Inexact shift invert Arnoldi method for evolution equations

Linear and nonlinear evolution equations with a first order time derivative, such as the heat equation, the Burgers equation, and the reaction diffusion equation have been used to solve problems in various fields of science. Differential algebraic equations of the first order are derived after space discretization. In the simplest case, the computation of one matrix exponential with a special form is required. In the most complex case, the computation of matrix functions related to its exponentials need to be implemented repeatedly. When computing large matrix functions, the Krylov subspace methods is a viable alternative. The most well known method is the Arnoldi method, but it may require a number of iterations depending on the condition of the matrix. As a solution to this issue, we propose the Inexact Shift invert Arnoldi method to do this more efficiently. As the result, the numerical solution of evolution equations can be computed efficiently with this method. Pertinent numerical experiments establish the effectiveness of this proposed algorithm. References Benzi, M. and Boito, P. Decay properties for functions of matrices over (C^ algebras. Linear Algebra and its Applications 456(1) 174 198, 2014. doi:10.1016/j.laa.2013.11.027 Butcher, J. C. Numerical methods for ordinary differential equations, second edition John Wiley and Sons, Chichester, England, 2008. Caratheodory, C. Uber die gegenseitige Beziehung der Rander bei der konformen Abbildung des Inneren einer Jordanschen Kurve auf einen Kreis.  Carra, E. J. Turner, I. W. and Perre, P. A variable stepsize Jacobian free exponential integrator for simulating transport in heterogeneous porous media: Application to wood drying. Journal of Computational Physics 233: 66 82, 2013. doi:10.1016/j.jcp.2012.07.024 Gang, W. Feng, T. and Yimin, W. An inexact shift and invert Arnoldi algorithm for Toeplitz matrix exponential. Numerical Linear Algebra with Applications 22(4) 777 792, 2015. doi:10.1002/nla.1992 Gallopoulos, E. and Saad, Y. Efficient solution of parabolic equations by Krylov approximation methods.  1992. doi:10.1137/0913071 Gockler, T. Rational Krylov subspace methods for \phi functions in exponential integrators. em Karlsruher Instituts fur Technologie, 2014, Ph.D. thesis. http:/d nb.info/1060425408/34 Guttel, S. Rational Krylov methods for operator functions. em Technischen Universitat Bergakademie Freiberg, 2010, Ph.D. thesis. http:/www.qucosa.de/fileadmin/data/qucosa/documents/2764/diss_guettel.pdf Hashimoto, Y. and Nodera, T. Inexact shift invert Arnoldi method for linear evolution equations (Japanese) IPSJ Journal 57(10) 2250 2259, 2016. https:/ipsj.ixsq.nii.ac.jp/ej/?action=pages_view_main&active_action=repository_view_main_item_detail&item_id=175055&item_no=1&page_id=13&block_id=8 Higham, N. J. The scaling and squaring method for the matrix exponential revisited. ) 1179 1193, 2005. doi:10.1137/04061101X Hochbruck, M. A short course on exponential integrators. Series in Contemporary Applied Mathematics 17: 29 49, 2015. http:/www.siam.org/students/g2s3/2013/lecturers/Hochbruck/Summary_Hochbruck.pdf Hochbruck, M. and Ostermann, A. Exponential Runge Kutta methods for parabolic problems. Applied Numerical Mathematics 53(2 4) 323 339, 2005.  Hochbruck, M. and Ostermann, A. Exponential integrators.  , M. and Lubich, C. On Krylov subspace approximations to the matrix exponential Operator.  1911 1925, 1997.  Exponential integrators for large systems of differential equations.  Hongqing, Z. Huazhong, S. and Meiyu, D. Numerical solutions of two dimensional Burgers' equations by discrete Adomian decomposition method. Computers and Mathematics with Applications 60(3) 840 848, 2010.  Kamel, A. K. Numerical study of Fisher's reaction diffusion equation by the Sinc collocation method. Journal of Computational and Applied Mathematics 137(2) 245 255, 2001. doi:10.1016/S0377 0427(01)00356 9 Lee, S. Pang, H. and Sun, H. Shift invert Arnoldi approximation to the Toeplitz matrix exponential.  2010. doi:10.1137/090758064 Moret, I. and Novati, P. RD rational approximations of the matrix exponential. BIT Numerical Mathematics 44(3) 595 615, 2004. BITN.0000046805.27551.3b Moler, C. and Van Loan, C. F. Nineteen dubious ways to compute the exponential of a matrix, Twenty Five Years Later.  2003.  Using the restricted denominator rational Arnoldi method for exponential Integrators.  2011. doi:10.1137/100814202 Svoboda, Z. The convective diffusion equation and its use in building physics.  2000.  A fast and smoothly converging variant of Bi CG for the solution of nonsymmetric linear systems.