Numerical Analysis of a Variable-Order Time-Fractional Incompressible Magnetohydrodynamics System

Abstract

We consider an incompressible magnetohydrodynamics (MHD) model in which the classical first-order time derivatives in the momentum and magnetic induction equations are replaced by variable-order Caputo time-fractional derivatives. This formulation allows the memory effect to vary during the evolution and represents a time-fractional generalization of the incompressible MHD system with nonstationary memory. To approximate the problem, we use a fully discrete scheme combining a finite element discretization in space with an L1-type approximation of the variable-order Caputo operators in time. For this discretization, we establish a discrete stability estimate and also derive an auxiliary corrected discrete energy estimate for the fully discrete solution. Convergence is proved by showing that the kernels generated by the variable-order L1 approximation satisfy the assumptions of an abstract discrete fractional Gr\"{o}nwall theorem, which is then applied to the coupled MHD system. The numerical study consists of four parts. First, representative order profiles are used to examine temporal convergence. Second, consistency with the classical incompressible MHD equations is studied as the fractional orders approach one, using norms of solution differences and deviations in kinetic and magnetic energies. Third, the influence of the variable-order fractional terms on nonlinear evolution is investigated through the periodic divergence-free vortex benchmark, with comparisons based on energy and enstrophy histories, divergence errors, Reynolds-number dependence, and time-integrated diagnostics. Fourth, parameter-space maps show how the parameters defining the variable orders affect global indicators. The results show that the variable orders can noticeably affect the evolution of the energy, enstrophy, and current enstrophy even when the Reynolds number is fixed.