Stability of Finite-Difference Approximations for Hyperbolic Initial-Boundary-Value Problems

Abstract

We consider the stability of finite-difference approximations to hyperbolic initial-boundary-value problems (IBVPs) in one spatial dimension. A complication is the fact that generally more boundary conditions are required for the discrete problem than are specified for the partial differential equation. Consequently, additional “numerical” boundary conditions are required and improper treatment of these additional conditions can lead to instability and/or inaccuracy. For a linear homogeneous IBVP, a finite-difference approximation with requisite numerical boundary conditions can be written in vector-matrix form as u ^{n+ 1} = Cu ⁿ where C is a matrix operator. Lax-Richtmyer stability requires a uniform bound on C ⁿ (i.e., C to the nth power) in some matrix norm for 0 ≤ t = nΔt ≤ T. One would like to have an algebraic test for Lax-Richtmyer stability. For a matrix C of dimension J (denoted by C _J ), a theorem in linear algebra relates ‖C ⁿ _J ‖ to the spectral radius of C _J as n → ∞, with J fixed. We state a conjecture which extends this theorem to difference approximations for IBVPs where the matrix size J increases linearly with n as n → ∞ which corresponds to mesh refinement in both space and time. The asymptotic behavior of ‖C ⁿ _J ‖ is related directly to the eigenvalues from the von Neumann analysis of the Cauchy problem and the eigenvalues from the normal mode analysis of Gustafsson, Kreiss, and Sundström for the left- and right-quarter plane problems. The conjecture is corroborated by examples where the matrix norm of C ⁿ _J is computed numerically at a fixed time as the mesh is refined. An additional conjecture relates the spectral radius of the matrix C _J as J → ∞ to the spectral radius of an auxiliary Dirichlet problem and the eigenvalues from the normal mode analysis of the left- and right-quarter plane problems.