Highly accurate numerical solution of time-fractional wave propagation using order-dependent enrichments

We develop a high-accuracy numerical solver for wave propagation problems governed by Caputo time-fractional derivatives using a fractional partition of unity finite element method. The time integration is achieved by implementing a quadratic spline scheme for the Caputo fractional derivative, providing accurate and stable numerical differentiation. For the spatial discretization, the finite element space is enriched with plane waves incorporating fractional orders in both directional and radial forms, enabling an efficient representation of oscillatory solutions. The proposed approach is designed to accurately resolve time-fractional wave phenomena on coarse meshes using low-order polynomial shape functions, even at high wavenumbers, while substantially reducing the number of degrees of freedom compared with standard finite element methods. The enrichment functions are designed to capture the intrinsic fractional behavior of the solution, leading to enhanced accuracy in highly oscillatory regimes. The performance of the method is first assessed using fractional ordinary differential equations with known analytical solutions, demonstrating favorable accuracy and computational efficiency relative to existing numerical schemes. A systematic comparison with the conventional finite element method further confirms the superior accuracy of the proposed partition of unity finite element solver for time-fractional wave equations at high wavenumbers. Finally, the methodology is applied to the propagation of a Hanning-windowed wave, illustrating its effectiveness in time-domain simulations. The proposed fractional partition of unity finite element method delivers high accuracy and computational efficiency, highlighting its strong potential for large-scale simulations of time-fractional wave propagation in high-frequency regimes.