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 $\alpha(t)$, 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.

翻译：本文针对变指数二维非线性扩散波方程提出了一种空间双网格紧致差分格式，该方程可用于描述具有变化材料特性的黏弹性介质中机械扩散波的传播。基于卷积方法的思想，首先将扩散波模型转化为等价形式。随后，通过在空间上应用紧致差分近似，并结合时间方向上的平均乘积积分规则与线性插值求积，构建了全离散格式。通过求解粗网格上的小规模非线性系统与细网格上的大规模线性化系统，并采用双三次样条插值算子将粗网格解投影至细网格，构造了一种高效的高阶双网格算法。在对变指数 $\alpha(t)$ 的温和假设下，严格证明了该空间双网格紧致差分格式的稳定性与收敛性。最后通过数值实验验证了所提方法的精度与效率。