Finite Difference Method for Global Stabilization of the Viscous Burgers' Equation with Nonlinear Neumann Boundary Feedback Control

TL;DR

This paper develops a finite difference scheme for the viscous Burgers' equation with nonlinear Neumann boundary feedback control, proving stability and convergence, and validating results through numerical experiments.

Contribution

It introduces a unified $ heta$-scheme for discretization, establishing stability and convergence results for nonlinear boundary control of the Burgers' equation.

Findings

The scheme is conditionally stable for $0 \\leq \theta < 1/2$ and unconditionally stable for $\theta \geq 1/2$.

First-order convergence in various norms for the state variable when $\theta \geq 1/2$.

Numerical experiments confirm theoretical stability and effectiveness of the method.

Abstract

This article focuses on a nonlinear Neumann boundary feedback control formulation for the viscous Burgers' equation and develops a class of finite difference schemes to achieve global stabilization. The proposed procedure, known as the $\theta$-scheme with $\theta \in [0,1]$, unifies explicit and implicit time discretizations and is suitable for handling the nonlinear boundary feedback control problem. Using the discrete energy method, we prove that the proposed difference scheme is conditionally stable for $0 \leq \theta < \frac{1}{2}$ and unconditionally stable for $\theta \geq \frac{1}{2}$. In addition, we establish the exponential stability of the fully discrete solution. The error analysis shows a first-order convergence rate of the state variable in the discrete $L^{2}$-, $H^{1}$-, and $L^{\infty}$-norms for $\theta \geq \frac{1}{2}$, while preserving the exponential stability property. A first-order convergence rate for the boundary control inputs is also obtained. Numerical experiments are conducted to validate the theoretical findings and to demonstrate the effectiveness of the method for the inhomogeneous nonlinear Neumann boundary feedback control of the viscous Burgers' equation.