  In this article, we present a simple technique for boosting the order of
accuracy of finite difference schemes for time dependent partial differential
equations by optimally selecting the time step used to advance the numerical
solution and adding defect correction terms in a non-iterative manner. The
power of the technique is its ability 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 combined with non-iterative defect correction (NIDC) on several
different types of finite difference schemes for a wide array of classical
linear and semilinear PDEs in one and more space dimensions on both regular and
irregular domains.
