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In this paper, we propose a local discontinuous Galerkin (LDG) method for nonlinear and possibly degenerate parabolic stochastic partial differential equations, which is a high-order numerical scheme. It extends the discontinuous Galerkin (DG) method for purely hyperbolic equations to parabolic equations and shares with the DG method its advantage and flexibility. We prove theL2-stability of the numerical scheme for fully nonlinear equations. Optimal error estimates (O(h(k+1))) for smooth solutions of semi-linear stochastic equations is shown if polynomials of degreekare used. We use an explicit derivative-free order 1.5 time discretization scheme to solve the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples are given to display the performance of the LDG method.

When interpolating data with certain regularity, spline functions are useful. They are defined as piecewise polynomials that satisfy certain regularity conditions at the joints. In the literature about splines it is possible to find several references that study the apparition of Gibbs phenomenon close to jump discontinuities in the results obtained by spline interpolation. This work is devoted to the construction and analysis of a new nonlinear technique that allows to improve the accuracy of splines near jump discontinuities eliminating the Gibbs phenomenon. The adaption is easily attained through a nonlinear modification of the right hand side of the system of equations of the spline, that contains divided differences. The modification is based on the use of a new limiter specifically designed to attain adaption close to jumps in the function. The new limiter can be seen as a nonlinear weighted mean that has better adaption properties than the linear weighted mean. We will prove that the nonlinear modification introduced in the spline keeps the maximum theoretical accuracy in all the domain except at the intervals that contain a jump discontinuity, where Gibbs oscillations are eliminated. Diffusion is introduced, but this is fine if the discontinuity appears due to a discretization of a high gradient with not enough accuracy. The new technique is introduced for cubic splines, but the theory presented allows to generalize the results very easily to splines of any order. The experiments presented satisfy the theoretical aspects analyzed in the paper. We would like to thank the anonymous referees for their valuable comments, which have helped to significantly improve this work. This work was funded by project 20928/PI/18 (Proyecto financiado por la Comunidad Autónoma de la Región de Murcia a través de la convocatoria de Ayudas a proyectos para el desarrollo de investigación científica y técnica por grupos competitivos, incluida en el Programa Regional de Fomento de la Investigación Científica y Técnica (Plan de Actuación 2018) de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia), by the national research project MTM2015- 64382-P (MINECO/FEDER) and by NSF grant DMS-1719410.

In this paper, we study the central discontinuous Galerkin (DG) method on overlapping meshes for second order wave equations. We consider the first order hyperbolic system, which is equivalent to the second order scalar equation, and construct the corresponding central DG scheme. We then provide the stability analysis and the optimal error estimates for the proposed central DG scheme for one- and multi-dimensional cases with piecewise Pk elements. The optimal error estimates are valid for uniform Cartesian meshes and polynomials of arbitrary degree k ≥ 0. In particular, we adopt the techniques in Liu et al. (SIAM J. Numer. Anal. 56 (2018) 520–541; ESAIM: M2AN 54 (2020) 705–726) and obtain the local projection that is crucial in deriving the optimal order of convergence. The construction of the projection here is more challenging since the unknowns are highly coupled in the proposed scheme. Dispersion analysis is performed on the proposed scheme for one dimensional problems, indicating that the numerical solution with P1 elements reaches its minimum with a suitable parameter in the dissipation term. Several numerical examples including accuracy tests and long time simulation are presented to validate the theoretical results.

In this work, we propose a novel framework for the numerical solution of time-dependent conservation laws with implicit schemes via primal-dual hybrid gradient methods. We solve an initial value problem (IVP) for the partial differential equation (PDE) by casting it as a saddle point of a min-max problem and using iterative optimization methods to find the saddle point. Our approach is flexible with the choice of both time and spatial discretization schemes. It benefits from the implicit structure and gains large regions of stability, and overcomes the restriction on the mesh size in time by explicit schemes from Courant--Friedrichs--Lewy (CFL) conditions (really via von Neumann stability analysis). Nevertheless, it is highly parallelizable and easy-to-implement. In particular, no nonlinear inversions are required! Specifically, we illustrate our approach using the finite difference scheme and discontinuous Galerkin method for the spatial scheme; backward Euler and backward differentiation formulas for implicit discretization in time. Numerical experiments illustrate the effectiveness and robustness of the approach. In future work, we will demonstrate that our idea of replacing an initial-value evolution equation with this primal-dual hybrid gradient approach has great advantages in many other situations.

In this paper, we study superconvergence properties of the ultraweak-local discontinuous Galerkin (UWLDG) method in Tao et al. [To appear in Math. Comput. DOI: https://doi.org/10.1090/mcom/3562 (2020).] for an one-dimensional linear fourth-order equation. With special initial discretizations, we prove the numerical solution of the semi-discrete UWLDG scheme superconverges to a special projection of the exact solution. The order of this superconvergence is proved to be k + min(3, k) when piecewise ℙk polynomials with k ≥ 2 are used. We also prove a 2k-th order superconvergence rate for the cell averages and for the function values and derivatives of the UWLDG approximation at cell boundaries. Moreover, we prove superconvergence of (k + 2)-th and (k + 1)-th order of the function values and the first order derivatives of the UWLDG solution at a class of special quadrature points, respectively. Our proof is valid for arbitrary non-uniform regular meshes and for arbitrary k ≥ 2. Numerical experiments verify that all theoretical findings are sharp.

