Spatial two-grid compact difference scheme for two-dimensional nonlinear diffusion-wave equations with variable exponent
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| Format: | Preprint |
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2025
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| _version_ | 1866917011634782208 |
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| author | Zhang, Hao Li, Kexin Qiu, Wenlin |
| author_facet | Zhang, Hao Li, Kexin Qiu, Wenlin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_12188 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spatial two-grid compact difference scheme for two-dimensional nonlinear diffusion-wave equations with variable exponent Zhang, Hao Li, Kexin Qiu, Wenlin Numerical Analysis 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. |
| title | Spatial two-grid compact difference scheme for two-dimensional nonlinear diffusion-wave equations with variable exponent |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2510.12188 |