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Abstract: In this paper, we extend a class of the Hermite weighted essentially non-oscillatory (HWENO) schemes for solving the Hamilton–Jacobi equations by Qiu and Shu (2005) [24] to two dimensional unstructured meshes. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first two derivative values are evolved via time advancing and used in the reconstructions, while only the function values are evolved and used in the original WENO schemes which are nodal based approximations. The third and fourth order HWENO schemes using the combinations of second order approximations with nonlinear weights and TVD Runge–Kutta time discretization method are used here. Comparing with the original WENO schemes for Hamilton–Jacobi equations, one major advantage of HWENO schemes presented here is its compactness in the reconstructions. Extensive numerical tests are performed to illustrate the capability and high order accuracy of the methodologies. [Copyright &y& Elsevier]

Abstract: In [J. Qiu, C.-W. Shu, Runge–Kutta discontinuous Galerkin method using WENO limiters, SIAM Journal on Scientific Computing 26 (2005) 907–929], Qiu and Shu investigated using weighted essentially non-oscillatory (WENO) finite volume methodology as limiters for the Runge–Kutta discontinuous Galerkin (RKDG) methods for solving nonlinear hyperbolic conservation law systems on structured meshes. In this continuation paper, we extend the method to solve two-dimensional problems on unstructured meshes, with the goal of obtaining a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, nonoscillatory shock transition for RKDG methods. Numerical results are provided to illustrate the behavior of this procedure. [Copyright &y& Elsevier]

Abstract: Runge–Kutta discontinuous Galerkin (RKDG) method is a high order finite element method for solving hyperbolic conservation laws employing useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes, TVD Runge–Kutta time discretizations, and limiters. In most of the RKDG papers in the literature, the Lax–Friedrichs numerical flux is used due to its simplicity, although there are many other numerical fluxes which could also be used. In this paper, we systematically investigate the performance of the RKDG method based on different numerical fluxes, including the first-order monotone fluxes such as the Godunov flux, the Engquist–Osher flux, etc., and second-order TVD fluxes, with the objective of obtaining better performance by choosing suitable numerical fluxes. The detailed numerical study is mainly performed for the one dimensional system case, addressing the issues of CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities. Numerical tests are also performed for two dimensional systems. [Copyright &y& Elsevier]

Abstract: In this paper we develop a Lax–Wendroff time discretization procedure for the discontinuous Galerkin method (LWDG) to solve hyperbolic conservation laws. This is an alternative method for time discretization to the popular total variation diminishing (TVD) Runge–Kutta time discretizations. The LWDG is a one step, explicit, high order finite element method. The limiter is performed once every time step. As a result, LWDG is more compact than Runge–Kutta discontinuous Galerkin (RKDG) and the Lax–Wendroff time discretization procedure is more cost effective than the Runge–Kutta time discretizations for certain problems including two-dimensional Euler systems of compressible gas dynamics when nonlinear limiters are applied. [Copyright &y& Elsevier]

Abstract: In this paper, a class of weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving Hamilton–Jacobi equations is presented. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and used in the reconstruction, while only the function values are evolved and used in the original WENO schemes. Comparing with the original WENO schemes of Jiang and Peng [Weighted ENO schemes for Hamilton–Jacobi equations, SIAM Journal on Scientific Computing 21 (2000) 2126] for Hamilton–Jacobi equations, one major advantage of HWENO schemes is its compactness in the reconstruction. Extensive numerical experiments are performed to illustrate the capability of the method. [Copyright &y& Elsevier]

In this paper, a class of fifth-order weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving one-dimensional nonlinear hyperbolic conservation law systems is presented. The construction of HWENO schemes is based on a finite volume formulation, Hermite interpolation, and nonlinearly stable Runge–Kutta methods. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and used in the reconstruction, while only the function values are evolved and used in the original WENO schemes. Comparing with the original WENO schemes of Liu et al. [J. Comput. Phys. 115 (1994) 200] and Jiang and Shu [J. Comput. Phys. 126 (1996) 202], one major advantage of HWENO schemes is its compactness in the reconstruction. For example, five points are needed in the stencil for a fifth-order WENO (WENO5) reconstruction, while only three points are needed for a fifth-order HWENO (HWENO5) reconstruction. For this reason, the HWENO finite volume methodology is more suitable to serve as limiters for the Runge–Kutta discontinuous Galerkin (RKDG) methods, than the original WENO finite volume methodology. Such applications in one space dimension is also developed in this paper. [Copyright &y& Elsevier]

In this paper, we construct a high order weighted essentially non-oscillatory (WENO) finite difference discretization for compressible Navier-Stokes (NS) equations, which is rendered positivity-preserving of density and internal energy by a positivity-preserving flux splitting and a scaling positivity-preserving limiter. The novelty of this paper is WENO reconstruction performed on variables from a positivity-preserving convection diffusion flux splitting, which is different from conventional WENO schemes solving compressible NS equations. The core advantages of our proposed method are robustness and efficiency, which especially are suitable for solving tough demanding problems of both compressible Euler and NS equation including low density and low pressure flow regime. Moreover, in terms of computational cost, it is more efficient and easier to implement and extend to multi-dimensional problems than the positivity-preserving high order discontinuous Galerkin schemes and finite volume WENO scheme for solving compressible NS equations on rectangle domain. Benchmark tests demonstrate that the proposed positivity-preserving WENO schemes are high order accuracy, efficient and robust without excessive artificial viscosity for demanding problems involving with low density, low pressure, and fine structure. • A positivity-preserving high order accurate finite difference WENO scheme for compressible N-S equations. • The method is the high order accuracy and positivity-preserving. • WENO reconstruction is performed on variables from a positivity-preserving convection diffusion flux splitting. • The method is robust and suitable for solving the compressible flow with low density and low pressure. [ABSTRACT FROM AUTHOR]

