely. Compared with previous results, a significant improvement of numerical efficiency can be observed from plenty of numerical experiments, which make the new method more suitable for the practical problems. • An accelerating multilevel correction AFEM is designed to solve Kohn-Sham equation. • Solving large-scale nonlinear eigenvalue problem is avoided. • A small-scale Kohn-Sham equation and large-scale boundary value problem are required. • Hartree potential and exchange-correlation potential are handling in a nested scheme. • The novel algorithm can be performed in an eigenpairwise approach. [ABSTRACT FROM AUTHOR]

• Asymptotic expansion regularization for inverse problems in PDEs. • First combination of asymptotic analysis and regularization theory. • A posteriori error estimation of the method under a priori information of solution. • Existence and uniqueness of a continuous solution with an internal transition layer. In this paper, by employing the asymptotic expansion method, we prove the existence and uniqueness of a smoothing solution for a time-dependent nonlinear singularly perturbed partial differential equation (PDE) with a small-scale parameter. As a by-product, we obtain an approximate smooth solution, constructed from a sequence of reduced stationary PDEs with vanished high-order derivative terms. We prove that the accuracy of the constructed approximate solution can be in any order of this small-scale parameter in the whole domain, except a negligible transition layer. Furthermore, based on a simpler link equation between this approximate solution and the source function, we propose an efficient algorithm, called the asymptotic expansion regularization (AER), for solving nonlinear inverse source problems governed by the original PDE. The convergence-rate results of AER are proven, and the a posteriori error estimation of AER is also studied under some a priori assumptions of source functions. Various numerical examples are provided to demonstrate the efficiency of our new approach. [ABSTRACT FROM AUTHOR]

• Analysis of multiscale methods for quasi-gas dynamics. • The multiscale setup of quasi-gas dynamics. • Constraint energy minimizing basis construction for quasi-gas dynamics. In this paper, we consider the quasi-gas-dynamic (QGD) model in a multiscale environment. The model equations can be regarded as a hyperbolic regularization and are derived from kinetic equations. So far, the research on QGD models has been focused on problems with constant coefficients. In this paper, we investigate the QGD model in multiscale media, which can be used in porous media applications. This multiscale problem is interesting from a multiscale methodology point of view as the model problem has a hyperbolic multiscale term, and designing multiscale methods for hyperbolic equations is challenging. In the paper, we apply the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) combined with the central difference scheme in time to solve this problem. The CEM-GMsFEM provides a flexible and systematical framework to construct crucial multiscale basis functions for approximating the solution to the problem with reduced computational cost. With this approach of spatial discretization, we establish the stability of the fully discretized scheme, based on the coarse grid, under a coarse-scale CFL condition. Complete convergence analysis of the proposed method is presented. Numerical results are provided to illustrate and verify the theoretical findings. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a High-Order/Low-Order radiation-hydrodynamics method that is second-order accurate in both space and time and uses the Variable Eddington Factor (VEF) method to couple a high-order set of 1-D slab-geometry grey S n radiation transport equations with a low-order set of radiation moment and hydrodynamics equations. The S n equations are spatially discretized with a lumped linear-discontinuous Galerkin scheme, while the low-order radiation-hydrodynamics equations are spatially discretized with a constant-linear mixed finite-element scheme. Both the high-order and low-order equations are discretized in time using a trapezoidal BDF-2 method. One manufactured solution is used to demonstrate that the scheme is second-order accurate for smooth solutions, and another one is used to demonstrate that the scheme is asymptotic-preserving in the equilibrium-diffusion limit. Calculations are performed for radiative shock problems and compared with semi-analytic solutions. In a previous paper it was shown that the pure radiative transfer scheme (the S n equations coupled to the radiation moment equations and a material temperature equation rather than the hydrodynamics equations) is asymptotic-preserving in the equilibrium-diffusion limit, is well-behaved with unresolved spatial boundary layers in that limit, and yields accurate Marshak wave speeds even with strongly temperature-dependent opacities and relatively coarse meshes. These same properties carry over to our radiation-hydrodynamics scheme. • We have developed a High-Order/Low-Order (HOLO) radiation-hydrodynamics scheme, based upon the Variable Eddington Factor (VEF) method. • We have computationally demonstrated that this scheme is second order accurate in time and space for smooth solutions. • We have computationally demonstrated that this scheme preserves the asymptotic thick diffusion limit. • The method produces radiative shock solutions in excellent agreement with semi-analytical solutions. [ABSTRACT FROM AUTHOR]

