The Finite Volume WENO with Lax–Wendroff Scheme for Nonlinear System of Euler Equations

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.

A Fifth Order Alternative Mapped WENO Scheme for Nonlinear Hyperbolic Conservation Laws

“…The finite volume schemes are flexible and robust compared to the finite difference schemes [10]. Several finite volume WENO schemes have been designed and applied to curvilinear grid [11,12], non-uniform grids [13], unstructured grids [14] and structured grids [15][16][17]. The alternative WENO scheme developed by Shu and Osher [18] is an efficient scheme in the finite difference framework.…”

The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach

“…Many works on nonlinear evolution equations have been studied, such as the Hamiltonian structure [1,2], the infinite conservation laws [3,4], the Bäcklund transformation [5,6] and so on [7][8][9]. Besides, the exact solution of these equations, which can be expressed in various forms by different methods, is also a significant subject of soliton research [10][11][12][13][14][15][16][17][18][19][20][21][22]. In recent years, with the development of soliton theory, more and more researchers pay attention to the Riemann-Hilbert approach.…”