Domain decomposition-based exponential time differencing methods for semilinear parabolic equations

Li, Xiao; Ju, Lili; Hoang, Thi-Thao-Phuong

doi:10.1007/s10543-020-00817-0

Domain decomposition-based exponential time differencing methods for semilinear parabolic equations

Journal Article · Fri May 01 00:00:00 EDT 2020 · BIT Numerical Mathematics

DOI:https://doi.org/10.1007/s10543-020-00817-0· OSTI ID:1631280

Li, Xiao ^[1]; Ju, Lili ^[2]; Hoang, Thi-Thao-Phuong ^[3]

Hong Kong Polytechnic Univ. (Hong Kong); University of South Carolina
Univ. of South Carolina, Columbia, SC (United States)
Auburn Univ., AL (United States)

The localized exponential time differencing method based on overlapping domain decomposition has been recently introduced and successfully applied to parallel computations for extreme-scale numerical simulations of coarsening dynamics based on phase field models. In this paper, we focus on numerical solutions of a class of semilinear parabolic equations with the well-known Allen-Cahn equation as a special case. We initially study the semi-discrete system under the standard central difference spatial discretization and prove the equivalence between the monodomain problem and the corresponding multidomain problem obtained by the Schwarz waveform relaxation iteration. Then we develop the fully discrete localized exponential time differencing schemes and, by establishing the maximum bound principle, prove the convergence of the fully discrete localized solutions to the exact semi-discrete solution and the convergence of the iterative solutions. Numerical experiments are carried out to confirm the theoretical results in one-dimensional space and test the convergence and accuracy of the proposed algorithms with different numbers of subdomains in two-dimensional space.

View Accepted Manuscript (DOE)

Research Organization:: University of South Carolina, Columbia, SC (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE Office of Science (SC), Biological and Environmental Research (BER). Climate and Environmental Sciences Division; National Natural Science Foundation of China (NSFC); National Science Foundation (NSF)

Grant/Contract Number:: SC0020270; SC0016540

OSTI ID:: 1631280

Alternate ID(s):: OSTI ID: 1852289

Journal Information:: BIT Numerical Mathematics, Journal Name: BIT Numerical Mathematics Vol. 61; ISSN 0006-3835

Publisher:: Springer NatureCopyright Statement

Country of Publication:: United States

Language:: English

References (29)

