Linear stability of the hopscotch scheme

Abstract

423This paper is devoted to the hopscotch scheme, which is a numerical integration technique for time-dependent partial differential equations. We examine its linear stability properties. A general theorem is presented which provides sufficient conditions for boundedness of the numerical solution during time stepping on a fixed space-time mesh. This theorem has applications in the field of parabolic problems. For the one-space-dimensional convection-diffusion equation we present a detailed stability analysis of the odd-even scheme combined with central and one-sided finite differences. We compare stability based on the spectral condition with von Neumann stability.

Strong stability results for the hopscotch method with applications to bending beam equations

“…In [1], [2], [3] and [4], a stability analysis is given for hopscotch discretizations of second order parabolic PDEs, but it is not clear how these analysis techniques can be applied in the situation of a bending beam equation. In [7], an analysis of the eigenvalues of the companion matrix of (R) led to conditions that are necessary and sufficient to have a stable recursion (R).…”