| Abstract: |
Gradient Flow Models serve as essential mathematical tools for describing evolutionary laws in
nature, with extensive applications in the heat equation, phase-field equations, Ricci flow, surface
minimization problems, and stochastic gradient descent algorithms in machine learning. In numerical
simulations, the rigorous preservation of the original energy dissipation law remains a core challenge
in this field. While classical algorithms-such as stabilization methods, convex splitting, invariant energy
quadraturization methods, scalar auxiliary variable methods, and implicit-explicit/exponentialtime-differencing
Runge-Kutta methods-have achieved significant success for constant-coefficient
models, existing algorithms face limitations in balancing structure preservation and computational
efficiency when dealing with nonlinear variable mobility. To address these challenges, this report proposes
a suite of innovative high-order Product-type Runge-Kutta (P-RK) methods. By investigating
three representative nonlinear models, we demonstrate the superior performance and flexibility of
the proposed methods in structure preservation:
- Dirichlet Harmonic Map Gradient Flow: By constructing appropriate P-RK methods combined
with a normalization post-processing step, we achieve dual structure preservation of the energy
dissipation law and the unit-length constraint (|u| = 1). To the best of our knowledge, this is the
first second-order scheme that maintains both structures while maintaining linear computational
complexity.
- Anisotropic Dendritic Growth Phase-field Model: By introducing P-RK methods with multi-block
Butcher tableaus, we develop an efficient decoupled algorithm that strictly follows the original
energy dissipation structure while significantly enhancing computational efficiency.
- Variable-mobility Allen-Cahn Equation: By constructing appropriate P-RK methods combined
with a cut-off post-processing technique, we achieve unconditional preservation of both the original
energy dissipation and the Maximum Bound Principle (MBP). To overcome the difficulties in
optimal error estimation caused by truncation post-processing, we propose a time two-grid method,
which successfully restores the optimal convergence order.
Both theoretical analysis and numerical experiments demonstrate that the proposed methods possess
significant advantages in structure-preserving properties, numerical stability, and computational
accuracy. This work provides a new theoretical framework and numerical tools for the efficient and
high-performance simulation of complex gradient flow systems. |
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