Semidiscrete formulations for transient transport at small time steps

摘要

Solutions of direct time-integration schemes for transient advection-diffusion-reaction problems that converge in time to conventional semidiscrete formulations may be polluted at small time steps by spurious spatial oscillations. This degradation is not an artifact of the time-marching scheme, but rather a property of the solution of the semidiscrete Galerkin approximation itself. An analogy to steady advection- diffusion-reaction problems with a modified reaction coefficient by the Rothe method of discretizing in time prior to spatial discretization provides an upper bound on the time step for the onset of spatial instability. Spatial stabilization removes this pathology, leading to stabilized implicit time-integration schemes that are free of spurious oscillations at small time steps.