View PDF HTML (experimental)

Abstract: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 $\alpha:[0,T] \rightarrow (0,1)$ 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.

Submission history

From: Lu Zhang [view email]
[v1] Mon, 25 May 2026 17:15:14 UTC (1,059 KB)