Advances in Numerical Methods for Stochastic Differential Equations and Phase Field Models

The field of numerical methods for stochastic differential equations and phase field models is experiencing significant growth, with a focus on developing innovative and efficient algorithms for solving complex problems. Recent research has emphasized the importance of structure-preserving schemes, which maintain the underlying properties of the continuous models, such as energy stability and maximum bound principles. Notable progress has been made in the development of high-order temporal parametric finite element methods, Milstein-type schemes for McKean-Vlasov SDEs, and unconditionally stable variable step methods for phase field models. These advancements have the potential to significantly impact the accuracy and efficiency of simulations in various fields, including materials science and fluid dynamics. Noteworthy papers include: The paper on dynamic-stabilization-based linear schemes for the Allen-Cahn equation, which introduces a novel approach to guarantee unconditional preservation of the maximum bound principle and energy stability. The paper on a linear unconditionally structure-preserving L1 scheme for the time-fractional Allen-Cahn equation, which develops and analyzes linear time-stepping schemes that preserve the discrete maximum bound principle and variational energy dissipation law.

Sources

Parameter-related strong convergence rate and polynomial stability of a Euler's type method for time-changed stochastic differential equations

Dynamic-stabilization-based linear schemes for the Allen-Cahn equation with degenerate mobility: MBP and energy stability

A linear unconditionally structure-preserving L1 scheme for the time-fractional Allen-Cahn equation

High-order temporal parametric finite element methods for simulating solid-state dewetting

Strong error analysis and first-order convergence of Milstein-type schemes for McKean-Vlasov SDEs with superlinear coefficients

Unconditionally Stable, Variable Step DLN Methods for the Allen-Cahn Active Fluid Model: A Divergence-free Preserving Approach

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