Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations

Lili Ju; Xinfeng Liu; Wei Leng

doi:10.3934/dcdsb.2014.19.1667

Article Contents

2014, Volume 19, Issue 6: 1667-1687. Doi: 10.3934/dcdsb.2014.19.1667

This issue Previous Article The linear hyperbolic initial and boundary value problems in a domain with corners Next Article Numerical study of blow-up in solutions to generalized Kadomtsev-Petviashvili equations

Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations

1.
Department of Mathematics, University of South Carolina, Columbia, SC 29208
2.
State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences, Beijing, 100190

Received: November 2013

Revised: January 2014

Published: August 2014

Abstract / Introduction Related Papers Cited by

Abstract

When developing efficient numerical methods for solving parabolic types of equations, severe temporal stability constraints on the time step are often required due to the high-order spatial derivatives and/or stiff reactions. The implicit integration factor (IIF) method, which treats spatial derivative terms explicitly and reaction terms implicitly, can provide excellent stability properties in time with nice accuracy. One major challenge for the IIF is the storage and calculation of the dense exponentials of the sparse discretization matrices resulted from the linear differential operators. The compact representation of the IIF (cIIF) can overcome this shortcoming and greatly save computational cost and storage. On the other hand, the cIIF is often hard to be directly applied to deal with problems involving cross derivatives. In this paper, by treating the discretization matrices in diagonalized forms, we develop an efficient cIIF method for solving a family of semilinear fourth-order parabolic equations, in which the bi-Laplace operator is explicitly handled and the computational cost and storage remain the same as to the classic cIIF for second-order problems. In particular, the proposed method can deal with not only stiff nonlinear reaction terms but also various types of homogeneous or inhomogeneous boundary conditions. Numerical experiments are finally presented to demonstrate effectiveness and accuracy of the proposed method.

Keywords:

Integration factor method,
compact representation,
implicit scheme,
bi-Laplace operator,
semilinear fourth-order parabolic equations.

Mathematics Subject Classification: 65M06, 35K25.

Citation:

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