Peer-reviewed veterinary case report
Numerical method for time-fractional incompressible
By Abidin MZ.·2026·Institute of Basic Mathematics, China·View original on Europe PMC →
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Original publication title: Energy-dissipative adaptive-step L1 discretisation for the Caputo time-fractional incompressible magnetohydrodynamic system.
Plain-English summary
This study looks at a complex system involving fluids and magnetic fields, specifically how they behave over time when certain mathematical techniques are applied. The researchers developed a new method to solve the equations that describe this system, ensuring that the results are accurate and stable. They tested their approach with various simulations and found that it worked well, showing that energy levels decreased as expected and that the method was efficient. Overall, the new technique proved to be effective in modeling the behavior of this fluid-magnetic system.
Abstract
We study the incompressible magnetohydrodynamic system endowed with a Caputo time-fractional derivative of order [Formula: see text]. First, we reformulate the coupled momentum-induction equations by applying a double-curl projection that eliminates both the hydrodynamic pressure and the magnetic pseudo-pressure, producing a divergence-free velocity-magnetic field pair. Next, we then discretise the reformulated problem by combining a divergence-free Fourier spectral approximation in space with a variable-step L1 convolution discretisation in time. Our fully discrete method satisfies a discrete fractional kinetic-magnetic energy inequality, enforces the divergence constraints to machine precision, and reduces to the classical magnetohydrodynamic energy law as [Formula: see text], thereby ensuring asymptotic compatibility. Using discrete orthogonal-complementary convolution identities together with discrete Sobolev embeddings, we derive optimal error bounds of order [Formula: see text] on arbitrarily graded time meshes. Finally, we present numerical experiments, including fractional magnetic Taylor-Green and Orszag-Tang vortices, which confirm the theoretical convergence rates, demonstrate monotone energy decay, and highlight the efficiency of the adaptive time-stepping strategy.
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Search related cases →Original publication on Europe PMC: https://europepmc.org/article/MED/41813788