Abstract
We develop a Lax–Wendroff scheme on time discretization procedure for finite volume weighted essentially non-oscillatory schemes, which is used to simulate hyperbolic conservation law. We put more focus on the implementation of one-dimensional and two-dimensional nonlinear systems of Euler functions. The scheme can keep avoiding the local characteristic decompositions for higher derivative terms in Taylor expansion, even omit partly procedure of the nonlinear weights. Extensive simulations are performed, which show that the fifth order finite volume WENO (Weighted Essentially Non-oscillatory) schemes based on Lax–Wendroff-type time discretization provide a higher accuracy order, non-oscillatory properties and more cost efficiency than WENO scheme based on Runge–Kutta time discretization for certain problems. Those conclusions almost agree with that of finite difference WENO schemes based on Lax–Wendroff time discretization for Euler system, while finite volume scheme has more flexible mesh structure, especially for unstructured meshes.
Highlights
The scheme is adopted to simulate one-dimensional and two-dimensional nonlinear Euler systems of compressible gas dynamics; the scheme and results can be generalized to other hyperbolic conservation laws
The Lax–Wendroff time discretization formula based on finite volume is easier than that of the finite difference, but the computation is little more complex than that of finite difference method, as cell averages have to be calculated for finite volume method, while finite difference method only needs to calculate the point value
The main work of this paper is organized as follows, in Section 2, we describe details of the discretization for Euler system with finite volume WENO schemes and Lax–Wendroff-type time discretization
Summary
A new scheme is proposed using Lax–Wendroff scheme to discretize the time derivative and finite volume WENO (Weighted Essentially Non-oscillatory) scheme to discretize the spatial derivative of PDEs (partial differential equations). The scheme is adopted to simulate one-dimensional and two-dimensional nonlinear Euler systems of compressible gas dynamics; the scheme and results can be generalized to other hyperbolic conservation laws
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