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摘要
The alternating evolution (AE) system of Liu [J. Hyperbolic Differ. Equ., 5 (2008), pp. 421-447], partial derivative(t)u + partial derivative(x)f(v) = 1/c (v - u), partial derivative(t)v + partial derivative(x)f(u) = 1/c (u - v), serves as a refined description of systems of hyperbolic conservation laws partial derivative(t)phi + partial derivative(x)f(phi) = 0, phi(x, 0) = phi(0)(x). The solution of conservation laws is precisely captured when two components take the same initial value as phi(0), or is approached by two components exponentially fast when epsilon down arrow 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 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 +/- Delta(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 epsilon 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 conservation laws, we prove the numerical stability in the sense of satisfying the maximum principle and total variation diminishing property. The formulation procedure of AE schemes in multiple dimensions 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.
- 出版日期2011
