In this paper, we further analyze, test, modify and improve the high order WENO (weighted essentially non-oscillatory) nite di erence schemes of Liu, Osher and Chan [9]. It was shown by Liu et al. that WENO schemes constructed from the r order (in L norm) ENO schemes are (r+1) order accurate. We propose a new way of measuring the smoothness of a numerical solution, emulating the idea of minimiz...

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 pape...

The weighted essentially non-oscillatory (WENO) schemes are a popular class of high order numerical methods for hyperbolic partial differential equations (PDEs). While WENO schemes on structured meshes are quite mature, the development of finite volume WENO schemes on unstructured meshes is more difficult. A major difficulty is how to design a robust WENO reconstruction procedure to deal with d...

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...

Abstract A new class of high order weighted essentially non-oscillatory (WENO) schemes [J. Comput. Phys., 318 (2016), 110-121] is applied to solve Euler equations with steady state solutions. It is known that the classical WENO schemes [J. Comput. Phys., 126 (1996), 202-228] might suffer from slight post-shock oscillations. Even though such post-shock oscillations are small enough in magnitude ...

ENO (essentially nonoscillatory) and weighted ENO (WENO) schemes were designed for high resolution of discontinuities, such as shock waves, while optimized schemes such as the DRP (dispersion–relation–preserving) schemes were optimized for short waves (with respect to the grid spacing 1x , e.g., waves that are 6–81x in wavelength) in the wavenumber space. In this paper, we seek to unite the adv...

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...

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...

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...