Avoiding spurious modes of time discretized operators in transport problems

The solution of the Crank–Nicolson (CN) method may present spurious modes, which mostly happens in systems with an initial singularity. In this work, we analyze the onset of spurious oscillations for the CN method for large time steps. This analysis is performed on the general one-dimensional linear advection–diffusion–reaction transport equation with initial singularity that models a broad range of mass transfer phenomena in natural and engineering sciences. The discrete problem is obtained using a stable finite element method in space and the generalized trapezoidal family of methods in time. Depending on the range of parameters, either the Galerkin or the streamline upwind Petrov–Galerkin methods are used to guarantee stability in space. We derive a stability analysis of the fully discrete method and investigate the techniques proposed in the literature to damp oscillations. We propose a new threshold of time oscillations to overcome the spurious modes, which is also extended to deal with nonlinear problems. Copyright © 2008 John Wiley & Sons, Ltd.

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