Dice Question Streamline Icon: https://streamlinehq.com

Stability of the 2D nonlinear compact difference scheme for Burgers-type nonlinearity

Establish a rigorous stability result for the coarse-grid nonlinear compact difference scheme used in the first step of the space-time two-grid compact difference method for the two-dimensional viscous Burgers' equation with periodic boundary conditions, specifically the nonlinear compact difference systems defined by equations (3.12)–(3.16) (coarse grid) and equivalently (3.7)–(3.11) (fine grid formulation), which approximate u_t + u(u_x + u_y) − λΔu = 0. The goal is to derive a stability bound analogous to the established one-dimensional case, overcoming the obstacles posed by the failure of the classical Sobolev embedding in two dimensions.

Information Square Streamline Icon: https://streamlinehq.com

Background

The paper develops a space-time two-grid compact difference scheme (ST-TGCD) for the 2D viscous Burgers' equation, proving unique solvability and unconditional convergence with second-order temporal and fourth-order spatial accuracy. Stability analysis is provided only for the fine-grid correction (step 3), where a perturbation bound is derived.

For the first step (the coarse-grid nonlinear compact difference scheme), an analogous stability proof to the one-dimensional case is obstructed by the failure of the classical Sobolev embedding theorem in two dimensions. The authors note that, to the best of their knowledge, a stability proof for compact difference schemes with 2D Burgers-type nonlinearity is presently unavailable, and they show only boundedness in L2 for the coarse-grid stage.

References

In 2D case, however, the classical Sobolev embedding theorem |v|_\infty \leq \widetilde{C}|v|_1 fails, precluding an analogous stability proof for the first step of ST-TGCD (or NCD) scheme eq3.7-eq3.11. To the best of our knowledge, at present, there is no proof of the stability of the compact difference scheme for PDEs with 2D Burgers’ type nonlinearity. Although we can't prove the stability of the first step of ST-TGCD scheme here, we are able to establish its boundedness in L2-norm.

— An Efficient Space-Time Two-Grid Compact Difference Scheme for the Two-Dimensional Viscous Burgers' Equation (2510.15274 - Peng et al., 17 Oct 2025) in Section 4.3 (Stability)