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.
Highlights
Many physical and mathematical models can be described by the partial differential equations (PDEs) in many work and technical problems, and the basic equations of many natural science problems are PDEs, which play a very important role in these fields [1]
We focus on applying precise integration methods (PIMs) to obtain numerical solutions of high accuracy
6 Conclusion This paper presents two high-order exponential time differencing precise integration method schemes in combination with a spatially global sixth-order compact finitedifference scheme, which have been developed for the numerical solutions of onedimensional and multi-dimensional parabolic equations
Summary
Many physical and mathematical models can be described by the partial differential equations (PDEs) in many work and technical problems, and the basic equations of many natural science problems are PDEs, which play a very important role in these fields [1]. The parabolic equation, as a kind of PDE, is often used to study diffusion and heat conduction problems. Due to the complexity of practical problems, many solutions of the PDEs are numerical. There are many numerical methods for heat transfer problems [2, 3], such as the finite-difference method (FDM), the finite element method (FEM), the finite volume method (FVM) and the spectrum method. Li [4] presented the useful Crank– Nicolson Galerkin FEM for the nonlinear parabolic problem. An important measure to improve the accu-
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