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

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.