A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations Gander, Martin J. Numerical Linear Algebra with Applications, Vol. 6, Issue 2 https://doi.org/10.1002/(SICI)1099-1506(199903)6:2<125::AID-NLA152>3.0.CO;2-4	journal	March 1999
Phase transitions and generalized motion by mean curvature Evans, L. C.; Soner, H. M.; Souganidis, P. E. Communications on Pure and Applied Mathematics, Vol. 45, Issue 9 https://doi.org/10.1002/cpa.3160450903	journal	October 1992
SERK2v2: A new second-order stabilized explicit Runge-Kutta method for stiff problems: SERK2v2 Kleefeld, B.; Martín-Vaquero, J. Numerical Methods for Partial Differential Equations, Vol. 29, Issue 1 https://doi.org/10.1002/num.21704	journal	February 2012
Exponential Time Differencing for Stiff Systems Cox, S. M.; Matthews, P. C. Journal of Computational Physics, Vol. 176, Issue 2 https://doi.org/10.1006/jcph.2002.6995	journal	March 2002
High order explicit methods for parabolic equations Medovikov, Alexei A. BIT Numerical Mathematics, Vol. 38, Issue 2 https://doi.org/10.1007/BF02512373	journal	June 1998
Fast Explicit Integration Factor Methods for Semilinear Parabolic Equations Ju, Lili; Zhang, Jian; Zhu, Liyong Journal of Scientific Computing, Vol. 62, Issue 2 https://doi.org/10.1007/s10915-014-9862-9	journal	May 2014
Fast High-Order Compact Exponential Time Differencing Runge–Kutta Methods for Second-Order Semilinear Parabolic Equations Zhu, Liyong; Ju, Lili; Zhao, Weidong Journal of Scientific Computing, Vol. 67, Issue 3 https://doi.org/10.1007/s10915-015-0117-1	journal	October 2015
Nonoverlapping Localized Exponential Time Differencing Methods for Diffusion Problems Hoang, Thi-Thao-Phuong; Ju, Lili; Wang, Zhu Journal of Scientific Computing, Vol. 82, Issue 2 https://doi.org/10.1007/s10915-020-01136-w	journal	February 2020
A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening Allen, Samuel M.; Cahn, John W. Acta Metallurgica, Vol. 27, Issue 6 https://doi.org/10.1016/0001-6160(79)90196-2	journal	June 1979
Efficient implementation of essentially non-oscillatory shock-capturing schemes Shu, Chi-Wang; Osher, Stanley Journal of Computational Physics, Vol. 77, Issue 2 https://doi.org/10.1016/0021-9991(88)90177-5	journal	August 1988
Characteristic exponents and diagonally dominant linear differential systems Lazer, A. C. Journal of Mathematical Analysis and Applications, Vol. 35, Issue 1 https://doi.org/10.1016/0022-247X(71)90246-0	journal	July 1971
Fast and accurate algorithms for simulating coarsening dynamics of Cahn–Hilliard equations Ju, Lili; Zhang, Jian; Du, Qiang Computational Materials Science, Vol. 108 https://doi.org/10.1016/j.commatsci.2015.04.046	journal	October 2015
Thin film epitaxy with or without slope selection Li, Bo; Liu, Jian-Guo European Journal of Applied Mathematics, Vol. 14, Issue 6 https://doi.org/10.1017/S095679250300528X	journal	January 1999
Exponential integrators Hochbruck, Marlis; Ostermann, Alexander Acta Numerica, Vol. 19 https://doi.org/10.1017/S0962492910000048	journal	May 2010
Free Energy of a Nonuniform System. I. Interfacial Free Energy Cahn, John W.; Hilliard, John E. The Journal of Chemical Physics, Vol. 28, Issue 2 https://doi.org/10.1063/1.1744102	journal	February 1958
Analytical inversion of symmetric tridiagonal matrices Hu, G. Y.; O'Connell, R. F. Journal of Physics A: Mathematical and General, Vol. 29, Issue 7 https://doi.org/10.1088/0305-4470/29/7/020	journal	April 1996
Energy stability and error estimates of exponential time differencing schemes for the epitaxial growth model without slope selection Ju, Lili; Li, Xiao; Qiao, Zhonghua Mathematics of Computation, Vol. 87, Issue 312 https://doi.org/10.1090/mcom/3262	journal	September 2017
Modeling Elasticity in Crystal Growth Elder, K. R.; Katakowski, Mark; Haataja, Mikko Physical Review Letters, Vol. 88, Issue 24 https://doi.org/10.1103/PhysRevLett.88.245701	journal	June 2002
Extreme-Scale Phase Field Simulations of Coarsening Dynamics on the Sunway TaihuLight Supercomputer Zhang, Jian; Zhou, Chunbao; Wang, Yangang SC16: International Conference for High Performance Computing, Networking, Storage and Analysis https://doi.org/10.1109/SC.2016.3	conference	November 2016
Explicit Exponential Runge--Kutta Methods for Semilinear Parabolic Problems Hochbruck, Marlis; Ostermann, Alexander SIAM Journal on Numerical Analysis, Vol. 43, Issue 3 https://doi.org/10.1137/040611434	journal	January 2005
Generalized Runge-Kutta Processes for Stable Systems with Large Lipschitz Constants Lawson, J. Douglas SIAM Journal on Numerical Analysis, Vol. 4, Issue 3 https://doi.org/10.1137/0704033	journal	September 1967
Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators Al-Mohy, Awad H.; Higham, Nicholas J. SIAM Journal on Scientific Computing, Vol. 33, Issue 2 https://doi.org/10.1137/100788860	journal	January 2011
Analysis and Approximation of the Ginzburg–Landau Model of Superconductivity Du, Qiang; Gunzburger, Max D.; Peterson, Janet S. SIAM Review, Vol. 34, Issue 1 https://doi.org/10.1137/1034003	journal	March 1992
Strong Stability Preserving Integrating Factor Runge--Kutta Methods Isherwood, Leah; Grant, Zachary J.; Gottlieb, Sigal SIAM Journal on Numerical Analysis, Vol. 56, Issue 6 https://doi.org/10.1137/17M1143290	journal	January 2018
Maximum Principle Preserving Exponential Time Differencing Schemes for the Nonlocal Allen--Cahn Equation Du, Qiang; Ju, Lili; Li, Xiao SIAM Journal on Numerical Analysis, Vol. 57, Issue 2 https://doi.org/10.1137/18M118236X	journal	January 2019
On Krylov Subspace Approximations to the Matrix Exponential Operator Hochbruck, Marlis; Lubich, Christian SIAM Journal on Numerical Analysis, Vol. 34, Issue 5 https://doi.org/10.1137/S0036142995280572	journal	October 1997
Strong Stability-Preserving High-Order Time Discretization Methods Gottlieb, Sigal; Shu, Chi-Wang; Tadmor, Eitan SIAM Review, Vol. 43, Issue 1 https://doi.org/10.1137/S003614450036757X	journal	January 2001
Space-Time Continuous Analysis of Waveform Relaxation for the Heat Equation Gander, Martin J.; Stuart, Andrew M. SIAM Journal on Scientific Computing, Vol. 19, Issue 6 https://doi.org/10.1137/S1064827596305337	journal	November 1998
Overlapping localized exponential time differencing methods for diffusion problems Hoang, Thi-Thao-Phuong; Ju, Lili; Wang, Zhu Communications in Mathematical Sciences, Vol. 16, Issue 6 https://doi.org/10.4310/CMS.2018.v16.n6.a3	journal	January 2018

Similar Records

Overlapping localized exponential time differencing methods for diffusion problems

Journal Article · Wed Feb 06 23:00:00 EST 2019 · Communications in Mathematical Sciences · OSTI ID:1593992

Maximum bound principles for a class of semilinear parabolic equations and exponential time differencing schemes

Journal Article · Fri May 01 00:00:00 EDT 2020 · SIAM Review · OSTI ID:1631279

Nonoverlapping Localized Exponential Time Differencing Methods for Diffusion Problems

Journal Article · Tue Jan 28 23:00:00 EST 2020 · Journal of Scientific Computing · OSTI ID:1594011

Related Subjects

97 MATHEMATICS AND COMPUTING
localized exponential time differencing
overlapping domain decompositio
parallel Schwarz iteration
semilinear parabolic equation

Domain decomposition-based exponential time differencing methods for semilinear parabolic equations

Citation Formats

References (29)

Similar Records

Related Subjects