Using Optimal Time Step Selection and Non-Iterated Defect Correction to Boost the Accuracy of FD Schemes for Variable-Coefficient PDEs and Systems of PDEs

Bay Area Scientific Computing Day 2009, Berkeley, CA, May 9, 2009



For scalar time-dependent PDEs where the leading-order spatial derivative term has a constant coefficient, optimal time step (OTS) selection and non-iterated defect correction make it possible to transform formally low-order finite-difference schemes into high-order numerical methods in any number of space dimensions on both regular and irregular domains. For example, OTS selection can be used to obtain a fourth-order accurate solution to the 2D diffusion equation on an irregular domain using only forward Euler time integration and the second-order nine-point stencil for the Laplacian. Similar boosts in the order of accuracy have been obtained for the linear advection equation, viscous Burgers equation, fourth-order parabolic equation, and second-order wave equation (using the two-step Kreiss-Petersson-Ystrom scheme). In this poster, we present recent extensions of OTS selection and non-iterated defect correction to PDEs with a variable-coefficient leading-order spatial derivative term and systems of PDEs. In the former case, an \emph{optimal grid} is used in conjunction with OTS selection and defect correction to achieve a boost in the accuracy. In the latter case, temporal interpolation can be used to synchronize the numerical solution of individual PDEs (each advanced with its own optimal time step) without introducing any low-order terms to the local truncation error.