On Pitfalls in Accuracy Verification Using Time-Dependent Problems

In this short note, we discuss the circumstances that can lead to a failure to observe design order of discretization error convergence in accuracy verification when solving a time-dependent problem. Intuitively, one would expect to observe a design spatial order of accuracy when the discretization error is measured on a series of consistently refined grids after one extremely small time step because the time integration is then nearly exact. However, in reality, one observes one-order lower discretization error convergence than the design order. This loss of accuracy is not necessarily resolved even if the time step is consistently reduced along with the grid refinement. This can cause a serious problem because then one would wind up trying to find a coding error that does not exist. This short note clarifies the mechanism causing this failure to observe a design order of discretization error convergence in accuracy verification when solving time-dependent problems, and provides a guide for avoiding such pitfalls.