This paper investigated a numerical method for a two-dimensional variable coefficient evolution equation (VCEE), utilizing the alternating direction implicit (ADI) method and an extrapolation formula. The time derivative was discretised by the backward Euler (BE) scheme on a uniform mesh and the finite difference method (FDM) was applied to spatial discretization. We proved an priori estimate and the error bound of the solution to the difference scheme using the energy analysis method, and verified the uniqueness, stability, and convergence of the proposed scheme. To further improve numerical accuracy, we introduced a Richardson extrapolation method, which enhances the global accuracy to fourth order. Finally, some numerical examples were provided to demonstrate the validity of the theoretical analysis.

This paper conducts a study on the alternating direction implicit (ADI) difference schemes for a class of three dimensional hyperbolic equations with constant coefficients. The central difference methods are employed in the temporal and spatial direction. The solvability, stability, and convergence of the proposed ADI schemes are proven. Moreover, the Richardson extrapolation method is established to enhance the accuracy of the algorithm. Numerical examples are presented for the errors and convergence orders of the established ADI schemes and extrapolation schemes. By comparing the results of numerical examples, it can be concluded that the proposed Richardson extrapolation method can effectively improve the accuracy of the numerical solutions and reduce the errors.

This work focuses on exploring pointwise error estimate of three-level conservative difference scheme for supergeneralized viscous Burgers' equation. The cut-off function method plays an important role in constructing difference scheme and presenting numerical analysis. We study the conservative invariant of proposed method, which is energy-preserving for all positive integers $p$ and $q$. Meanwhile, one could apply the discrete energy argument to the rigorous proof that the three-level scheme has unique solution combining the mathematical induction. In addition, we prove the $L_2$-norm and $L_{\mathrm{\infty }}$-norm convergence of proposed scheme in pointwise sense with separate and different ways, which is different from previous work in ^[1]. Numerical results verify the theoretical conclusions.