Abstract
In this note we present an issue in the verification of order of convergence for finite difference approximations to partial differential equations discretized in time and space. When one refines both space and time together, the convergence rate for a correct implementation will match that expected from an analysis of the asymptotic convergence rate (or local truncation error) of the method. Sometimes, however, only the time-step size is reduced, while the spatial mesh is held fixed. The observed rate of convergence in this case may then differ from the formal order of accuracy of the method in space–time. In particular, one class of methods, second-order Lax–Wendroff time-differencing, produces only first-order when the time step is refined at fixed spatial mesh resolution because the time and space differencing are intrinsically linked. Our method to arrive at this result is through analyzing the error at a defined point in time associated with many time steps. We demonstrate our results computationally.
Published Version
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