In this paper, a family of high order numerical methods are designed to solve the Hamilton-Jacobi equation for the viscosity solution. In particular, the methods start with a hyperbolic conservation law system closely related to the Hamilton-Jacobi equation. The compact one-step one-stage Lax-Wendroff type time discretization is then applied together with the local-structure-preserving disconti...

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, n...

Abstract. In this paper, we give a survey and discuss new developments and computational results for a high order accurate numerical boundary condition based on finite difference methods for solving hyperbolic equations on Cartesian grids, while the physical domain can be arbitrarily shaped. The challenges are the wide stencil for the high order scheme and the fact that the physical boundary do...

In this paper, we propose a class of high-order schemes for solving oneand two-dimensional hyperbolic conservation laws. The methods are formulated in a central finite volume framework on staggered meshes, and they involve Hermite WENO (HWENO) reconstructions in space, and Lax-Wendroff type discretizations or the natural continuous extension of Runge-Kutta methods in time. Compared with central...

We study the stability of mesh refinement in space and time for several different interface equations and finite-difference approximations. First, we derive a root condition which implies stability for the initial-boundary value problem for this type of interface. From the root condition, we prove the stability of several interface equations using the maximum principle. In some cases, the final...

The quest continues for accurate and efficient computational fluid dynamics (CFD) algorithms for convection-dominated flows. The boundary value ‘optimal’ Galerkin weak statement invariably requires manipulation to handle the disruptive character introduced by the discretized first-order convection term. An incredible variety of methodologies have been derived and examined to address this issue,...

We prove in this paper the Lax–Wendroff consistency of a general finite volume convection operator acting on discrete functions which are possibly not piecewise-constant over cells mesh and time steps. It yields an extension theorem for colocated or non-colocated schemes. This result is obtained polygonal polyhedral meshes, under assumptions which, usual practical cases, essentially boil down t...

In this paper, we study the stability of a numerical boundary treatment high order compact finite difference methods for parabolic equations. The schemes could achieve very accuracy with relatively small stencils. To match convergence in interior domain, take simplified inverse Lax–Wendroff (SILW) procedure (Tan et al., 2012; Li 2017) as our treatment. third total variation diminishing (TVD) Ru...

This paper is devoted to the construction of numerical fluxes for hyperbolic systems. We first present a GFORCE numerical flux, which is a weighted average of the Lax-Friedrichs and Lax-Wendroff fluxes. For the linear advection equation with constant coefficient, the new flux reduces identically to that of the Godunov first order upwind method. Then we incorporate GFORCE in the framework of the...

We have developed a new computer program in Fortran 90, in order to obtain numerical solutions of a system of Relativistic Magnetohydrodynamics partial differential equations with predetermined gravitation (GRMHD), capable of simulating the formation of relativistic jets from the accretion disk of matter up to his ejection. Initially we carried out a study on numerical methods of unidimensional...