In this paper, we study the optimal error estimates of the classical discontinuous Galerkin method for time-dependent 2-D hyperbolic equations using Pk elements on uniform Cartesian meshes, and prove that the error in the L2 norm achieves optimal (k + 1)th order convergence when upwind fluxes are used. For the linear constant coefficient case, the results hold true for arbitrary piecewise polynomials of degree k ≥ 0. For variable coefficient and nonlinear cases, we give the proof for piecewise polynomials of degree k = 0, 1, 2, 3 and k = 2, 3, respectively, under the condition that the wind direction does not change. The theoretical results are verified by numerical examples.

One- and multi-dimensional stochastic Maxwell equations with additive noise are considered in this paper. It is known that such system can be written in the multi-symplectic structure, and the stochastic energy increases linearly in time. High order discontinuous Galerkin methods are designed for the stochastic Maxwell equations with additive noise, and we show that the proposed methods satisfy the discrete form of the stochastic energy linear growth property and preserve the multi-symplectic structure on the discrete level. Optimal error estimate of the semi-discrete DG method is also analyzed. The fully discrete methods are obtained by coupling with symplectic temporal discretizations. One- and two-dimensional numerical results are provided to demonstrate the performance of the proposed methods, and optimal error estimates and linear growth of the discrete energy can be observed for all cases.

In this paper, we continue our work in [46] and propose a new type of high-order finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes to solve hyperbolic conservation laws on triangular meshes. Although termed “multi-resolution WENO schemes”, we only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. We construct new third-order, fourth-order, and fifth-order WENO schemes using three or four unequal-sized central spatial stencils, different from the classical WENO procedure using equal-sized biased/central spatial stencils for the spatial reconstruction. The new WENO schemes could obtain the optimal order of accuracy in smooth regions, and could degrade gradually to first-order of accuracy so as to suppress spurious oscillations near strong discontinuities. This is the first time that only a series of unequal-sized hierarchical central spatial stencils are used in designing arbitrary high-order finite volume WENO schemes on triangular meshes. The main advantages of these schemes are their compactness, robustness, and their ability to maintain good convergence property for steady-state computation. The linear weights of such WENO schemes can be any positive numbers on the condition that they sum to one. Extensive numerical results are provided to illustrate the good performance of these new finite volume WENO schemes.

In this paper, we discuss the stability and error estimates of the fully discrete schemes for linear conservation laws, which consists of an arbitrary Lagrangian–Eulerian discontinuous Galerkin method in space and explicit total variation diminishing Runge–Kutta (TVD-RK) methods up to third order accuracy in time. The scaling arguments and the standard energy analysis are the key techniques used in our work. We present a rigorous proof to obtain stability for the three fully discrete schemes under suitable CFL conditions. With the help of the reference cell, the error equations are easy to establish and we derive the quasi-optimal error estimates in space and optimal convergence rates in time. For the Euler-forward scheme with piecewise constant elements, the second order TVD-RK method with piecewise linear elements and the third order TVD-RK scheme with polynomials of any order, the usual CFL condition is required, while for other cases, stronger time step restrictions are needed for the results to hold true. More precisely, the Euler-forward scheme needs τ ≤ ρh2 and the second order TVD-RK scheme needs $ \tau \le \rho {h}^{\frac{4}{3}}$ for higher order polynomials in space, where τ and h are the time and maximum space step, respectively, and ρ is a positive constant independent of τ and h.

In this paper, a new type of high-order finite difference and finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes is presented for solving hyperbolic conservation laws. We only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. These new WENO schemes use the same large stencils as the classical WENO schemes in [25] , [45] , could obtain the optimal order of accuracy in smooth regions, and could simultaneously suppress spurious oscillations near discontinuities. The linear weights of such WENO schemes can be any positive numbers on the condition that their sum equals one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite difference and finite volume WENO schemes. These new WENO schemes are simple to construct and can be easily implemented to arbitrary high order of accuracy and in higher dimensions. Benchmark examples are given to demonstrate the robustness and good performance of these new WENO schemes.

Air pollution has become a serious issue over the past few decades, and the transport sector is an important emission source. In this study, we model and simulate the dispersion of vehicle exhaust ...

