The convergence to steady state solutions of the Euler equations for high order weighted essentially non-oscillatory (WENO) finite difference schemes with the Lax-Friedrichs flux splitting [Jiang G.-S. and Shu C.-W. (1996), J. Comput. Phys. 126, 202-228] is investigated. Numerical evidence in [Zhang S. and Shu C.-W. (2007), J. Sci. Comput. 31, 273-305] indicates that there exist slight post-sho...

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 ...

Weighted essentially non-oscillatory (WENO) methods can simultaneously provide the high order of accuracy, high bandwidth-resolving efficiency, and shock-capturing capability required for the detailed simulation of compressible turbulence. However, rigorous analysis of the actual versus theoretical error properties of these non-linear numerical methods is difficult. We use a bandwidth-optimized...

High order accurate weighted essentially non-oscillatory (WENO) schemes are usually designed to solve hyperbolic conservation laws or to discretize the first derivative convection terms in convection dominated partial differential equations. In this paper we discuss a high order WENO finite difference discretization for nonlinear degenerate parabolic equations which may contain discontinuous so...

The weighted essentially non-oscillatory method (WENO) is an excellent spatial discretization for hyperbolic partial differential equations with discontinuous solutions. However, the time-step restriction associated with explicit methods may pose severe limitations on their use in applications requiring large scale computations. An efficient implicit WENO method is necessary. In this paper, we ...

Abstract. This paper develops Runge-Kutta PK-based central discontinuous Galerkin (CDG) methods with WENO limiter to the oneand two-dimensional special relativistic hydrodynamical (RHD) equations, K = 1,2,3. Different from the non-central DG methods, the Runge-Kutta CDG methods have to find two approximate solutions defined on mutually dual meshes. For each mesh, the CDG approximate solutions o...

An important property for finite difference schemes designed on curvilinear meshes is the exact preservation of free-stream solutions. This property is difficult to fulfill for high order conservative essentially non-oscillatory (WENO) finite difference schemes. In this paper we explore an alternative flux formulation for such finite difference schemes [5] which can preserve free-stream solutio...

In {J. Comput. Phys. 229 (2010) 8105-8129}, we studied hybrid weighted essentially non-oscillatory (WENO) schemes with different indicators for hyperbolic conservation laws on uniform grids for Cartesian domains. In this paper, we extend the schemes to solve two-dimensional systems of hyperbolic conservation laws on curvilinear grids for non-Cartesian domains. Our goal is to obtain similar adva...

A general approach is given to extend WENO reconstructions to a class of numerical schemes that use different types of moments (i.e., multi-moments) simultaneously as the computational variables, such as point values and grid cell averages. The key is to re-map the multi-moment values to single moment values (e.g., cell average or point values), which can then be used to invoke known, standard ...