Multistep finite difference schemes for the variable coefficient delay parabolic equations

Abstract

In this paper, we study the high efficient numerical methods to solve a class of variable coefficient delay parabolic differential equations. We propose two types of multistep finite difference schemes while prove the solvability, convergence and stability of both schemes. The convergence orders are $\mathcal{O}({\mathit{\tau }}^2+h^2)$ and $\mathcal{O}({\mathit{\tau }}^2+h^4)$, respectively, in the sense of $L_{\infty }$-norm. By some new analytical transformations, we extend our schemes to solve a more general variable coefficient delay convection–diffusion–reaction equations and also apply them to the higher dimensional cases. Furthermore, multistep finite difference method and compact multistep finite difference method for two dimensional variable coefficient diffusion–reaction equations are proposed and the suitable alternate direction implicit (ADI) technique is constructed for the multistep finite difference scheme. At last, several numerical experiments are carried out to illustrate the effectiveness of both schemes.

Keywords:

• Variable coefficient delay parabolic differential equations
• multistep finite difference scheme
• ADI scheme
• solvability
• stability
• convergence

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by NSFC [grant number 11501514], [grant number 11201161], [grant number 11571128], [grant number 11471287]; Natural Sciences Foundation of Zhejiang Province [grant number LQ16A010007]; Zhejiang Province Ministry of Education [grant number Y201533134]; School Initiation Funds in ZSTU [grant number 11432932611470].