The numerical solutions for the nonhomogeneous Burgers' equation with the generalized Hopf-Cole transformation

Tong Yan; Tong Yan

doi:10.3934/nhm.2023014

Networks and Heterogeneous Media

2023, Volume 18, Issue 1: 359-379. doi: 10.3934/nhm.2023014

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The numerical solutions for the nonhomogeneous Burgers' equation with the generalized Hopf-Cole transformation

Tong Yan ^,

Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, 310018, China

Received: 25 August 2022 Revised: 10 November 2022 Accepted: 01 December 2022 Published: 26 December 2022

In this paper, with the help of the generalized Hopf-Cole transformation, we first convert the nonhomogeneous Burgers' equation into an equivalent heat equation with the derivative boundary conditions, in which Neumann boundary conditions and Robin boundary conditions can be viewed as its special cases. For easy derivation and numerical analysis, the reduction order method is used to convert the problem into an equivalent first-order coupled system. Next, we establish a box scheme for this first-order system. By the technical energy analysis method, we obtain the prior estimate of the numerical solution for the box scheme. Furthermore, the solvability and convergence are obtained directly from the prior estimate. The extensive numerical examples are carried out, which verify the developed box scheme can achieve global second-order accuracy for both homogeneous and nonhomogeneous Burgers' equations.
- Nonhomogeneous Burgers' equation,
- Reduction order method,
- Difference scheme,
- Prior estimate,
- The generalized Hopf-Cole transformation
Citation: Tong Yan. The numerical solutions for the nonhomogeneous Burgers' equation with the generalized Hopf-Cole transformation[J]. Networks and Heterogeneous Media, 2023, 18(1): 359-379. doi: 10.3934/nhm.2023014

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Abstract

In this paper, with the help of the generalized Hopf-Cole transformation, we first convert the nonhomogeneous Burgers' equation into an equivalent heat equation with the derivative boundary conditions, in which Neumann boundary conditions and Robin boundary conditions can be viewed as its special cases. For easy derivation and numerical analysis, the reduction order method is used to convert the problem into an equivalent first-order coupled system. Next, we establish a box scheme for this first-order system. By the technical energy analysis method, we obtain the prior estimate of the numerical solution for the box scheme. Furthermore, the solvability and convergence are obtained directly from the prior estimate. The extensive numerical examples are carried out, which verify the developed box scheme can achieve global second-order accuracy for both homogeneous and nonhomogeneous Burgers' equations.

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