Numerical solution of the Burgers' equation with high order splitting methods

TL;DR

This paper explores high order splitting and extrapolation methods with complex coefficients to efficiently and accurately solve the Burgers' equation, overcoming limitations of traditional real-coefficient methods.

Contribution

It introduces the use of complex coefficient splitting and extrapolation methods for Burgers' equation, enabling high order accuracy despite the equation's time-irreversibility.

Findings

Complex coefficient methods yield highly accurate solutions.

Extrapolation methods improve efficiency and accuracy.

High order methods outperform traditional approaches.

Abstract

In this work, high order splitting methods have been used for calculating the numerical solutions of the Burgers' equation in one space dimension with periodic and Dirichlet boundary conditions. However, splitting methods with real coefficients of order higher than two necessarily have negative coefficients and can not be used for time-irreversible systems, such as Burgers equations, due to the time-irreversibility of the Laplacian operator. Therefore, the splitting methods with complex coefficients and extrapolation methods with real and positive coefficients have been employed. If we consider the system as the perturbation of an exactly solvable problem(or can be easily approximated numerically), it is possible to employ highly efficient methods to approximate Burgers' equation. The numerical results show that the methods with complex time steps having one set of coefficients real and positive, say $a_i\in\mathbb{R}^+$ and $b_i\in\mathbb{C}^+$, and high order extrapolation methods derived from a lower order splitting method produce very accurate solutions of the Burgers' equation.