Stability of Galerkin multistep procedures in time-homogeneous hyperbolic problems

Abstract

Linear and time-homogeneous hyperbolic initial boundary value problems are approximated using Galerkin procedures for the space directions and linear multistep methods for the time direction. At first error bounds are proved for multistep methods having a stability interval [−ω, 0], 0<ω, and systemsY″=AY+C(t) under the condition that\(\Delta t^2 \left\| A \right\| \leqslant \omega \) Δt time step. Then these error bounds are applied to derive bounds for the error in hyperbolic problems. The result shows that the initial error and the discretization error grow liket andt2 respectively. But the initial error is multiplied with a factor which becomes large if the mesh width of the space discretization is small.