Well balanced and free energy dissipative first- and second-order accurate finite volume schemes are proposed for a general class of hydrodynamic systems with linear and nonlinear damping. The natural Liapunov functional of the system, given by its free energy, allows for a characterization of the stationary states by its variation. An analog property at the discrete level enables us to preserve stationary states at machine precision while keeping the dissipation of the discrete free energy. These schemes allow for analysing accurately the stability properties of stationary states in challeging problems such as: phase transitions in collective behavior, generalized Euler-Poisson systems in chemotaxis and astrophysics, and models in dynamic density functional theories; having done a careful validation in a battery of relevant test cases., Videos from the simulations of this work are available at https://figshare.com/projects/Well-balanced_finite_volume_schemes_for_hydrodynamic_equations_with_general_free_energy/60122

International audience; Two new classes of compact weighted essentially nonoscillatory (WENO) polynomial limiters are presented for second-, third-, fourth-, and fifth-order discontinuous Galerkin (DG) schemes on irregular simplex elements. The presented WENO-DG procedures are extensions of the high-order WENO finite-volume and finite-difference schemes of Zhu and Shu (2017) [25], (2019) [26] to high-order unstructured DG schemes. A compact positivity preserving limiter is applied to the solutions to ensure pressure and density remain within physical ranges at all time. It is then verified that the bounded WENO-DG maintains the formal order of accuracy of the underlying DG schemes in the smooth regions. The performance of the proposed WENO-DG is also demonstrated with inviscid test cases including the classical Riemann problems, shock-turbulence interaction, scramjet, blunt body flows, and the double Mach Reflection problems.

So far the theory and numerical practice of stochastic partial differential equations (SPDEs) have dealt almost exclusively with Gaussian noise or Levy noise. Recently, Mikulevicius and Rozovskii (Stoch Partial Differ Equ Anal Comput 4:319–360, 2016) proposed a distribution-free Skorokhod–Malliavin calculus framework that is based on generalized stochastic polynomial chaos expansion, and is compatible with arbitrary driving noise. In this paper, we conduct systematic investigation on numerical results of these newly developed distribution-free SPDEs, exhibiting the efficiency of truncated polynomial chaos solutions in approximating moments and distributions. We obtain an estimate for the mean square truncation error in the linear case. The theoretical convergence rate, also verified by numerical experiments, is exponential with respect to polynomial order and cubic with respect to number of random variables included.

In this paper, we develop bound-preserving modified exponential Runge–Kutta (RK) discontinuous Galerkin (DG) schemes to solve scalar hyperbolic equations with stiff source terms by extending the idea in Zhang and Shu [43] . Exponential strong stability preserving (SSP) high order time discretizations are constructed and then modified to overcome the stiffness and preserve the bound of the numerical solutions. It is also straightforward to extend the method to two dimensions on rectangular and triangular meshes. Even though we only discuss the bound-preserving limiter for DG schemes, it can also be applied to high order finite volume schemes, such as weighted essentially non-oscillatory (WENO) finite volume schemes as well.

We present a discontinuous Galerkin (DG) scheme with suitable quadrature rules [15] for ideal compressible magnetohydrodynamic (MHD) equations on structural meshes. The semi-discrete scheme is analyzed to be entropy stable by using the symmetrizable version of the equations as introduced by Godunov [32] , the entropy stable DG framework with suitable quadrature rules [15] , the entropy conservative flux in [14] inside each cell and the entropy dissipative approximate Godunov type numerical flux at cell interfaces to make the scheme entropy stable. The main difficulty in the generalization of the results in [15] is the appearance of the non-conservative “source terms” added in the modified MHD model introduced by Godunov [32] , which do not exist in the general hyperbolic system studied in [15] . Special care must be taken to discretize these “source terms” adequately so that the resulting DG scheme satisfies entropy stability. Total variation diminishing / bounded (TVD/TVB) limiters and bound-preserving limiters are applied to control spurious oscillations. We demonstrate the accuracy and robustness of this new scheme on standard MHD examples.

In this paper, a new type of high-order finite volume and finite difference multi-resolution Hermite weighted essentially non-oscillatory (HWENO) schemes are designed for solving hyperbolic conservation laws on structured meshes. Here we only use the information defined on a hierarchy of nested central spatial stencils but do not introduce any equivalent multi-resolution representation, the terminology of multi-resolution HWENO follows that of the multi-resolution WENO schemes (Zhu and Shu, 2018) [29] . The main idea of our spatial reconstruction is derived from the original HWENO schemes (Qiu and Shu, 2004) [19] , in which both the function and its first-order derivative values are evolved in time and used in the reconstruction. Our HWENO schemes use the same large stencils as the classical HWENO schemes which are narrower than the stencils of the classical WENO schemes for the same order of accuracy. Only the function values need to be reconstructed by our HWENO schemes, the first-order derivative values are obtained from the high-order linear polynomials directly. Furthermore, the linear weights of such HWENO schemes can be any positive numbers as long as their sum equals one, and there is no need to do any modification or positivity-preserving flux limiting in our numerical experiments. Extensive benchmark examples are performed to illustrate the robustness and good performance of such finite volume and finite difference HWENO schemes.

In this paper, a high order finite difference conservative scheme is proposed to solve two-medium flows. Our scheme has four advantages: First, our scheme is conservative, which is important to ensure the numerical solution captures the main features properly. Second, our scheme directly applies the WENO interpolation method to the primitive variables so that it can maintain the equilibrium of velocity and pressure across the interface, which is very helpful to obtain a non-oscillatory solution. Third, the usage of nodal values enables us to manipulate algebraic functions easily. Fourth, the scheme can maintain high order accuracy when the solution is smooth. Extensive numerical experiments are performed to verify the high resolution and non-oscillatory performance of this new scheme.

