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Maximum bound principle preserving integrating factor Runge–Kutta methods for semilinear parabolic equations

Journal Article · · Journal of Computational Physics
 [1];  [2];  [2];  [3]
  1. Univ. of South Carolina, Columbia, SC (United States); University of South Carolina
  2. Hong Kong Polytechnic Univ., Kowloon (Hong Kong)
  3. Southern Univ. of Science and Technology, Shenzhen (China)
+ Show Author Affiliations

A large class of semilinear parabolic equations satisfy the maximum bound principle (MBP) in the sense that the time-dependent solution preserves for any time a uniform pointwise bound imposed by its initial and boundary conditions. Here, the MBP plays a crucial role in understanding the physical meaning and the well-posedness of the mathematical model. Investigation on numerical algorithms with preservation of the MBP has attracted increasingly attentions in recent years, especially for the temporal discretizations, since the violation of MBP may lead to nonphysical solutions or even blow-ups of the algorithms. In this paper, we study high-order MBP-preserving time integration schemes by means of the integrating factor Runge-Kutta (IFRK) method. Beginning with the space-discrete system of semilinear parabolic equations, we present the IFRK method in general form and derive the sufficient conditions for the method to preserve the MBP. In particular, we show that the classic four-stage, fourth-order IFRK scheme is MBP preserving for some typical semilinear systems although not strong stability preserving, which can be instantly applied to the Allen-Cahn type of equations. To our best knowledge, this is the first time to present a fourth-order linear numerical method preserving the MBP. In addition, convergence of these numerical schemes is proved theoretically and verified numerically, as well as their efficiency by simulations of 2D and 3D long-time evolutional behaviors. Numerical experiments are also carried out for a model which is not a typical gradient flow as the Allen-Cahn type of equations.

Research Organization:
Univ. of South Carolina, Columbia, SC (United States)
Sponsoring Organization:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR). Scientific Discovery through Advanced Computing (SciDAC); USDOE Office of Science (SC), Biological and Environmental Research (BER). Earth and Environmental Systems Science Division; US National Science Foundation; National Natural Science Foundation of China; Hong Kong Research Council
Grant/Contract Number:
SC0020270
OSTI ID:
1785013
Alternate ID(s):
OSTI ID: 23203759
Journal Information:
Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 439; ISSN 0021-9991
Publisher:
ElsevierCopyright Statement
Country of Publication:
United States
Language:
English

References (45)

Phase transitions and generalized motion by mean curvature journal October 1992
Strong Stability-Preserving High-Order Time Discretization Methods journal January 2001
Downwinding for Preserving Strong Stability in Explicit Integrating Factor Runge--Kutta Methods preprint January 2018
A second order explicit finite element scheme to multidimensional conservation laws and its convergence journal September 2000
A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation journal January 2014
Analysis and Applications of the Exponential Time Differencing Schemes and Their Contour Integration Modifications journal June 2005
The logarithmic norm. History and modern theory journal August 2006
Numerical Analysis of Fully Discretized Crank–Nicolson Scheme for Fractional-in-Space Allen–Cahn Equations journal February 2017
Analysis of Fully Discrete Approximations for Dissipative Systems and Application to Time-Dependent Nonlocal Diffusion Problems journal September 2018
A Stabilized Semi-Implicit Euler Gauge-Invariant Method for the Time-Dependent Ginzburg–Landau Equations journal May 2019
Time-Fractional Allen–Cahn Equations: Analysis and Numerical Methods journal November 2020
A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening journal June 1979
Efficient implementation of essentially non-oscillatory shock-capturing schemes journal August 1988
Numerical analysis of a stabilized Crank–Nicolson/Adams–Bashforth finite difference scheme for Allen–Cahn equations journal April 2020
A new second-order maximum-principle preserving finite difference scheme for Allen–Cahn equations with periodic boundary conditions journal June 2020
Hitchhikerʼs guide to the fractional Sobolev spaces journal July 2012
High order integration factor methods for systems with inhomogeneous boundary conditions journal March 2019
An integration factor method for stochastic and stiff reaction–diffusion systems journal August 2015
Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends journal December 2016
Stabilized linear semi-implicit schemes for the nonlocal Cahn–Hilliard equation journal June 2018
A New Two-Constant Equation of State journal February 1976
Numerical approximations of the Ginzburg–Landau models for superconductivity journal September 2005
A class of second order difference approximations for solving space fractional diffusion equations journal January 2015
Total variation diminishing Runge-Kutta schemes journal January 1998
Discrete gauge invariant approximations of a time dependent Ginzburg-Landau model of superconductivity journal July 1998
Energy stability and error estimates of exponential time differencing schemes for the epitaxial growth model without slope selection journal September 2017
A diffusion generated method for orthogonal matrix-valued fields journal September 2019
Comparison principle for some nonlocal problems journal January 1992
Asymptotically compatible discretization of multidimensional nonlocal diffusion models and approximation of nonlocal Green’s functions journal April 2018
Stability Analysis of Large Time‐Stepping Methods for Epitaxial Growth Models journal January 2006
An Energy-Stable and Convergent Finite-Difference Scheme for the Phase Field Crystal Equation journal January 2009
An Adaptive Time-Stepping Strategy for the Molecular Beam Epitaxy Models journal January 2011
Analysis and Approximation of the Ginzburg–Landau Model of Superconductivity journal March 1992
Second-order Convex Splitting Schemes for Gradient Flows with Ehrlich–Schwoebel Type Energy: Application to Thin Film Epitaxy journal January 2012
Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints journal January 2012
Two-Phase Fluid Simulation Using a Diffuse Interface Model with Peng--Robinson Equation of State journal January 2014
Strong Stability Preserving Integrating Factor Runge--Kutta Methods journal January 2018
A New Class of Efficient and Robust Energy Stable Schemes for Gradient Flows journal January 2019
Maximum Principle Preserving Exponential Time Differencing Schemes for the Nonlocal Allen--Cahn Equation journal January 2019
Maximum Bound Principles for a Class of Semilinear Parabolic Equations and Exponential Time-Differencing Schemes journal January 2021
Strong Stability-Preserving High-Order Time Discretization Methods journal January 2001
Maximum Principles for Discrete and Semidiscrete Reaction-Diffusion Equation journal January 2015
Numerical approximations of Allen-Cahn and Cahn-Hilliard equations journal June 2010
Stabilized Crank-Nicolson/Adams-Bashforth Schemes for Phase Field Models journal February 2013
On the maximum principle preserving schemes for the generalized Allen–Cahn equation journal January 2016

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Related Subjects

97 MATHEMATICS AND COMPUTING
Allen-Cahn equations
High-order numerical methods
Integrating factor Runge-Kutta method
Maximum bound principle
Semilinear parabolic equation