The high-order maximum-principle-preserving integrating factor Runge-Kutta methods for nonlocal Allen-Cahn equation

Nan, Caixia; Song, Huailing

doi:10.1016/j.jcp.2022.111028

Cited by 11 publications

(4 citation statements)

References 46 publications

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“…It is worth noting that the conditions used in the above theorem are the same as the ones of [13, Theorem 3.1], which indicates that the SRK scheme is MBP-preserving if and only if the IFRK scheme is MBP-preserving with the same coefficients. Hence, one can propose some specific time semidiscrete MBP-preserving SRK schemes up to fourth order by choosing exactly the same coefficients of the MBP-preserving IFRK schemes presented in [13,17].…”

Section: The Time Semidiscrete Mbpmentioning

confidence: 99%

“…Therefore, the MBP is an indispensable tool to study physical features of semilinear parabolic equations, including the aspects of mathematical analysis and numerical simulation. Up to now, great efforts have been made in developing MBP-preserving numerical methods for equations like (1.1), such as the stablized linear semi-implicit method [24,25], the nonlinear second-order method [9,10], the exponential time differencing method [7,8], the integrating factor method [13,16,17], the exponential cut-off method [15,29], and the exponential-SAV method [11,12]. As for the spatial discretizations, a partial list includes the works for finite element method [2,5,15,27,28,30], finite difference method [3,4,26], and finite volume method [21,22].…”

Section: Introductionmentioning

confidence: 99%

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Stochastic Runge-Kutta Methods for Preserving Maximum Bound Principle of Semilinear Parabolic Equations. Part I: Gaussian Quadrature Rule

Yabing Sun,

Weidong Zhao

2024

CSIAM-AM

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In this paper, we propose a class of stochastic Runge-Kutta (SRK) methods for solving semilinear parabolic equations. By using the nonlinear Feynman-Kac formula, we first write the solution of the parabolic equation in the form of the backward stochastic differential equation (BSDE) and then deduce an ordinary differential equation (ODE) containing the conditional expectations with respect to a diffusion process. The time semidiscrete SRK methods are then developed based on the corresponding ODE. Under some reasonable constraints on the time step, we theoretically prove the maximum bound principle (MBP) of the proposed methods and obtain their error estimates. By combining the Gaussian quadrature rule for approximating the conditional expectations, we further propose the first-and second-order fully discrete SRK schemes, which can be written in the matrix form. We also rigorously analyze the MBP-preserving and error estimates of the fully discrete schemes. Some numerical experiments are carried out to verify our theoretical results and to show the efficiency and stability of the proposed schemes.

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Section: The Time Semidiscrete Mbpmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stochastic Runge-Kutta Methods for Preserving Maximum Bound Principle of Semilinear Parabolic Equations. Part I: Gaussian Quadrature Rule

Yabing Sun,

Weidong Zhao

2024

CSIAM-AM

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“…A series of structure-preserving exponential time differencing (ETD) and integrating factor Runge-Kutta (IFRK) methods were also investigated for a class of semilinear parabolic equations in [3,7,19,20,23,26,27,30,31], all of them unconditionally and conditionally preserve the discrete MBP. In recent work [8], Du et al established an abstract framework of MBP investigation for problem (1.1), where sufficient conditions on linear and nonlinear operators are given such that the equation satisfies MBP and the corresponding MBP preserving first-order ETD and second-order ETD Runge-Kutta (ET-DRK) schemes were developed and analyzed.…”

Section: Introductionmentioning

confidence: 99%

A Linear Doubly Stabilized Crank-Nicolson Scheme for the Allen–Cahn Equation with a General Mobility

Dianming Hou,

Lili Ju,

Zhonghua Qiao

2024

AAMM

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In this paper, a linear second order numerical scheme is developed and investigated for the Allen-Cahn equation with a general positive mobility. In particular, our fully discrete scheme is mainly constructed based on the Crank-Nicolson formula for temporal discretization and the central finite difference method for spatial approximation, and two extra stabilizing terms are also introduced for the purpose of improving numerical stability. The proposed scheme is shown to unconditionally preserve the maximum bound principle (MBP) under mild restrictions on the stabilization parameters, which is of practical importance for achieving good accuracy and stability simultaneously. With the help of uniform boundedness of the numerical solutions due to MBP, we then successfully derive H 1 -norm and L ∞ -norm error estimates for the Allen-Cahn equation with a constant and a variable mobility, respectively. Moreover, the energy stability of the proposed scheme is also obtained in the sense that the discrete free energy is uniformly bounded by the one at the initial time plus a constant. Finally, some numerical experiments are carried out to verify the theoretical results and illustrate the performance of the proposed scheme with a time adaptive strategy.

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“…To obtain accurate and stable approximate solutions to the model equation ( 1), various numerical schemes have been proposed, including the stablized semi-implicit method [9,31], fully implicit scheme [26], implicit-explicit scheme [8,27], semi-analytical Fourier spectral method [15], operator splitting scheme [4,16,28], exponential time differencing (ETD) schemes [3,11,13,33] and integrating factor Runge-Kutta (IFRK) methods [12,18,22,32]. Among these methods, exponential integrators [10,14] such as ETD and IFRK schemes have been shown to be more effective when solving problems with strong stiff terms.…”

Section: Introductionmentioning

confidence: 99%

Fully discrete error analysis of first‐order low regularity integrators for the Allen‐Cahn equation

Doan

Hoang

2023

Numerical Methods Partial

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The Allen-Cahn equation satisfies the maximum bound principle, that is, its solution is uniformly bounded for all time by a positive constant under appropriate initial and/or boundary conditions. It has been shown recently that the time-discrete solutions produced by low regularity integrators (LRIs) are likewise bounded in the infinity norm; however, the corresponding fully discrete error analysis is still lacking. This work is concerned with convergence analysis of the fully discrete numerical solutions to the Allen-Cahn equation obtained based on two first-order LRIs in time and the central finite difference method in space. By utilizing some fundamental properties of the fully discrete system and the Duhamel's principle, we prove optimal error estimates of the numerical solutions in time and space while the exact solution is only assumed to be continuous in time. Numerical results are presented to confirm such error estimates and show that the solution obtained by the proposed LRI schemes is more accurate than the classical exponential time differencing (ETD) scheme of the same order.

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The high-order maximum-principle-preserving integrating factor Runge-Kutta methods for nonlocal Allen-Cahn equation

Cited by 11 publications

References 46 publications

Stochastic Runge-Kutta Methods for Preserving Maximum Bound Principle of Semilinear Parabolic Equations. Part I: Gaussian Quadrature Rule

Stochastic Runge-Kutta Methods for Preserving Maximum Bound Principle of Semilinear Parabolic Equations. Part I: Gaussian Quadrature Rule

A Linear Doubly Stabilized Crank-Nicolson Scheme for the Allen–Cahn Equation with a General Mobility

Fully discrete error analysis of first‐order low regularity integrators for the Allen‐Cahn equation

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