A new high-order compact finite difference scheme based on precise integration method for the numerical simulation of parabolic equations

Abstract

This paper presents two high-order exponential time differencing precise integration methods (PIMs) in combination with a spatially global sixth-order compact finite difference scheme (CFDS) for solving parabolic equations with high accuracy. One scheme is a modification of the compact finite difference scheme of precise integration method (CFDS-PIM) based on the fourth-order Taylor approximation and the other is a modification of the CFDS-PIM based on a (4, 4)-Padé approximation. Moreover, on coupling with the Strang splitting method, these schemes are extended to multi-dimensional problems, which also have fast computational efficiency and high computational accuracy. Several numerical examples are carried out in order to demonstrate the performance and ability of the proposed schemes. Numerical results indicate that the proposed schemes improve remarkably the computational accuracy rather than the empirical finite difference scheme. Moreover, these examples show that the CFDS-PIM based on the fourth-order Taylor approximation yields more accurate results than the CFDS-PIM based on the (4, 4)-Padé approximation.

Radiation fluxes of gravitational, electromagnetic, and scalar perturbations in type-D black holes: an exact approach

“…Therefore, we introduce the ROC of the floating-point numbers 7 N to evaluate the speed of convergence for the computational methods. And the ROC of floating-point numbers is defined as [135][136][137]…”

Variational Solutions for Resonances by a Finite-Difference Grid Method

“…An important improvement of the accuracy and stability of the FDM has been recently described in Ref. [38] by combining two high-order exponential time differencing precise integration methods (PIMs) with a spatially global sixth-order compact finite difference scheme (CFDS). In addition, by modifying the representation of the Laplacian operator one can obtain a rigorous upper bound estimate of the true kinetic energy [39].…”

Fourth-Order Compact Finite Difference Method Combined with Richardson Extrapolation for OneDimensional Heat Equation

“…Partial differential equations (PDEs) are mathematical equations that play a significant role in modelling physical phenomena that occur in nature (sun et al, 2010). Because of the complex nature of many practical problems, the majority of the solutions of the PDEs are numerical (Chen et al, 2020). The parabolic equation is used to study diffusion and heat conduction problems.…”

“…The usual FDM shows shortcomings in computational accuracy. One technique to overcome such shortcomings is to refine the mesh which in turn leads to a large system of equations and hence will increase the amount of storage and CPU time (Chen et al, 2020). Another natural way to increase the accuracy and improve the order of convergence of the numerical solution is through the application of the Richardson extrapolation technique (Wang and Zhang, 2009, Gonzalez et al, 2010and Liao and Sun, 2011.…”

“…The time discretization methods such as the Runge-Kutta, Predictor-Corrector, Cubic splines, etc. may be used to fully discretize the resulting ODEs (Chen et al, 2020). This paper aims to apply the fourth-order compact finite difference method combined with the Richardson extrapolation technique to the one-dimensional heat equations to generate a more accurate numerical solution.…”