TITLE:
Discrete Transparent Boundary Conditions for Burgers’ Equation
AUTHORS:
Xinyue Chen, Wenqing Zhang, Yuchen Wu
KEYWORDS:
Burgers’ Equation, Cole-Hopf Transformation, Discrete Transparent Boundary Conditions
JOURNAL NAME:
Engineering,
Vol.16 No.12,
December
30,
2024
ABSTRACT: This paper is devoted to numerically solving the Burgers’ equation in unbounded region, which describes the nonlinear wave propagation and diffusion effect. How to numerically and efficiently solve this problem remains a challenge due to the principal difficulties of not only the unboundedness but also nonlinearity. This paper aims to design the discrete transparent boundary conditions of Burgers’ equation to overcome the unboundedness. The presence of a nonlinear term in the equation makes it challenging to derive suitable artificial boundary conditions. To deal with the nonlinear term, a linear parabolic equation is obtained by applying the well-known Cole-Hopf transformation to the Burgers’ equation. Employing the
Z
-transformation, we then establish the discrete transparent boundary conditions for the linearized problem. Subsequently, the original equation is reduced to an initial boundary value problem, that can be efficiently solved by the finite difference method. The numerical analysis is reported rigorously. Some numerical results are given to demonstrate the accuracy and feasibility of the proposed method.