Resolution of high order WENO schemes for complicated flow structures

doi:10.1016/S0021-9991(03)00094-9

Journal of Computational Physics

Volume 186, Issue 2, 10 April 2003, Pages 690-696

https://doi.org/10.1016/S0021-9991(03)00094-9 Get rights and content

Abstract

In this short note we address the issue of numerical resolution and efficiency of high order weighted essentially non-oscillatory (WENO) schemes for computing solutions containing both discontinuities and complex solution features, through two representative numerical examples: the double Mach reflection problem and the Rayleigh–Taylor instability problem. We conclude that for such solutions with both discontinuities and complex solution features, it is more economical in CPU time to use higher order WENO schemes to obtain comparable numerical resolution.

Introduction

The WENO schemes we use in this paper are the fifth order finite difference version developed by Jiang and Shu in [7] and the ninth order finite difference version developed by Balsara and Shu in [1]. We will only give a very rough sketch of the algorithms and refer to [7] and [1], and also to the lecture notes [10], for most details. For a conservation laws system $u_{t} +f(u)_{x} +g(u)_{y} =0$ the conservative finite difference schemes we use approximate the point values u_ij at a uniform (or smoothly varying) grid (x_i,y_j) in a conservative fashion. Namely, the derivative f(u)_x at (x_i,y_j) is approximated along the line y=y_j by a conservative flux difference $f(u)_{x} |_{x=x_{i}} ≈ 1 Δ x f ̂_{i+1/2} − f ̂_{i−1/2},$ where for the fifth order WENO scheme the numerical flux $f ̂_{i+1/2}$ depends on 5 point values f(u_kj), k=i−2,…,i+2, when the wind is positive (i.e., when f^′(u)⩾0 for the scalar case, or when the corresponding eigenvalue is positive for the system case with a local characteristic decomposition). This numerical flux $f ̂_{i+1/2}$ is written as a convex combination of three third order numerical fluxes based on three different sub-stencils of three points each, and the combination coefficients depend on a “smoothness indicator” measuring the smoothness of the solution in each stencil. The resulting scheme can be proven uniformly fifth order accurate in smooth regions including at any smooth extrema. For discontinuities the solution is essentially non-oscillatory and gives sharp shock transitions. The ninth order WENO schemes follow a similar recipe, with 9 points in the stencil and 5 sub-stencils of 5 points each. The “monotonicity preserving limiters” in [1] is not used in this paper. We do not observe a need to further limit the solution beyond the WENO recipe for the test cases here. Time discretization is via the third order TVD Runge–Kutta method in [11]. The CFL number is taken as 0.6 for all the runs. We have not observed significant improvement for the numerical resolution when the time step is reduced or when the order of accuracy in time is increased for these test cases.

WENO finite difference schemes are relatively easy to code and have excellent parallel efficiency. However, because of the local characteristic decomposition and the evaluation of the non-linear smooth indicators, the methods are CPU time costly, especially for the higher order versions. A natural question is whether it is worthwhile to use such high order methods. The answer to this question is problem dependent. For many problems containing only simple shocks with almost linear smooth solutions in between, such as the solutions to most Riemann problems (shock tube problems), a good second order method, such as PPM [4] or other TVD [6] methods, would be the optimal choice. However, when the solution contains both discontinuities and complex solution structures in the smooth regions, a higher order method may be more economical in CPU time, as demonstrated by the examples in this paper.

Section snippets

Two numerical examples

In this section, we use the double Mach reflection problem and the Rayleigh–Taylor instability problem as examples for problems with both discontinuities and complex solution structures to demonstrate the resolution of high order WENO schemes. Both problems are for the two dimensional Euler equations of compressible gas dynamics, namely Eq. (1.1) with $u=(ρ,ρ u ̄,ρ v ̄,E)^{T},$ $f(u)=(ρ u ̄,ρ u ̄^{2} +p,ρ u ̄ v ̄, u ̄ (E+p))^{T},$ $g(u)=(ρ v ̄,ρ u ̄ v ̄,ρ v ̄^{2} +p, v ̄ (E+p))^{T} .$ Here ρ is the density, $(u ̄, v ̄)$ is the velocity, E is the total

Concluding remarks

We have performed a numerical study on the resolution of high order WENO finite difference methods on problems containing both discontinuities and complex solution structures, using the double Mach reflection problem and the Rayleigh–Taylor instability problem as examples. We have concluded that the ninth order WENO method needs only about half the number of mesh points in each direction compared with the fifth order WENO method to achieve a comparable numerical resolution and hence to have the

Acknowledgements

Research supported by LLNL Subcontract B513236, ARO Grant DAAD19-00-1-0405, NSF Grants DMS-9804985 and DMS-0207451, NASA Langley Grant NCC1-01035 and AFOSR Grant F49620-02-1-0113.

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