In this paper, we study the superconvergence behavior of discontinuous Galerkin methods using upwind numerical fluxes for one-dimensional linear hyperbolic equations with degenerate variable coefficients. The study establishes superconvergence results for the flux function approximation as well as for the DG solution itself. To be more precise, we first prove that the DG flux function is superconvergent towards a particular flux function of the exact solution, with an order of O (h k +2 ), when piecewise polynomials of degree k are used. We then prove that the highest superconvergence rate of the DG solution itself is O (h k +3/2 ) as the variable coefficient degenerates or achieves the value zero in the domain. As byproducts, we obtain superconvergence properties for the DG solution and the DG flux function at special points and for cell averages. All theoretical findings are confirmed by numerical experiments.

A new class of high order weighted essentially non-oscillatory (WENO) schemes (Zhu and Qiu, 2016, [50] ) is applied to solve Euler equations with steady state solutions. It is known that the classical WENO schemes (Jiang and Shu, 1996, [23] ) might suffer from slight post-shock oscillations. Even though such post-shock oscillations are small enough in magnitude and do not visually affect the essentially non-oscillatory property, they are truly responsible for the residue to hang at a truncation error level instead of converging to machine zero. With the application of this new class of WENO schemes, such slight post-shock oscillations are essentially removed and the residue can settle down to machine zero in steady state simulations. This new class of WENO schemes uses a convex combination of a quartic polynomial with two linear polynomials on unequal size spatial stencils in one dimension and is extended to two dimensions in a dimension-by-dimension fashion. By doing so, such WENO schemes use the same information as the classical WENO schemes in Jiang and Shu (1996) [23] and yield the same formal order of accuracy in smooth regions, yet they could converge to steady state solutions with very tiny residue close to machine zero for our extensive list of test problems including shocks, contact discontinuities, rarefaction waves or their interactions, and with these complex waves passing through the boundaries of the computational domain.

We introduce a new troubled-cell indicator for the discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws. This indicator can be defined on unstructured meshes for high order DG methods and depends only on data from the target cell and its immediate neighbors. It is able to identify shocks without PDE sensitive parameters to tune. Extensive one- and two-dimensional simulations on the hyperbolic systems of Euler equations indicate the good performance of this new troubled-cell indicator coupled with a simple minmod-type TVD limiter for the Runge–Kutta DG (RKDG) methods.

The main purpose of this paper is to give stability analysis and error estimates of the local discontinuous Galerkin (LDG) methods coupled with three specific implicit-explicit (IMEX) Runge–Kutta time discretization methods up to third order accuracy, for solving one-dimensional time-dependent linear fourth order partial differential equations. In the time discretization, all the lower order derivative terms are treated explicitly and the fourth order derivative term is treated implicitly. By the aid of energy analysis, we show that the IMEX-LDG schemes are unconditionally energy stable, in the sense that the time step τ is only required to be upper-bounded by a constant which is independent of the mesh size h . The optimal error estimate is also derived by the aid of the elliptic projection and the adjoint argument. Numerical experiments are given to verify that the corresponding IMEX-LDG schemes can achieve optimal error accuracy.

It is well known that semi-discrete high order discontinuous Galerkin (DG) methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws (Jiang and Shu (1994) [39] ) and symmetric hyperbolic systems (Hou and Liu (2007) [36] ), in any space dimension and for any triangulations. However, this property holds only for the square entropy and the integrations in the DG methods must be exact. It is significantly more difficult to design DG methods to satisfy entropy inequalities for a non-square convex entropy, and/or when the integration is approximated by a numerical quadrature. In this paper, we develop a unified framework for designing high order DG methods which will satisfy entropy inequalities for any given single convex entropy, through suitable numerical quadrature which is specific to this given entropy. Our framework applies from one-dimensional scalar cases all the way to multi-dimensional systems of conservation laws. For the one-dimensional case, our numerical quadrature is based on the methodology established in Carpenter et al. (2014) [5] and Gassner (2013) [19] . The main ingredients are summation-by-parts (SBP) operators derived from Legendre Gauss–Lobatto quadrature, the entropy conservative flux within elements, and the entropy stable flux at element interfaces. We then generalize the scheme to two-dimensional triangular meshes by constructing SBP operators on triangles based on a special quadrature rule. A local discontinuous Galerkin (LDG) type treatment is also incorporated to achieve the generalization to convection–diffusion equations. Extensive numerical experiments are performed to validate the accuracy and shock capturing efficacy of these entropy stable DG methods.

In this paper, we analyze the Lax–Wendroff discontinuous Galerkin (LWDG) method for solving linear conservation laws. The method was originally proposed by Guo et al . in [W. Guo, J.-M. Qiu and J. Qiu, J. Sci. Comput. 65 (2015) 299–326], where they applied local discontinuous Galerkin (LDG) techniques to approximate high order spatial derivatives in the Lax–Wendroff time discretization. We show that, under the standard CFL condition τ ≤ λh (where τ and h are the time step and the maximum element length respectively and λ > 0 is a constant) and uniform or non-increasing time steps, the second order schemes with piecewise linear elements and the third order schemes with arbitrary piecewise polynomial elements are stable in the L 2 norm. The specific type of stability may differ with different choices of numerical fluxes. Our stability analysis includes multidimensional problems with divergence-free coefficients. Besides solving the equation itself, the LWDG method also gives approximations to its time derivative simultaneously. We obtain optimal error estimates for both the solution u and its first order time derivative u t in one dimension, and numerical examples are given to validate our analysis.

