Verification of BDF Time-Integration Method with Variable Step Size

Solution methods for systems of ordinary differential equations with variable step size base their choice for the step size on (an estimate of) the local truncation error. Significant savings in computational effort can be obtained as the step size is only refined when necessary. We first summarize the information that is needed for a variable step size BDF2 method in a way that is directly suitable for implementation in an existing code. We then propose a solution verification procedure that does not require direct control of the step size to carry out refinement studies and apply it to a textbook example. Finally, we explore the use of the variable step size BDF2 method for time integration of the Navier- Stokes equations in the CFD package REFRESCO and find plausible results for vortex shedding in the wake of a cylinder, but no major savings for this particular case.