Boosting the Accuracy of Finite Difference Schemes via Optimal Time Step Selection

In this article, we present a novel technique for boosting the order of accuracy of formally low-order finite difference schemes for time dependent partial differential equations by optimally selecting the time step used to advance the numerical solution. This technique allows us to extract as much accuracy as possible from existing finite difference schemes with minimal additional effort. Through straightforward numerical analysis arguments, we explain the origin of the boost in accuracy and estimate the computational cost of the resulting numerical method. We demonstrate the utility of optimal time step (OTS) selection on a wide array of classical linear and semilinear PDEs in one and more space dimensions on both regular and irregular domains.