Alternating Evolution Schemes for Hyperbolic Conservation Laws

The alternating evolution (AE) system of Liu [25] ∂tu +∇x · f(v) = 1 2 (v − u), ∂tv +∇x · f(u) = 1 2 (u− v) serves as a refined description of systems of hyperbolic conservation laws ∂tφ +∇x · f(φ) = 0, φ(x, 0) = φ0(x). The solution of conservation laws is precisely captured when two components take the same initial value as φ0, or approached by two components exponentially fast when 2 ↓ 0 if two initial states are sufficiently close. This nice property enables us to construct novel shock capturing schemes by sampling the AE system on alternating grids. In this paper we develop a class of local Alternating Evolution (AE) schemes by taking advantage of the AE system. Our approach is based on an average of the AE system over a hypercube centered at x with vertices at x ±∆x. The numerical scheme is then constructed by sampling the averaged system over alternating grids. Higher order accuracy is achieved by a combination of high-order nonoscillatory polynomial reconstruction from the obtained averages and a matching Runge-Kutta solver in time discretization. Local AE schemes are made possible by letting the scale parameter 2 reflect the local distribution of nonlinear waves. The AE schemes have the advantage of easier formulation and implementation, and efficient computation of the solution. For the first and second order local AE schemes applied to one-dimensional scalar laws, we prove the numerical stability in the sense of satisfying the maximum principle and total variation diminishing (TVD) property. The formulation procedure of AE schemes in multi-dimension is given, followed by both the first and second order AE schemes for two-dimensional conservation laws. Numerical experiments for both scalar conservation laws and compressible Euler equations are presented to demonstrate the high order accuracy and capacity of these AE schemes.