Numerical analysis and simulation of the compact difference scheme for the pseudo-parabolic Burgers' equation

In this paper, we analyzed and tested a nonlinear implicit compact finite difference scheme for the pseudo-parabolic Burgers' equation. The discrete conservation laws and boundedness of the scheme were rigorously established. We then proved the unique solvability of the numerical scheme by reformulating it as an equivalent system. Furthermore, using the energy method, we derived an error estimate for the proposed scheme, achieving a convergence order of $\mathcal{O}({\tau }^2+h^4)$ under the discrete $L^{\mathrm{\infty }}$-norm. The stability of the compact finite difference scheme was subsequently proven using a similar approach. Finally, a series of numerical experiments were performed to validate the theoretical findings.