Abstract: A novel implicit second-order accurate immersed boundary method (IBM) for simulating the flow around arbitrary stationary bodies is developed, implemented and validated in this paper. The IBM is used to efficiently take into account the existence of bodies within the fluid domain. The flow domain consist of simple Cartesian cells whereas the body can be arbitrary. At the triangulated interface of the body and the fluid, the immersed boundary, the coefficients obtained from discretizing the Navier–Stokes equations are closed with a second-order accurate interpolation arising from the immersed boundary condition employed at the interface. Two different conditions are developed in this paper and it is shown that for the mirroring method the resulting coefficients lead to a well-posed and diagonally dominant system which can be efficiently solved with a preconditioned Krylov sub-space solver. The immersed boundary condition generates a fictitious reversed velocity field inside the immersed boundary, which is excluded from the continuity equation to account for the presence of the IB in the pressure correction equation, resulting in no mass flux over the IB. The force acting on the object from the fluid is determined by integrating the pressure and the viscous forces over the object. The method is validated by simulating the flow around a sphere for a range of Re numbers. It is shown that the drag is very well in agreement with experimental data. Accuracy and convergence studies are employed, proving the second-order accuracy of the method and showing the superiority in convergence rate compared to other IBM. Finally the drag force of a cluster of non-spherical particles is employed to show the generality and potential of the method. [Copyright &y& Elsevier]

Abstract: The scope of the present study is Eulerian modeling and simulation of polydisperse liquid sprays undergoing droplet coalescence and evaporation. The fundamental mathematical description is the Williams spray equation governing the joint number density function of droplet volume and velocity. Eulerian multi-fluid models have already been rigorously derived from this equation in Laurent et al. [F. Laurent, M. Massot, P. Villedieu, Eulerian multi-fluid modeling for the numerical simulation of coalescence in polydisperse dense liquid sprays, J. Comput. Phys. 194 (2004) 505–543]. The first key feature of the paper is the application of direct quadrature method of moments (DQMOM) introduced by Marchisio and Fox [D.L. Marchisio, R.O. Fox, Solution of population balance equations using the direct quadrature method of moments, J. Aerosol Sci. 36 (2005) 43–73] to the Williams spray equation. Both the multi-fluid method and DQMOM yield systems of Eulerian conservation equations with complicated interaction terms representing coalescence. In order to focus on the difficulties associated with treating size-dependent coalescence and to avoid numerical uncertainty issues associated with two-way coupling, only one-way coupling between the droplets and a given gas velocity field is considered. In order to validate and compare these approaches, the chosen configuration is a self-similar 2D axisymmetrical decelerating nozzle with sprays having various size distributions, ranging from smooth ones up to Dirac delta functions. The second key feature of the paper is a thorough comparison of the two approaches for various test-cases to a reference solution obtained through a classical stochastic Lagrangian solver. Both Eulerian models prove to describe adequately spray coalescence and yield a very interesting alternative to the Lagrangian solver. The third key point of the study is a detailed description of the limitations associated with each method, thus giving criteria for their use as well as for their respective efficiency. [Copyright &y& Elsevier]

Abstract: This paper extends the gas-kinetic scheme for one-dimensional inviscid shallow water equations (K. Xu, A well-balanced gas-kinetic scheme for the shallow-water equations with source terms, J. Comput. Phys. 178 (2002) 533–562) to multidimensional gas dynamic equations under gravitational fields. Four important issues in the construction of a well-balanced scheme for gas dynamic equations are addressed. First, the inclusion of the gravitational source term into the flux function is necessary. Second, to achieve second-order accuracy of a well-balanced scheme, the Chapman–Enskog expansion of the Boltzmann equation with the inclusion of the external force term is used. Third, to avoid artificial heating in an isolated system under a gravitational field, the source term treatment inside each cell has to be evaluated consistently with the flux evaluation at the cell interface. Fourth, the multidimensional approach with the inclusion of tangential gradients in two-dimensional and three-dimensional cases becomes important in order to maintain the accuracy of the scheme. Many numerical examples are used to validate the above issues, which include the comparison between the solutions from the current scheme and the Strang splitting method. The methodology developed in this paper can also be applied to other systems, such as semi-conductor device simulations under electric fields. [Copyright &y& Elsevier]

