Stability of semidiscrete formulations for parabolic problems at small time steps

Abstract

Solutions of direct time-integration schemes that converge in time to conventional semidiscrete formulations may be polluted at small time steps by spurious spatial oscillations, along with attendant overshoot in time. This degradation is not an artifact of the time-marching scheme, but rather a property of the solution of the semidiscrete formulation itself. A critical time step for the onset of spatial oscillations is derived by analogy to singularly perturbed elliptic problems. We then propose a simple procedure of spatial stabilization to remove this pathology from implicit time-integration schemes, without affecting unconditional temporal stability. Spatially stabilized implicit time integration is free of spurious spatial oscillations at small time steps, and numerical experience points to improved temporal accuracy as well.