A high-order compact finite difference scheme and precise integration method based on modified Hopf-Cole transformation for numerical simulation of n-dimensional Burgers’ system

Changkai Chen a, Xiaohua Zhang a , b , ∗ , Zhang Liu a

a College of Science, China Three Gorges University, Yichang 443002, China

b Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China

Abstract

This paper modifies a n-dimensional Hopf-Cole transformation to the n-dimensional Burgers’system. We obtain the n-dimensional heat conduction equation through the

modification of the Hopf-Cole transformation. Then the fourth-order precise integration method (PIM) in combination with a spatially global sixth-order compact finite difference (CFD) scheme is presented to solve the equation with high accuracy. Moreover, coupling with the Strang splitting method, the scheme is extended to multi-dimensional (two, three-dimensional) Burgers’ system. Numerical results show that the proposed method appreciably improves the computational accuracy compared with the existing numerical method. Moreover, the two-dimensional and three-dimensional examples demonstrate excellent adaptability, and the numerical simulation results also have very high accuracy in medium Reynolds numbers.

Keywords:

n-dimensional Hopf-Cole transformation

n-dimensional Burges’ system

Compact finite difference

Precise integration method

Strang splitting method