On the Accuracy and Stability of Explicit Schemes for Multidimensional Linear Homogeneous Advection Equations

Abstract

In [1] Roe provided, in a simple form, conditions which determine the accuracy of a numerical scheme for the solution of the one-dimensional linear advection equation u{sub 1} + au{sub x} = 0, where a is a constant wavespeed, by considering a general scheme for (1) in the form u{sub i}{sup n+1} = {Sigma} A{sub a}U{sub i+a}{sup n}, a where u{sub i}{sup n} = u(i {Delta}{chi}, n {Delta}t), (A{sub a}) is a finite set of constant, nonzero coefficients, {Delta}{chi} is the constant mesh spacing, and {Delta}t is the timestep. Define the Courant number as {nu} = a {Delta}t/{Delta}{chi}. Roe proved the following two theorems. THEOREM 1. If u{sub i}{sup n} is a polynomial of degree p in i, scheme (2) will give the exact solution to (1) if and only if {Sigma} a{sup 1}A{sub a} = (-{nu}){sup q} a for all integers q such that 0 {le} q {le} p. THEOREM 2. If scheme (2) meets the conditions of Theorem 1, the leading term in its pointwise error is a{Delta}{chi}{sup p}/{nu}(p + 1)! [(-{nu}){sup p+1}- {Sigma} a{sup p+1}Aa] {partial_derivative}{sup p+1}/{partial_derivative}{chi}{sup p+1} u. These two theorems enabled the following definition. DEFINITION 1. Any Scheme of the form (2) for the one-dimensionalmore » linear advection equation (1) that satisfies the conditions in Theorem 1 is called p{sup th}-order accurate in space and time. The aim of this paper is to extend Roe`s result to two and three dimensions. Methods of finding stability restrictions on multidimensional schemes are also discussed. 5 refs.« less