August  2014, 19(6): 1667-1687. doi: 10.3934/dcdsb.2014.19.1667

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, United States

2. 

State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences, Beijing, 100190, China

Received  November 2013 Revised  January 2014 Published  June 2014

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.
Citation: Lili Ju, Xinfeng Liu, Wei Leng. Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1667-1687. doi: 10.3934/dcdsb.2014.19.1667
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X.-D. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes,, Journal of Computational Physics, 115 (1994), 200.  doi: 10.1006/jcph.1994.1187.  Google Scholar

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Q. Nie, F. Wan, Y.-T. Zhang and X. Liu, Compact integration factor methods in high spatial dimensions,, Journal of Computational Physics, 277 (2008), 5238.  doi: 10.1016/j.jcp.2008.01.050.  Google Scholar

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Q. Nie, Y.-T. Zhang and R. Zhao, Efficient semi-implicit schemes for stiff systems,, Journal of Computational Physics, 214 (2006), 521.  doi: 10.1016/j.jcp.2005.09.030.  Google Scholar

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Y. Saad, Analysis of some krylov subspace approximations to the matrix exponential operator,, SIAM Journal on Numerical Analysis, 29 (1992), 209.  doi: 10.1137/0729014.  Google Scholar

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J. Shen and H. Yu, Efficient spectral sparse grid methods and applications to high-dimensional elliptic problems,, SIAM Journal on Scientific Computing, 32 (2010), 3228.  doi: 10.1137/100787842.  Google Scholar

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A. Wiegmann, Fast Poisson, Fast Helmholtz and Fast Linear Elastostatic Solvers on Rectangular Parallelepipeds,, Lawrence Berkeley National Laboratory, (1999).  doi: 10.2172/982430.  Google Scholar

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S. Zhao, J. Ovadia, X. Liu, Y. Zhang and Q. Nie, Operator splitting implicit integration factor methods for stiff reaction-diffusion-advection systems,, Journal of Computational Physics, 230 (2011), 5996.  doi: 10.1016/j.jcp.2011.04.009.  Google Scholar

show all references

References:
[1]

M. Berger and P. Colella, Local adaptive mesh refinement for shock hydrodynamics,, Journal of Computational Physics, 82 (1989), 64.  doi: 10.1016/0021-9991(89)90035-1.  Google Scholar

[2]

M. Berger and J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations,, Journal of Computational Physics, 53 (1984), 484.  doi: 10.1016/0021-9991(84)90073-1.  Google Scholar

[3]

E. O. Brigham, The Fast Fourier Transform and its Applications,, Prentice Hall, (1988).   Google Scholar

[4]

S. Chen and Y.-T. Zhang, Krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes: Application to discontinuous Galerkin methods,, Journal of Computational Physics, 230 (2011), 4336.  doi: 10.1016/j.jcp.2011.01.010.  Google Scholar

[5]

S. M. Cox and P. C. Matthews, Exponential time differencing for stiff systems,, Journal of Computational Physics, 176 (2002), 430.  doi: 10.1006/jcph.2002.6995.  Google Scholar

[6]

Q. Du and W. Zhu, Stability analysis and applications of the exponential time differencing schemes,, Journal of Computational Mathematics, 22 (2004), 200.   Google Scholar

[7]

Q. Du and W. Zhu, Modified exponential time differencing schemes: Analysis and applications,, BIT Numerical Mathematics, 45 (2005), 307.  doi: 10.1007/s10543-005-7141-8.  Google Scholar

[8]

R. Eymard, T. Gallouët and R. Herbin, Finite volume methods,, Handbook of Numerical Analysis, 7 (2000), 713.   Google Scholar

[9]

B. Gustafsson, H.-O. Kreiss and J. Oliger, Time Dependent Problems and Difference Methods, volume 67., Wiley New York, (1995).   Google Scholar

[10]