Abstract: In this paper we extend the special relativistic hydrodynamic (SRHD) equations [L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon, New York, 1987] and as a limiting case the ultra-relativistic hydrodynamic equations [M. Kunik, S. Qamar, G. Warnecke, J. Comput. Phys. 187 (2003) 572–596] to the special relativistic magnetohydrodynamics (SRMHD). We derive a flux splitting method based on gas-kinetic theory in order to solve these equations in one space dimension. The scheme is based on the direct splitting of macroscopic flux functions with consideration of particle transport. At the same time, particle “collisions” are implemented in the free transport process to reduce numerical dissipation. To achieve high-order accuracy we use a MUSCL-type initial reconstruction and Runge–Kutta time stepping method. For the direct comparison of the numerical results, we also solve the SRMHD equations with the well-developed second-order central schemes. The 1D computations reported in this paper have comparable accuracy to the already published results. The results verify the desired accuracy, high resolution, and robustness of the kinetic flux splitting method and central schemes. [Copyright &y& Elsevier]

Abstract: In this paper numerical results are reviewed [D.L. Ropp, J.N. Shadid, C.C. Ober, Studies of the accuracy of time integration methods for reaction-diffusion equations, J. Comput. Phys. 194 (2) (2004) 544–574] that demonstrate that common second-order operator-splitting methods can exhibit instabilities when integrating the Brusselator equations out to moderate times of about seven periods of oscillation. These instabilities are manifested as high wave number spatial errors. In this paper, we further analyze this problem, and present a theorem for stability of operator-splitting methods applied to linear reaction-diffusion equations with indefinite reaction terms which controls both low and high wave number instabilities. A corollary shows that if L-stable methods are used for the diffusion term the high wave number instability will be controlled more easily. In the absence of L-stability, an additional time step condition that suppresses the high wave number modes appears to guarantee convergence at the asymptotic order for the operator-splitting method. Numerical results for a model problem confirm this theory, and results for the Brusselator problem agree as well. [Copyright &y& Elsevier]

Abstract: Our purpose is to derive a model for charged particles which combines a kinetic description of the fast particles with a fluid description of the slow ones. In a previous paper, a similar model was derived from a kinetic BGK equation that uses a constant relaxation time and does not include the effect of an electric field. In this paper, we consider a more general kinetic equation including an electric field and a varying relaxation time. Fast particles will be described through a collisional kinetic equation of Vlasov–BGK type while thermal particles will be modeled by a hydrodynamic model. Then, we construct a numerical scheme for this model and validate the approach by presenting various numerical tests. [Copyright &y& Elsevier]

Compact finite-difference schemes have been recently used in several Direct Numerical Simulations of turbulent flows, since they can achieve high-order accuracy and high resolution without exceedingly increasing the size of the computational stencil. The development of compact finite-volume schemes is more involved, due to the appearance of surface and volume integrals. While Pereira et al. [J. Comput. Phys. 167 (2001)] and Smirnov et al. [AIAA Paper, 2546, 2001] focused on collocated grids, in this paper we use the staggered grid arrangement. Compact schemes can be tuned to achieve very high resolution for a given formal order of accuracy. We develop and test high-resolution schemes by following a procedure proposed by Lele [J. Comput. Phys. 103 (1992)] which, to the best of our knowledge, has not yet been applied to compact finite-volume methods on staggered grids. Results from several one- and two-dimensional simulations for the scalar transport and Navier–Stokes equations are presented, showing that the proposed method is capable to accurately reproduce complex steady and unsteady flows. [Copyright &y& Elsevier]

