VODE: a variable-coefficient ODE solver
SIAM Journal on Scientific and Statistical Computing
A Test for Instability in the Numerical Solution of Ordinary Differential Equations
Journal of the ACM (JACM)
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Weakly damped oscillatory modes in ODE systems, such as are generated from advection-dominated PDE problems, frequently cause difficulty for BDF integrators, because of their absolute stability regions at high orders. We describe a procedure to detect the stability limit responsible for the difficulty. From it, we form a detection algorithm, which we have added to the order selection logic in an experimental version of the ODE solver VODE. Tests on an advection-diffusion problem demonstrate the effectiveness of this algorithm in lowering order to avoid the stability limit.