M. Hochbruck and C. Lubich, On krylov subspace approximations to the matrix exponential operator,, SIAM Journal on Numerical Analysis, 34 (1997), 1911.  doi: 10.1137/S0036142995280572.  Google Scholar

[11]

A. Jameson, W. Schmidt and E. Turkel, Numerical Solutions of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes,, The 14th AIAA Fluid and Plasma Dynamics Conference, (1981).  doi: 10.2514/6.1981-1259.  Google Scholar

[12]

G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes,, Journal of Computational Physics, 126 (1996), 202.  doi: 10.1006/jcph.1996.0130.  Google Scholar

[13]

L. Ju, J. Zhang, L. Zhu and Q. Du, Fast Explicit Integration Factor Methods for Semilinear Parabolic Equations,, Journal of Scientific Computing, (2014).  doi: 10.1007/s10915-014-9862-9.  Google Scholar

[14]

A.-K. Kassam and L. N. Trefethen, Fourth-order time stepping for stiff PDEs,, SIAM Journal on Scientific Computing, 26 (2005), 1214.  doi: 10.1137/S1064827502410633.  Google Scholar

[15]

B. Kleefeld, A. Khaliq and B. Wade, An ETD Crank-Nicolson method for reaction-diffusion systems,, Numerical Methods for Partial Differential Equations, 28 (2012), 1309.  doi: 10.1002/num.20682.  Google Scholar

[16]

S. Krogstad, Generalized integrating factor methods for stiff PDEs,, Journal of Computational Physics, 203 (2005), 72.  doi: 10.1016/j.jcp.2004.08.006.  Google Scholar

[17]

R. LeVeque, Numerical Methods for Conservation Laws,, Birkhauser, (1992).  doi: 10.1007/978-3-0348-8629-1.  Google Scholar

[18]

X. Liu and Q. Nie, Compact integration factor methods for complex domains and adaptive mesh refinement,, Journal of computational physics, 229 (2010), 5692.  doi: 10.1016/j.jcp.2010.04.003.  Google Scholar

[19]

X.-D. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes,, Journal of Computational Physics, 115 (1994), 200.  doi: 10.1006/jcph.1994.1187.  Google Scholar

[20]

Q. Nie, F. Wan, Y.-T. Zhang and X. Liu, Compact integration factor methods in high spatial dimensions,, Journal of Computational Physics, 277 (2008), 5238.  doi: 10.1016/j.jcp.2008.01.050.  Google Scholar

[21]

Q. Nie, Y.-T. Zhang and R. Zhao, Efficient semi-implicit schemes for stiff systems,, Journal of Computational Physics, 214 (2006), 521.  doi: 10.1016/j.jcp.2005.09.030.  Google Scholar

[22]

Y. Saad, Analysis of some krylov subspace approximations to the matrix exponential operator,, SIAM Journal on Numerical Analysis, 29 (1992), 209.  doi: 10.1137/0729014.  Google Scholar

[23]

J. Shen and H. Yu, Efficient spectral sparse grid methods and applications to high-dimensional elliptic problems,, SIAM Journal on Scientific Computing, 32 (2010), 3228.  doi: 10.1137/100787842.  Google Scholar

[24]

C. Van Loan, Computational Frameworks for the Fast Fourier Transform, volume 10., SIAM, (1992).  doi: 10.1137/1.9781611970999.  Google Scholar

[25]

A. Wiegmann, Fast Poisson, Fast Helmholtz and Fast Linear Elastostatic Solvers on Rectangular Parallelepipeds,, Lawrence Berkeley National Laboratory, (1999).  doi: 10.2172/982430.  Google Scholar

[26]

S. Zhao, J. Ovadia, X. Liu, Y. Zhang and Q. Nie, Operator splitting implicit integration factor methods for stiff reaction-diffusion-advection systems,, Journal of Computational Physics, 230 (2011), 5996.  doi: 10.1016/j.jcp.2011.04.009.  Google Scholar

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