In this paper, a new type of finite difference Hermite weighted essentially non-oscillatory (HWENO) schemes are constructed for solving Hamilton–Jacobi (HJ) equations. Point values of both the solution and its first derivatives are used in the HWENO reconstruction and evolved via time advancing. While the evolution of the solution is still through the classical numerical fluxes to ensure convergence to weak solutions, the evolution of the first derivatives of the solution is through a simple dimension-by-dimension non-conservative procedure to gain efficiency. The main advantages of this new scheme include its compactness in the spatial field and its simplicity in the reconstructions. Extensive numerical experiments in one and two dimensional cases are performed to verify the accuracy, high resolution and efficiency of this new scheme.

Simulation of electromagnetic wave propagation in metamaterials leads to more complicated time domain Maxwells equations than the standard Maxwells equations in free space. In this paper, we develop and analyze a non-dissipative discontinuous Galerkin (DG) method for solving the Maxwells equations in Drude metamaterials. Previous discontinuous Galerkin methods in the literature for electromagnetic wave propagation in metamaterials were either non-dissipative but sub-optimal, or dissipative and optimal. Our method uses a different and simple choice of numerical fluxes, achieving provable non-dissipative stability and optimal error estimates simultaneously. We prove the stability and optimal error estimates for both semi- and fully discrete DG schemes, with the leap-frog time discretization for the fully discrete case. Numerical results are given to demonstrate that the DG method can solve metamaterial Maxwells equations effectively.

In this paper, we develop a second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin (DG) scheme for the Kerr–Debye model. By using the approach first introduced by Zhang and Shu in [Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin methods for scalar conservation laws, SIAM J. Numer. Anal. 42 (2004) 641–666.] with an energy estimate and Taylor expansion, the asymptotic-preserving property of the semi-discrete DG methods is proved rigorously. In addition, we propose a class of unconditional positivity-preserving implicit–explicit (IMEX) Runge–Kutta methods for the system of ordinary differential equations arising from the semi-discretization of the Kerr–Debye model. The new IMEX Runge–Kutta methods are based on the modification of the strong-stability-preserving (SSP) implicit Runge–Kutta method and have second-order accuracy. The numerical results validate our analysis.

A maximum-principle-satisfying space-time conservation element and solution element (CE/SE) scheme is constructed to solve a reduced five-equation model coupled with the stiffened equation of state for compressible multifluids. We first derive a sufficient condition for CE/SE schemes to satisfy maximum-principle when solving a general conservation law. And then we introduce a slope limiter to ensure the sufficient condition which is applicative for both central and upwind CE/SE schemes. Finally, we implement the upwind maximum-principle-satisfying CE/SE scheme to solve the volume-fraction-based five-equation model for compressible multifluids. Several numerical examples are carried out to carefully examine the accuracy, efficiency, conservativeness and maximum-principle-satisfying property of the proposed approach.

Finite difference WENO schemes have established themselves as very worthy performers for entire classes of applications that involve hyperbolic conservation laws. In this paper we report on two major advances that make finite difference WENO schemes more efficient.The first advance consists of realizing that WENO schemes require us to carry out stencil operations very efficiently. In this paper we show that the reconstructed polynomials for any one-dimensional stencil can be expressed most efficiently and economically in Legendre polynomials. By using Legendre basis, we show that the reconstruction polynomials and their corresponding smoothness indicators can be written very compactly. The smoothness indicators are written as a sum of perfect squares. Since this is a computationally expensive step, the efficiency of finite difference WENO schemes is enhanced by the innovation which is reported here.The second advance consists of realizing that one can make a non-linear hybridization between a large, centered, very high accuracy stencil and a lower order WENO scheme that is nevertheless very stable and capable of capturing physically meaningful extrema. This yields a class of adaptive order WENO schemes, which we call WENO-AO (for adaptive order). Thus we arrive at a WENO-AO(5,3) scheme that is at best fifth order accurate by virtue of its centered stencil with five zones and at worst third order accurate by virtue of being non-linearly hybridized with an r=3 CWENO scheme. The process can be extended to arrive at a WENO-AO(7,3) scheme that is at best seventh order accurate by virtue of its centered stencil with seven zones and at worst third order accurate. We then recursively combine the above two schemes to arrive at a WENO-AO(7,5,3) scheme which can achieve seventh order accuracy when that is possible; graciously drop down to fifth order accuracy when that is the best one can do; and also operate stably with an r=3 CWENO scheme when that is the only thing that one can do. Schemes with ninth order of accuracy are also presented.Several accuracy tests and several stringent test problems are presented to demonstrate that the method works very well.

