Spatial two-grid compact difference scheme for two-dimensional nonlinear diffusion-wave equations with variable exponent

TL;DR

This paper addresses a 2D nonlinear diffusion-wave equation with a time-varying exponent $\alpha(t)\in(1,2)$ by transforming the model to an integro-differential form via a convolution-based approach. A spatial two-grid (STG) compact difference scheme is then developed, combining a compact spatial discretization with a temporal averaged product-integration scheme and a two-grid acceleration that solves a small nonlinear system on a coarse grid and a large linear system on a fine grid using a bicubic-spline projection. The authors prove stability and convergence under mild regularity assumptions on $\alpha(t)$, achieving second-order accuracy in time and fourth-order accuracy in space, and corroborate these results with numerical experiments demonstrating high efficiency and robustness. The work advances high-order, efficient numerical methods for variable-exponent diffusion-wave models, enabling accurate simulations in viscoelastic media with evolving material properties.

Abstract

This paper presents a spatial two-grid (STG) compact difference scheme for a two-dimensional (2D) nonlinear diffusion-wave equation with variable exponent, which describes, e.g., the propagation of mechanical diffusive waves in viscoelastic media with varying material properties. Following the idea of the convolution approach, the diffusion-wave model is first transformed into an equivalent formulation. A fully discrete scheme is then developed by applying a compact difference approximation in space and combining the averaged product integration rule with linear interpolation quadrature in time. An efficient high-order two-grid algorithm is constructed by solving a small-scale nonlinear system on the coarse grid and a large-scale linearized system on the fine grid, where the bicubic spline interpolation operator is used to project coarse-grid solutions to the fine grid. Under mild assumptions on the variable exponent $α(t)$, the stability and convergence of the STG compact difference scheme are rigorously established. Numerical experiments are finally presented to verify the accuracy and efficiency of the proposed method.

Theorems & Definitions (19)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 3.1
• Remark 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• Lemma 3.4