In the past various arguments have been used for applying kinetic energy-conserving advection schemes in numerical simulations of incompressible fluid flows. One argument is obeying the programmed dissipation by viscous stresses or by sub-grid stresses in Direct Numerical Simulation and Large Eddy Simulation, see e.g. [Phys. Fluids A 3 (7) (1991) 1766]. Another argument is that, according to e.g. [J. Comput. Phys. 6 (1970) 392; 1 (1966) 119], energy-conserving convection schemes are more stable i.e. by prohibiting a spurious blow-up of volume-integrated energy in a closed volume without external energy sources. In the above-mentioned references it is stated that nonlinear instability is due to spatial truncation rather than to time truncation and therefore these papers are mainly concerned with the spatial integration. In this paper we demonstrate that discretized temporal integration of a spatially variance-conserving convection scheme can induce non-energy conserving solutions. In this paper the conservation of the variance of a scalar property is taken as a simple model for the conservation of kinetic energy. In addition, the derivation and testing of a variance-conserving scheme allows for a clear definition of kinetic energy-conserving advection schemes for solving the Navier–Stokes equations. Consequently, we first derive and test a strictly variance-conserving space–time discretization for the convection term in the convection–diffusion equation. Our starting point is the variance-conserving spatial discretization of the convection operator presented by Piacsek and Williams [J. Comput. Phys. 6 (1970) 392]. In terms of its conservation properties, our variance-conserving scheme is compared to other spatially variance-conserving schemes as well as with the non-variance-conserving schemes applied in our shallow-water solver, see e.g. [Direct and Large-eddy Simulation Workshop IV, ERCOFTAC Series, Kluwer Academic Publishers, 2001, pp. 409–287]. [Copyright &y& Elsevier]

In 1994, Bérenger [Journal of Computational Physics 114 (1994) 185] proposed a new layer method: perfectly matched layer, PML, for electromagnetism. This new method is based on the truncation of the computational domain by a layer which absorbs waves regardless of their frequency and angle of incidence. Unfortunately, the technique proposed by Bérenger (loc. cit.) leads to a system which has lost the most important properties of the original one: strong hyperbolicity and symmetry. We present in this paper an algebraic technique leading to well-known PML model [IEEE Transactions on Antennas and Propagation 44 (1996) 1630] for the linearized Euler equations, strongly well-posed, preserving the advantages of the initial method, and retaining symmetry. The technique proposed in this paper can be extended to various hyperbolic problems. [Copyright &y& Elsevier]

Models for reacting flow are typically based on advection–diffusion–reaction (A–D–R) partial differential equations. Many practical cases correspond to situations where the relevant time scales associated with each of the three sub-processes can be widely different, leading to disparate time-step requirements for robust and accurate time-integration. In particular, interesting regimes in combustion correspond to systems in which diffusion and reaction are much faster processes than advection. The numerical strategy introduced in this paper is a general procedure to account for this time-scale disparity. The proposed methods are high-order multi-implicit generalizations of spectral deferred correction methods (MISDC methods), constructed for the temporal integration of A–D–R equations. Spectral deferred correction methods compute a high-order approximation to the solution of a differential equation by using a simple, low-order numerical method to solve a series of correction equations, each of which increases the order of accuracy of the approximation. The key feature of MISDC methods is their flexibility in handling several sub-processes implicitly but independently, while avoiding the splitting errors present in traditional operator-splitting methods and also allowing for different time steps for each process. The stability, accuracy, and efficiency of MISDC methods are first analyzed using a linear model problem and the results are compared to semi-implicit spectral deferred correction methods. Furthermore, numerical tests on simplified reacting flows demonstrate the expected convergence rates for MISDC methods of orders three, four, and five. The gain in efficiency by independently controlling the sub-process time steps is illustrated for nonlinear problems, where reaction and diffusion are much stiffer than advection. Although the paper focuses on this specific time-scales ordering, the generalization to any ordering combination is straightforward. [Copyright &y& Elsevier]

• A new conservative reconstruction for high-order semi-Lagrangian schemes. • Non-oscillatory and/or positive properties. • A compact form of the reconstruction and its mathematical properties. • Asymptotic preserving semi-Lagrangian schemes in the relaxation limit. • Applications to the Xin-Jin model and the Broadwell model. In this paper, we propose and analyze a reconstruction technique which enables one to design high-order conservative semi-Lagrangian schemes for kinetic equations. The proposed reconstruction can be obtained by taking the sliding average of a given polynomial reconstruction of the numerical solution. A compact representation of the high order conservative reconstruction in one and two space dimension is provided, and its mathematical properties are analyzed. To demonstrate the performance of proposed technique, we consider implicit semi-Lagrangian schemes for kinetic-like equations such as the Xin-Jin model and the Broadwell model, and then solve related shock problems which arise in the relaxation limit. Applications to BGK and Vlasov-Poisson equations will be presented in the second part of the paper. [ABSTRACT FROM AUTHOR]