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, stability is obtained using corrected discrete memory functionals adapted to the variable-order fractional terms. Convergence is proved by showing that the kernels generated by the variable-order L1 approximation satisfy the assumptions of an abstract discrete fractional Grönwall 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 in the limit as the fractional orders approach one, using norms of solution differences together with 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 changes in time-integrated diagnostics. Fourth, parameter-space maps are used to show how the parameters defining the variable orders affect global indicators. The results show that the variable fractional orders can noticeably affect the evolution of the energy, enstrophy, and current enstrophy even when the Reynolds number is fixed.

• The paper proposes a variable-order time-fractional modification of incompressible MHD equations to capture nonstationary memory effects.
• A fully discrete finite element–L1 scheme with divergence stabilization is employed, achieving first-order temporal and second-order spatial convergence.
• Numerical experiments reveal that variable-order profiles significantly affect kinetic and magnetic diagnostics, supporting applications in flow control and parameter identification.

Numerical Methods for Variable-Order Time-Fractional Incompressible MHD Systems

Problem Formulation

The paper formulates a variable-order time-fractional generalization of the incompressible magnetohydrodynamics (MHD) equations. Classical first-order time derivatives in the momentum and induction equations are replaced by Caputo derivatives with time-dependent orders $\alpha(t)$ and $\beta(t)$ in $(0,1)$. This replacement models nonstationary memory effects and anomalous diffusion in MHD flows, with the order functions encoding the strength and evolution of memory phenomena. Dimensionless variables are introduced via scaling, and spatial Sobolev spaces are used to define variational formulations. The weak forms retain the divergence-free constraints for velocity and magnetic field but, at the discrete level, incorporate stabilization terms to penalize divergence errors.

Discretization and Numerical Scheme

Spatial discretization employs conforming finite element spaces for velocity, pressure, and magnetic field—specifically Taylor-Hood $P_2/P_1$ elements for velocity-pressure and $P_2$ for magnetic field. Temporal discretization uses a variable-order L1-type scheme for the Caputo derivative, where the temporal kernel coefficients depend on the current fractional order. Corrected discrete memory functionals are introduced to ensure stability under non-monotone variable-order profiles. The coupled nonlinear discrete system is solved fully implicitly at each time-step via Picard iterations, with a block preconditioner applied to the monolithic velocity-pressure-magnetic field system. Magnetic divergence cleaning uses an auxiliary elliptic problem for post-step correction.

Analysis: Stability and Convergence

Rigorous analysis establishes stability and convergence of the finite element–L1 scheme even when fractional orders vary in time and are not monotone. The key technical challenge is extending fixed-order theory to variable-order kernels; coefficients of the L1 scheme must satisfy the discrete fractional Grönwall lemma assumptions [Liao_2019]. Stability is proven by bounding a corrected discrete memory functional, and convergence is demonstrated via a decomposition of the error into projection and discrete components. The optimal error estimate $$\|u^n-u_h^n\| + \|B^n-B_h^n\| \leq C(h^{\min\{l,m\}} + \tau)$$ is achieved under appropriate smoothness and step-size restrictions, with $l$, $m$ the finite element approximation orders.

Numerical Experiments: Convergence, Consistency, and Variable Order Effects

Temporal and Spatial Convergence

Systematic refinement studies for various representative variable-order profiles (linear ramp, periodic, smooth transition) demonstrate first-order temporal and second-order spatial convergence. Divergence norms for both velocity and magnetic field remain small, consistent with $H^1$-conforming discretization and stabilization.

Classical Limit: Recovery of Standard MHD

As the fractional orders approach unity, numerical experiments confirm that solutions and diagnostic norms (kinetic/magnetic energy differences) converge to those of the classical incompressible MHD system. Discrepancies decrease monotonically as the order functions approach $1$, with errors dropping to $\beta(t)$0 or lower in both state variables and energies.

Variable Order Impact on MHD Dynamics

Periodic divergence-free vortex benchmarks are used to probe the effect of different variable-order profiles (constant, ramp, step, sinusoidal, smooth step, asynchronous/unsymmetric). Magnetic diagnostics ($\beta(t)$1, $\beta(t)$2) are strongly sensitive to the order profile: variable orders cause significant reductions in amplitude and alter decay rates relative to the classical case. Kinetic diagnostics ($\beta(t)$3, $\beta(t)$4) reveal both accelerated dissipation and altered transient development in fractional cases, with late-time behavior diverging from classical profiles. Asynchronous evolution of $\beta(t)$5, $\beta(t)$6 produces distinct trends in kinetic and magnetic deviations, showing substantial impact on global diagnostic integrals.

Parametric and Diagnostic Mapping

Phase diagrams constructed as heatmaps in the $\beta(t)$7 plane (for linear ramps) allow fine-grained quantification of the influence of fractional order range on diagnostic quantities. The magnetic energy and current enstrophy show the largest sensitivity; their time-integrated values decrease as fractional orders move away from unity. Kinetic energy is less sensitive to order range but still exhibits measurable variation.

Practical and Theoretical Implications

The results substantiate that variable-order time-fractional modeling in incompressible MHD provides a tractable mechanism for representing nonstationary temporal memory—relevant for laboratory, industrial, and geophysical settings where diffusion and relaxation rates are not constant. The robust finite element–L1 discretization, with divergence stabilization and rigorous theoretical support, enables faithful simulation and analysis. Variable-order effects on dissipation, energy, and enstrophy measures suggest applications in parameter identification, flow control, and materials processing regimes sensitive to memory effects. The theoretical framework accommodates both constant and non-monotone variable order, with extension potential to spatially-dependent orders, distributed order, or more general MHD settings.

Potential future developments include exploration of spatially-variable or solution-dependent fractional orders, connection to experimental data and turbulence modeling, enhancement of solver efficiency for long-time simulation, and adaptation to three-dimensional and multi-physics scenarios. There is scope for deeper analysis of well-posedness and regularity in variable order regimes, as well as comparative study with distributed-order and alternative nonlocal operators.

Conclusion

This work develops and analyzes a fully discrete finite element–L1 scheme for the variable-order time-fractional incompressible MHD equations, proves stability and convergence for general order profiles, and conducts comprehensive numerical studies demonstrating consistency, sensitivity, and parameter effects. These results confirm the viability of variable-order fractional calculus for advanced modeling of incompressible MHD, with strong theoretical and computational assurances and broad applicability for systems exhibiting temporally varying memory or anomalous diffusion (2604.24030).