# American Institute of Mathematical Sciences

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
##### 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. [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. [3] E. O. Brigham, The Fast Fourier Transform and its Applications,, Prentice Hall, (1988). [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. [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. [6] Q. Du and W. Zhu, Stability analysis and applications of the exponential time differencing schemes,, Journal of Computational Mathematics, 22 (2004), 200. [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. [8] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods,, Handbook of Numerical Analysis, 7 (2000), 713. [9] B. Gustafsson, H.-O. Kreiss and J. Oliger, Time Dependent Problems and Difference Methods, volume 67., Wiley New York, (1995). [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. [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. [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. [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. [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. [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. [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. [17] R. LeVeque, Numerical Methods for Conservation Laws,, Birkhauser, (1992). doi: 10.1007/978-3-0348-8629-1. [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. [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. [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. [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. [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. [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. [24] C. Van Loan, Computational Frameworks for the Fast Fourier Transform, volume 10., SIAM, (1992). doi: 10.1137/1.9781611970999. [25] A. Wiegmann, Fast Poisson, Fast Helmholtz and Fast Linear Elastostatic Solvers on Rectangular Parallelepipeds,, Lawrence Berkeley National Laboratory, (1999). doi: 10.2172/982430. [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.

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##### 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. [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. [3] E. O. Brigham, The Fast Fourier Transform and its Applications,, Prentice Hall, (1988). [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. [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. [6] Q. Du and W. Zhu, Stability analysis and applications of the exponential time differencing schemes,, Journal of Computational Mathematics, 22 (2004), 200. [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. [8] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods,, Handbook of Numerical Analysis, 7 (2000), 713. [9] B. Gustafsson, H.-O. Kreiss and J. Oliger, Time Dependent Problems and Difference Methods, volume 67., Wiley New York, (1995). [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. [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. [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. [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. [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. [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. [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. [17] R. LeVeque, Numerical Methods for Conservation Laws,, Birkhauser, (1992). doi: 10.1007/978-3-0348-8629-1. [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. [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. [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. [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. [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. [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. [24] C. Van Loan, Computational Frameworks for the Fast Fourier Transform, volume 10., SIAM, (1992). doi: 10.1137/1.9781611970999. [25] A. Wiegmann, Fast Poisson, Fast Helmholtz and Fast Linear Elastostatic Solvers on Rectangular Parallelepipeds,, Lawrence Berkeley National Laboratory, (1999). doi: 10.2172/982430. [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.
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