A Fully Discrete Energy-Based Discontinuous Galerkin Method for Variable-Order Time-Fractional Wave Equations
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| Format: | Preprint |
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2026
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| _version_ | 1866917532292612096 |
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| author | Zhang, Lu |
| author_facet | Zhang, Lu |
| contents | Variable-order time-fractional wave equations provide a flexible model for wave phenomena with evolving memory effects and anomalous temporal dynamics. Their numerical approximation is challenging because the variable-order fractional derivative generates time-dependent history weights and therefore lacks the standard time-translation-invariant convolution structure of constant-order fractional operators. In this paper, we develop and analyze a fully discrete energy-based discontinuous Galerkin (DG) method for wave equations with a Caputo-type variable-order time-fractional derivative. The equation is reformulated as a reduced first-order-in-time system, discretized in space by an energy-based DG method, and advanced in time using a second-order approximation of the variable-order Caputo derivative at a specially chosen point in each time interval. The main analytical novelty is a cumulative weight-variation estimate for the variable-order memory weights, which requires only that the variable order $α:[0,T] \rightarrow (0,1)$ be Lipschitz continuous. Based on this estimate, we establish energy stability of the fully discrete scheme and derive second-order temporal convergence together with energy-norm spatial error estimates. The analysis gives suboptimal convergence on general affine simplicial or tensor-product meshes and optimal convergence under additional Cartesian and flux assumptions. Numerical experiments in one and two dimensions validate the theoretical findings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_26054 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Fully Discrete Energy-Based Discontinuous Galerkin Method for Variable-Order Time-Fractional Wave Equations Zhang, Lu Numerical Analysis 35R11, 65M12, 65M15, 65M60 Variable-order time-fractional wave equations provide a flexible model for wave phenomena with evolving memory effects and anomalous temporal dynamics. Their numerical approximation is challenging because the variable-order fractional derivative generates time-dependent history weights and therefore lacks the standard time-translation-invariant convolution structure of constant-order fractional operators. In this paper, we develop and analyze a fully discrete energy-based discontinuous Galerkin (DG) method for wave equations with a Caputo-type variable-order time-fractional derivative. The equation is reformulated as a reduced first-order-in-time system, discretized in space by an energy-based DG method, and advanced in time using a second-order approximation of the variable-order Caputo derivative at a specially chosen point in each time interval. The main analytical novelty is a cumulative weight-variation estimate for the variable-order memory weights, which requires only that the variable order $α:[0,T] \rightarrow (0,1)$ be Lipschitz continuous. Based on this estimate, we establish energy stability of the fully discrete scheme and derive second-order temporal convergence together with energy-norm spatial error estimates. The analysis gives suboptimal convergence on general affine simplicial or tensor-product meshes and optimal convergence under additional Cartesian and flux assumptions. Numerical experiments in one and two dimensions validate the theoretical findings. |
| title | A Fully Discrete Energy-Based Discontinuous Galerkin Method for Variable-Order Time-Fractional Wave Equations |
| topic | Numerical Analysis 35R11, 65M12, 65M15, 65M60 |
| url | https://arxiv.org/abs/2605.26054 |