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Adaptive discontinuous Galerkin finite element methods for the Allen-Cahn equation on polygonal meshes
In this paper, we develop a polygonal mesh adaptation algorithm for a fully implicit scheme based on discontinuous Galerkin (DG) finite element...
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Unconditional energy stability and maximum principle preserving scheme for the Allen-Cahn equation
In this paper, we propose a novel fully implicit numerical scheme that satisfies both nonlinear energy stability and maximum principle for the space...
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A multi-physical structure-preserving method and its analysis for the conservative Allen-Cahn equation with nonlocal constraint
The conservative Allen-Cahn equation satisfies three important physical properties, namely the mass conservation law, the energy dissipation law, and...
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Maximum bound principle for matrix-valued Allen-Cahn equation and integrating factor Runge-Kutta method
The matrix-valued Allen-Cahn (MAC) equation was first introduced as a model problem of finding the stationary points of an energy for orthogonal...
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Up to eighth-order maximum-principle-preserving methods for the Allen–Cahn equation
In this work, we develop a class of up to eighth-order maximum-principle-preserving (MPP) methods for the Allen–Cahn equation. Beginning with the...
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Third-order accurate, large time-step** and maximum-principle-preserving schemes for the Allen-Cahn equation
We present and evaluate several explicit, large time-step** algorithms for the Allen-Cahn equation. Our approach incorporates a stabilization...
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Adaptive Deep Learning Approximation for Allen-Cahn Equation
Solving general non-linear partial differential equations (PDE) precisely and efficiently has been a long-lasting challenge in the field of...
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A new linearized maximum principle preserving and energy stability scheme for the space fractional Allen-Cahn equation
In this paper, a numerical method is proposed to solve the space fractional Allen-Cahn equation. Based on Crank-Nicolson method for time...
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Verification of the Domains Tracking Algorithm for Solving the System of Allen-Cahn and Cahn-Hilliard Equations
The sintering process is widely used in modern industry because it allows for obtaining materials with predefined properties. Chemical or physical...
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Three decoupled, second-order accurate, and energy stable schemes for the conserved Allen–Cahn-type block copolymer (BCP) model
In this paper, the numerical approximations of a new, Allen–Cahn-type block copolymer (BCP) model describing the phase transition of the block...
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Highly efficient schemes for time-fractional Allen-Cahn equation using extended SAV approach
In this paper, we propose and analyze high-order efficient schemes for the time-fractional Allen-Cahn equation. The proposed schemes are based on the...
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Highly efficient variant of SAV approach for two-phase incompressible conservative Allen–Cahn fluids
Herein, we construct efficient linear, totally decoupled, and energy dissipative schemes for two-phase incompressible conservative Allen–Cahn (CAC)...
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On the preserving of the maximum principle and energy stability of high-order implicit-explicit Runge-Kutta schemes for the space-fractional Allen-Cahn equation
We put forward and analyze the high-order (up to fourth) strong stability-preserving implicit-explicit Runge-Kutta schemes for the time integration...
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Second-order maximum principle preserving Strang’s splitting schemes for anisotropic fractional Allen-Cahn equations
In this paper, we exploit the Strang splitting technique for solving the multidimensional Allen-Cahn equations with anisotropic spatial fractional...
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Efficient and structure-preserving time-dependent auxiliary variable method for a conservative Allen–Cahn type surfactant system
In this study, we establish a phase-field two-phase surfactant system using two conservative Allen–Cahn type equations. Two nonlocal Lagrange...
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High-order \(L^{2}\)-bound-preserving Fourier pseudo-spectral schemes for the Allen-Cahn equation
In this paper, we present a class of high-order, large time-step**, and delay-free stabilization schemes for the Allen-Cahn equation. First, we...
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Isogeometric analysis based mesh adaptation for time dependent problems
This article presents a new algorithm designed to create a dynamic r-adaptive mesh within the framework of isogeometric analysis. The approach is...
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Two novel numerical methods for gradient flows: generalizations of the Invariant Energy Quadratization method
In this paper, we conduct an in-depth investigation of the structural intricacies inherent to the Invariant Energy Quadratization (IEQ) method as...
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High order multiquadric trigonometric quasi-interpolation method for solving time-dependent partial differential equations
In this paper, we propose a high order multiquadric trigonometric quasi-interpolation method for function approximation and derivative approximation...
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Investigating Guiding Information for Adaptive Collocation Point Sampling in PINNs
Physics-informed neural networks (PINNs) provide a means of obtaining approximate solutions of partial differential equations and systems through the...