Grid adaptation and non-iterative defect correction for improved accuracy of numerical solutions of PDEs

In this work we present a computational approach for improving the order of accuracy of a given finite difference method for solution of linear and nonlinear hyperbolic partial differential equations. The methodology consists of analysis of leading order terms in the discretization error of any given finite difference method, leading to a modified version of the original partial differential equation. Singular perturbations of this modified equation are regularized using an adaptive grid distribution and a non-iterative defect correction method is used to eliminate the leading order, regular perturbation terms in the modified equation. Implementation of this approach on a low order finite difference scheme not only results in an increase in its order of accuracy but also results in an improvement in its numerical stability due to the regularization of singular perturbations. The proposed approach is applied to four different canonical problems including the numerical solution of (1) Liouville equation, (2) inviscid Burgers equation, (3) nonlinear reaction-advection equation and (4) a system of hyperbolic PDEs. When compared to exact solutions, the numerical results demonstrate the ability of this method in both boosting the accuracy of finite difference schemes up to the desired order and also providing a fully stable numerical solution.