On the accuracy and stability of explicit schemes for multidimensional liner homogeneous advection equations

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 u1 + aux = 0, where a is a constant wavespeed, by considering a general scheme for (1) in the form uin+1 = Σ AaUi+an, a where uin = u(i Δχ, n Δt), (Aa) is a finite set of constant, nonzero coefficients, Δχ is the constant mesh spacing, and Δt is the timestep. Define the Courant number as ν = a Δt/Δχ. Roe proved the following two theorems. THEOREM 1. If uin is a polynomial of degree p in i, scheme (2) will give the exact solution to (1) if and only if Σ a1Aa = (-ν)q a for all integers q such that 0 ≤ q ≤ p. THEOREM 2. If scheme (2) meets the conditions of Theorem 1, the leading term in its pointwise error is aΔχp/ν(p + 1)exclamation point [(-ν)p+1- Σ ap+1Aa] ∂p+1/∂χp+1 u. These two theorems enabled the following definition. DEFINITION 1. Any Scheme of the form (2) for the one-dimensional linear advection equation (1) that satisfies the conditions in Theorem 1 is called pth-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