# 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

## TL;DR

This paper introduces a high-order compact finite difference scheme combined with a precise integration method, based on a modified Hopf-Cole transformation, to accurately simulate multi-dimensional Burgers' systems.

## Contribution

It develops a novel high-order numerical scheme that enhances accuracy for n-dimensional Burgers' equations using a modified Hopf-Cole transformation and advanced integration techniques.

## Key findings

- Significantly improves computational accuracy over existing methods.
- Demonstrates excellent adaptability in 2D and 3D simulations.
- Achieves high accuracy in medium Reynolds number flows.

## 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.