A new interface treating method is proposed to simulate compressible two-medium problems with the Runge-Kutta discontinuous Galerkin (RKDG) method. In the present work, both the Euler equation and the level-set equation are discretized with the RKDG method which is compact and of high-order accuracy. The linearized interface inside an interface cell is recovered by the level-set function. The new solution of this cell is taken as a convex combination of two auxiliary solutions. One is the solution obtained by the RKDG method for a single-medium cell with proper numerical fluxes, and the other one is the intermediate state of the two-medium Riemann problem constructed in the normal direction. The weights of the two auxiliary solutions are carefully chosen according to the location of the interface inside the cell. Thus, it ensures a smooth transition when the interface leaves one cell and enters a neighboring cell. The entropy-fix technique is adopted to minimize the overshoots or undershoots in problems with large entropy ratio across the interface. The scheme is justified in a 1-dimensional situation and extended to 2-dimensional problems. Several 1-dimensional two-medium problems, including both smooth and discontinuous examples, are simulated and compared with exact solutions. Also, three 2-dimensional benchmark problems are simulated to validate the present method in two-medium problems.

In this paper, we reconsider the inverse Lax-Wendroff (ILW) procedure, which is a numerical boundary treatment for solving hyperbolic conservation laws, and propose a new approach to evaluate the values on the ghost points. The ILW procedure was firstly proposed to deal with the “cut cell” problems, when the physical boundary intersects with the Cartesian mesh in an arbitrary fashion. The key idea of the ILW procedure is repeatedly utilizing the partial differential equations (PDEs) and inflow boundary conditions to obtain the normal derivatives of each order on the boundary. A simplified ILW procedure was proposed in [28] and used the ILW procedure for the evaluation of the first order normal derivatives only. The main difference between the simplified ILW procedure and the proposed ILW procedure here is that we define the unknown u and the flux f ( u ) on the ghost points separately. One advantage of this treatment is that it allows the eigenvalues of the Jacobian f ′ ( u ) to be close to zero on the boundary, which may appear in many physical problems. We also propose a new weighted essentially non-oscillatory (WENO) type extrapolation at the outflow boundaries, whose idea comes from the multi-resolution WENO schemes in [32] . The WENO type extrapolation maintains high order accuracy if the solution is smooth near the boundary and it becomes a low order extrapolation automatically if a shock is close to the boundary. This WENO type extrapolation preserves the property of self-similarity, thus it is more preferable in computing the hyperbolic conservation laws. We provide extensive numerical examples to demonstrate that our method is stable, high order accurate and has good performance for various problems with different kinds of boundary conditions including the solid wall boundary condition, when the physical boundary is not aligned with the grids.

Radiation hydrodynamics (RH) describes the interaction between matter and radiation which affects the thermodynamic states and the dynamic flow characteristics of the matter-radiation system. Its application areas are mainly in high-temperature hydrodynamics, including gaseous stars in astrophysics, combustion phenomena, reentry vehicles fusion physics and inertial confinement fusion (ICF). Solving the radiation hydrodynamics equations (RHE), even in the equilibrium-diffusion limit, is a difficult task. In this paper, we will discuss the methodology to construct fully explicit and implicit-explicit (IMEX) high order Lagrangian schemes solving one dimensional RHE in the equilibrium-diffusion limit respectively, which can be used to simulate multi-material problems with the coupling of radiation and hydrodynamics. The schemes are based on the HLLC numerical flux, the essentially non-oscillatory (ENO) reconstruction for the advection term, ENO reconstruction or high order central reconstruction and interpolation for the radiation diffusion term, the Newton iteration method (for the IMEX scheme), and the strong stability preserving (SSP) high order time discretizations. The schemes can maintain conservation and uniformly high order accuracy both in space and time. The issue of positivity-preserving for the high order explicit Lagrangian scheme is also discussed. Various numerical tests for the high order Lagrangian schemes are provided to demonstrate the desired properties of the schemes such as high order accuracy, non-oscillation, and positivity-preserving.

We consider numerical boundary conditions for high order finite difference schemes for solving convection-diffusion equations on arbitrary geometry. The two main difficulties for numerical boundary conditions in such situations are: (1) the wide stencil of the high order finite difference operator requires special treatment for a few ghost points near the boundary; (2) the physical boundary may not coincide with grid points in a Cartesian mesh and may intersect with the mesh in an arbitrary fashion. For purely convection equations, the so-called inverse Lax-Wendroff procedure 28, in which we convert the normal derivatives into the time derivatives and tangential derivatives along the physical boundary by using the equations, has been quite successful. In this paper, we extend this methodology to convection-diffusion equations. It turns out that this extension is non-trivial, because totally different boundary treatments are needed for the diffusion-dominated and the convection-dominated regimes. We design a careful combination of the boundary treatments for the two regimes and obtain a stable and accurate boundary condition for general convection-diffusion equations. We provide extensive numerical tests for one- and two-dimensional problems involving both scalar equations and systems, including the compressible Navier-Stokes equations, to demonstrate the good performance of our numerical boundary conditions.