Abstract: Ideal MHD equations arise in many applications such as astrophysical plasmas and space physics, and they consist of a system of nonlinear hyperbolic conservation laws. The exact density ρ and pressure p should be non-negative. Numerically, such positivity property is not always satisfied by approximated solutions. One can encounter this when simulating problems with low density, high Mach number, or much large magnetic energy compared with internal energy. When this occurs, numerical instability may develop and the simulation can break down. In this paper, we propose positivity-preserving discontinuous Galerkin and central discontinuous Galerkin methods for solving ideal MHD equations by following [X. Zhang, C.-W. Shu, Journal of Computational Physics 229 (2010) 8918–8934]. In one dimension, the positivity-preserving property is established for both methods under a reasonable assumption. The performance of the proposed methods, in terms of accuracy, stability and positivity-preserving property, is demonstrated through a set of one and two dimensional numerical experiments. The proposed methods formally can be of any order of accuracy. [Copyright &y& Elsevier]

It is an important and challenging issue for the numerical solution of radiative transfer equations to maintain both high order accuracy and positivity. For the two-dimensional radiative transfer equations, Ling et al. give a counterexample (Ling et al. (2018) [13]) showing that unmodulated discontinuous Galerkin (DG) solver based either on the P k or Q k polynomial spaces could generate negative cell averages even if the inflow boundary value and the source term are both positive (and, for time dependent problems, also a nonnegative initial condition). Therefore the positivity-preserving frameworks in Zhang and Shu (2010) [28] and Zhang et al. (2012) [29] which are based on the value of cell averages being positive cannot be directly used to obtain a high order conservative positivity-preserving DG scheme for the radiative transfer equations neither on rectangular meshes nor on triangular meshes. In Yuan et al. (2016) [26] , when the cell average of DG schemes is negative, a rotational positivity-preserving limiter is constructed which could keep high order accuracy and positivity in the one-dimensional radiative transfer equations with P k polynomials and could be straightforwardly extended to two-dimensional radiative transfer equations on rectangular meshes with Q k polynomials (tensor product polynomials). This paper presents an extension of the idea of the above mentioned one-dimensional rotational positivity-preserving limiter algorithm to two-dimensional high order positivity-preserving DG schemes for solving steady and unsteady radiative transfer equations on triangular meshes with P k polynomials. The extension of this method is conceptually plausible but highly nontrivial. We first focus on finding a special quadrature rule on a triangle which should satisfy some conditions. The most important one is that the quadrature points can be arranged on several line segments, on which we can use the one-dimensional rotational positivity-preserving limiter. Since the number of the quadrature points is larger than the number of basis functions of P k polynomial space, we determine a k -th polynomial by a L 2 -norm Least Square subject to its cell average being equal to the weighted average of the values on the quadrature points after using the rotational positivity-preserving limiter. Since the weights used here are the quadrature weights which are positive, then the cell average of the modified polynomial is nonnegative. And the final modified polynomial can be obtained by using the two-dimensional scaling positivity-preserving limiter on the triangular element. We theoretically prove that our rotational positivity-preserving limiter on triangular meshes could keep both high order accuracy and positivity. It is relatively simple to implement, and also does not affect convergence to weak solutions. The numerical results validate the high order accuracy and the positivity-preserving properties of our schemes. The advantage of the triangular meshes on handling complex domain is also presented in our numerical examples. • High order positivity-preserving limiter for discontinuous Galerkin methods to solve radiative transfer equations on triangular meshes. • We theoretically proved that the rotational positivity-preserving limiter on triangular meshes could keep both high order accuracy and positivity. • The numerical results validate the high order accuracy and the positivity-preserving property of our schemes. [ABSTRACT FROM AUTHOR]

• A residual distribution method with a novel Hermite WENO scheme for steady state flow. • The new method is the sixth order accuracy in both one and two dimensions. • The new method is more efficient than the traditional HWENO scheme. • The new method is compact. In this paper, we develop a high order residual distribution (RD) method for solving steady state conservation laws in a novel Hermite weighted essentially non-oscillatory (HWENO) framework recently developed in [24]. In particular, we design a high order HWENO integration for the integrals of source term and fluxes based on the point value of the solution and its spatial derivatives, and the principles of residual distribution schemes are adapted to obtain steady state solutions. Two advantages of the novel HWENO framework have been shown in [24] : first, compared with the traditional HWENO framework, the proposed method does not need to introduce additional auxiliary equations to update the derivatives of the unknown variable, and just computes them from the current point value of the solution and its old spatial derivatives, which saves the computational storage and CPU time, and thereby improves the computational efficiency of the traditional HWENO framework. Second, compared with the traditional WENO method, reconstruction stencil of the HWENO methods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller at the same grid. Thus, it is also a compact scheme when we design the higher order accuracy, compared with that in [11] Chou and Shu proposed. Extensive numerical experiments for one- and two-dimensional scalar and systems problems confirm the high order accuracy and good quality of our scheme. [ABSTRACT FROM AUTHOR]