An Efficient Space-Time Two-Grid Compact Difference Scheme for the Two-Dimensional Viscous Burgers' Equation

TL;DR

This work develops a space-time two-grid compact difference method for the 2D viscous Burgers' equation, combining a coarse-grid nonlinear solve, time-space interpolation, and a fine-grid linearized correction to achieve second-order accuracy in time and fourth-order accuracy in space. The method reduces computational cost by solving a nonlinear system on a coarse grid and refining the solution on a fine grid via a linearized correction, with rigorous analysis proving unique solvability and unconditional convergence. Numerical experiments demonstrate that ST-TGCD achieves comparable accuracy to a traditional nonlinear compact scheme while delivering substantial CPU-time reductions (over 70%), validating its efficiency and robustness. The framework shows promise as a high-efficiency alternative for nonlinear convection-diffusion problems and can be extended to related PDEs requiring compact spatial discretization.

Abstract

This work proposes an efficient space-time two-grid compact difference (ST-TGCD) scheme for solving the two-dimensional (2D) viscous Burgers' equation subject to initial and periodic boundary conditions. The proposed approach combines a compact finite difference discretization with a two-grid strategy to achieve high computational efficiency without sacrificing accuracy. In the coarse-grid stage, a fixed-point iteration is employed to handle the nonlinear system, while in the fine-grid stage, linear temporal and cubic spatial Lagrange interpolations are used to construct initial approximations. The final fine-grid solution is refined through a carefully designed linearized correction scheme. Rigorous analysis establishes unconditional convergence of the method, demonstrating second-order accuracy in time and fourth-order accuracy in space. Numerical experiments verify the theoretical results and show that the ST-TGCD scheme reduces CPU time by more than 70\% compared with the traditional nonlinear compact difference (NCD) method, while maintaining comparable accuracy. These findings confirm the proposed scheme as a highly efficient alternative to conventional nonlinear approaches.

Figures (4)

• Figure 1: Spatial two-grid computing process, where we suppose $L_1=L_2,~M_1^c=M_2^c=4$, and the spatial step-size ratio $k_h=2$.
• Figure 2: The spatial convergence orders of ST-TGCD scheme for different spatiotemporal step-size ratios $k_h$ and $k_\tau$, but fixed $N^c=128$, $\lambda=1$.
• Figure 3: Comparison of CPU time cost between ST-TGCD scheme and NCD scheme, fixed $M^f=80$, $\lambda=1$, and space-time step-size ratios $k_h=2$ and $k_\tau=3$.
• Figure 4: (a) is the spatial convergence orders of ST-TGCD scheme for different space-time step-size ratios $k_h$ and $k_\tau$, but fixed $N^c=128$, $\lambda=0.5$. (b) is the comparison of CPU time cost between ST-TGCD scheme and NCD scheme, fixed $M^f=60$, $\lambda=0.1$, and spatiotemporal step-size ratios $k_h=2$ and $k_\tau=3$.

Theorems & Definitions (24)

• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• Lemma 2.7
• Remark 2.1