Boosting the accuracy of finite difference schemes via optimal time step selection and non-iterative defect correction

Abstract

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.

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

“…The leading order terms in the truncation error expressions can be classified into singular and regular perturbations of the original partial differential equation [18]. A simple means of improving the order of accuracy of finite difference schemes, using information from the truncation error expression, is to subtract the complete leading order expression from the original discretized equation.…”

“…As high order finite difference schemes typically are more complicated to develop and implement, when compared to lower order schemes, a number of methodologies have been developed which enhance the accuracy of underlying low-order schemes through choices made regarding the computational (spatial and temporal) grid. Such approaches are based upon reducing the truncation error of these schemes through application of various numerical techniques including adaptive grid refinement [12,13], Richardson extrapolation [14], defect correction [15,16,17] and optimal time-step selection combined with non-iterative defect correction (OTS-NIDC) [18].…”

“…As such, this approach is implicit. Moreover, it only reduces the spatial discretization error and the temporal accuracy is only affected by the order of temporal discretization of the underlying low-order scheme [18].…”

“…Recently, a non-iterative defect correction method in conjunction with the optimal time-step selection (OTS-NIDC) has been used by Chu [18] to increase the order of accuracy of finite difference schemes. In this method, the leading order terms in the truncation error expression are categorized into singular and regular perturbation terms.…”

Adaptive finite difference solutions of Liouville equations in computational uncertainty quantification

“…A truncation error analysis of the finite difference scheme will show that imposing the local CFL=1 condition results in an order of magnitude reduction in both the dispersion and diffusion errors. A similar concept was used by Chu to enhance the order of accuracy of finite difference solutions through optimal time-step selection [24]. However, Chu's approach can not easily be applied to Liouville equations which have spatially dependent drift coefficients as application of the method would lead to a time-step that varies throughout the computational domain.…”