For solving time-dependent convection-dominated partial differential equations (PDEs), which arise frequently in computational physics, high order numerical methods, including finite difference, finite volume, finite element and spectral methods, have been undergoing rapid developments over the past decades. In this article we give a brief survey of two selected classes of high order methods, namely the weighted essentially non-oscillatory (WENO) finite difference and finite volume schemes and discontinuous Galerkin (DG) finite element methods, emphasizing several of their recent developments: bound-preserving limiters for DG, finite volume and finite difference schemes, which address issues in robustness and accuracy; WENO limiters for DG methods, which address issues in non-oscillatory performance when there are strong shocks, and inverse Lax-Wendroff type boundary treatments for finite difference schemes, which address issues in solving complex geometry problems using Cartesian meshes.

The main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with implicit-explicit (IMEX) time discretization schemes, for solving multi-dimensional convection-diffusion equations with nonlinear convection. By establishing the important relationship between the gradient and the interface jump of the numerical solution with the independent numerical solution of the gradient in the LDG method, on both rectangular and triangular elements, we can obtain the same stability results as in the one-dimensional case [H.J. Wang, C.-W. Shu and Q. Zhang, SIAM J. Numer. Anal. 53 (2015) 206–227; H.J. Wang, C.-W. Shu and Q. Zhang, Appl. Math. Comput. 272 (2015) 237–258], i.e. , the IMEX LDG schemes are unconditionally stable for the multi-dimensional convection-diffusion problems, in the sense that the time-step τ is only required to be upper-bounded by a positive constant independent of the spatial mesh size h . Furthermore, by the aid of the so-called elliptic projection and the adjoint argument, we can also obtain optimal error estimates in both space and time, for the corresponding fully discrete IMEX LDG schemes, under the same condition, i.e. , if piecewise polynomial of degree k is adopted on either rectangular or triangular meshes, we can show the convergence accuracy is of order 𝒪( h k+1 + τ s ) for the s th order IMEX LDG scheme ( s = 1,2,3) under consideration. Numerical experiments are also given to verify our main results.

In this paper, we develop a discontinuous Galerkin (DG) method to solve the ideal special relativistic hydrodynamics (RHD) and design a bound-preserving (BP) limiter for this scheme by extending the idea in X. Zhang and C.-W. Shu, (2010) 56. For RHD, the density and pressure are positive and the velocity is bounded by the speed of light. One difficulty in numerically solving the RHD in its conservative form is that the failure of preserving these physical bounds will result in ill-posedness of the problem and blowup of the code, especially in extreme relativistic cases. The standard way in dealing with this difficulty is to add extra numerical dissipation, while in doing so there is no guarantee of maintaining the high order of accuracy. Our BP limiter has the following features. It can theoretically guarantee to preserve the physical bounds for the numerical solution and maintain its designed high order accuracy. The limiter is local to the cell and hence is very easy to implement. Moreover, it renders L 1 -stability to the numerical scheme. Numerical experiments are performed to demonstrate the good performance of this bound-preserving DG scheme. Even though we only discuss the BP limiter for DG schemes, it can be applied to high order finite volume schemes, such as weighted essentially non-oscillatory (WENO) finite volume schemes as well.

International audience; One of the main issues in the field of numerical schemes is to ally robustness with accuracy. Considering gas dynamics, numerical approximations may generate negative density or pressure, which may lead to nonlinear instability and crash of the code. This phenomenon is even more critical using a Lagrangian formalism, the grid moving and being deformed during the calculation. Furthermore, most of the problems studied in this framework contain very intense rarefaction and shock waves. In this paper, the admissibility of numerical solutions obtained by high-order finite-volume-scheme-based methods, such as the discontinuous Galerkin (DG) method, the essentially non-oscillatory (ENO) and the weighted ENO (WENO) finite volume schemes, is addressed in the one-dimensional Lagrangian gas dynamics framework. After briefly recalling how to derive Lagrangian forms of the 1D gas dynamics system of equations, a discussion on positivity-preserving approximate Riemann solvers, ensuring first-order finite volume schemes to be positive, is then given. This study is conducted for both ideal gas and non ideal gas equations of state (EOS), such as the Jones-Wilkins-Lee (JWL) EOS or the Mie-Grüneisen (MG) EOS, and relies on two different techniques: either a particular definition of the local approximation of the acoustic impedances arising from the approximate Riemann solver, or an additional time step constraint relative to the cell volume variation. Then, making use of the work presented in [89, 90, 22], this positivity study is extended to high-orders of accuracy, where new time step constraints are obtained, and proper limitation is required. Through this new procedure, scheme robustness is highly improved and hence new problems can be tackled. Numerical results are provided to demonstrate the effectiveness of these methods. This paper is the first part of a series of two. The whole analysis presented here is extended to the two-dimensional case in [85], and proves to fit a wide range of numerical schemes in the literature, such as those presented in [19, 64, 15, 82, 84].

For numerical simulation of detonation, computational cost using uniform meshes is large due to the vast separation in both time and space scales. Adaptive mesh refinement (AMR) is advantageous for problems with vastly different scales. This paper aims to propose an AMR method with high order accuracy for numerical investigation of multi-dimensional detonation. A well-designed AMR method based on finite difference weighted essentially non-oscillatory (WENO) scheme, named as AMR&WENO is proposed. A new cell-based data structure is used to organize the adaptive meshes. The new data structure makes it possible for cells to communicate with each other quickly and easily. In order to develop an AMR method with high order accuracy, high order prolongations in both space and time are utilized in the data prolongation procedure. Based on the message passing interface (MPI) platform, we have developed a workload balancing parallel AMR&WENO code using the Hilbert space-filling curve algorithm. Our numerical experiments with detonation simulations indicate that the AMR&WENO is accurate and has a high resolution. Moreover, we evaluate and compare the performance of the uniform mesh WENO scheme and the parallel AMR&WENO method. The comparison results provide us further insight into the high performance of the parallel AMR&WENO method.

We discuss the issue of conservation of the total mass for finite difference WENO schemes solving hyperbolic conservation laws on a Cartesian mesh using the inverse Lax-Wendroff boundary treatments in arbitrary physical domains whose boundaries do not coincide with grid lines. The numerical fluxes near the boundary are suitably modified so that strict conservation of the total mass is achieved and the high order accuracy and non-oscillatory performance are not compromised. The key point is a suitable definition of the total mass, which is consistent with the high order accuracy finite difference framework over an arbitrary domain with a boundary not necessarily coinciding with grid lines. Extensive numerical examples are provided to demonstrate that our modified method is strictly conservative, and is high order accurate and has as good performance as the original high order WENO schemes with the Lax-Wendroff boundary treatments, for both smooth problems and problems with discontinuities, in both one- and two-dimensional problems involving both scalar equations and systems.

In this continuing paper of [J. Comput. Phys., 375 (2018), 659-683; J. Comput. Phys., 392 (2019), 19-33], we design a new third-order finite volume multi-resolution weighted essentially non-oscillatory (WENO) scheme for solving hyperbolic conservation laws on tetrahedral meshes. We only use the information defined on a hierarchy of nested central spatial stencils without introducing any equivalent multi-resolution representation. Comparing with classical third-order finite volume WENO schemes [Commun. Comput. Phys., 5 (2009), 836-848] on tetrahedral meshes, the crucial advantages of such new multi-resolution WENO schemes are their simplicity and compactness with the application of only three unequal-sized central stencils for reconstructing unequal degree polynomials in the WENO type spatial procedures, their easy choice of arbitrary positive linear weights without considering the topology of the tetrahedral meshes, their optimal order of accuracy in smooth regions, and their suppression of spurious oscillations near strong discontinuities. The linear weights of such new multi-resolution WENO scheme can be any positive numbers on the condition that their sum is one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite volume WENO scheme on tetrahedral meshes. By performing such new spatial reconstruction procedures and adopting a third-order TVD Runge-Kutta method for time discretization, the occupied memory is decreased and the computing efficiency is increased. This new third-order finite volume multi-resolution WENO scheme is suitable for large scale engineering applications and could maintain good convergence property for steady-state problems on tetrahedral meshes. Benchmark examples are computed to demonstrate the robustness and good performance of these new finite volume WENO schemes.

In this paper, a new type of multi-resolution weighted essentially non-oscillatory (WENO) limiters for high-order Runge-Kutta discontinuous Galerkin (RKDG) methods is designed. This type of multi-resolution WENO limiters is an extension of the multi-resolution WENO finite volume and finite difference schemes developed in [43] . Such new limiters use information of the DG solution essentially only within the troubled cell itself, to build a sequence of hierarchical L 2 projection polynomials from zeroth degree to the highest degree of the RKDG method. The second-order, third-order, fourth-order, and fifth-order RKDG methods with these multi-resolution WENO limiters have been developed as examples, which could maintain the original order of accuracy in smooth regions and could simultaneously suppress spurious oscillations near strong discontinuities. The linear weights of such new multi-resolution WENO limiters can be any positive numbers on the condition that their sum equals one. This is the first time that a series of polynomials of different degrees within the troubled cell itself are applied in a WENO fashion to modify the DG solutions in the troubled cell. These new WENO limiters are very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions. Such spatial reconstruction methodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original DG methods. Benchmark examples are given to demonstrate the good performance of these RKDG methods with the associated multi-resolution WENO limiters.

We discuss the aeroacoustic resolution properties of the discontinuous Galerkin (DG) schemes in detail and compare their performance with the state of the art finite difference (FD) schemes, including the classical dispersion-relation-preserving (DRP) schemes. Analysis shows that, even though the DG schemes are slightly dissipative, their overall dispersion and dissipation properties are comparable with the corresponding DRP schemes on the same stencil. For the convenience of a direct comparison with FD schemes, we write the DG schemes in the form of block finite difference schemes. Ample numerical tests, including the tests on nozzle flow problems, are performed and we observe that the DG schemes and DRP schemes with the same stencil produce comparable numerical results. Since the DG schemes are flexible on non-uniform meshes and general unstructured meshes, they should be good candidates for computational aeroacoustics, especially those on complex geometry.