Fast numerical solution for fractional diffusion equations by exponential quadrature rule

After spatial discretization to the fractional diffusion equation by the shifted Grunwald formula, it leads to a system of ordinary differential equations, where the resulting coefficient matrix possesses the Toeplitz-like structure. An exponential quadrature rule is employed to solve such a system of ordinary differential equations. The convergence by the proposed method is theoretically studied. In practical computation, the product of a Toeplitz-like matrix exponential and a vector is calculated by the shift-invert Arnoldi method. Meanwhile, the coefficient matrix satisfies a condition that guarantees the fast approximation by the shift-invert Arnoldi method. Numerical results are given to demonstrate the efficiency of the proposed method. The matrix of the spatial discretization of the fractional diffusion equation is Toeplitz-like.An exponential quadrature rule is employed to solve the system of ordinary differential equations.The Toeplitz-like matrix exponential is calculated by the shift-invert Arnoldi method.The coefficient matrix satisfies a condition that guarantees the fast approximation by the shift-invert Arnoldi method.Numerical results show the efficiency of the exponential quadrature rule.

[1]  Weihua Deng,et al.  Finite Element Method for the Space and Time Fractional Fokker-Planck Equation , 2008, SIAM J. Numer. Anal..

[2]  Hai-Wei Sun,et al.  Fast Exponential Time Integration for Pricing Options in Stochastic Volatility Jump Diffusion Models , 2014 .

[3]  Hai-Wei Sun,et al.  Shift‐invert Lanczos method for the symmetric positive semidefinite Toeplitz matrix exponential , 2011, Numer. Linear Algebra Appl..

[4]  Siu-Long Lei,et al.  A circulant preconditioner for fractional diffusion equations , 2013, J. Comput. Phys..

[5]  Weihua Deng,et al.  Fourth Order Difference Approximations for Space Riemann-Liouville Derivatives Based on Weighted and Shifted Lubich Difference Operators , 2014 .

[6]  D. Benson,et al.  The fractional‐order governing equation of Lévy Motion , 2000 .

[7]  Hai-Wei Sun,et al.  Multigrid method for fractional diffusion equations , 2012, J. Comput. Phys..

[8]  Mihály Kovács,et al.  Numerical solutions for fractional reaction-diffusion equations , 2008, Comput. Math. Appl..

[9]  Xiao-Qing Jin,et al.  Developments and Applications of Block Toeplitz Iterative Solvers , 2003 .

[10]  Jian Bai,et al.  Fractional-Order Anisotropic Diffusion for Image Denoising , 2007, IEEE Transactions on Image Processing.

[11]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[12]  I. Moret,et al.  RD-Rational Approximations of the Matrix Exponential , 2004 .

[13]  I. Podlubny.  Fractional differential equations , 1998 .

[14]  Enrico Scalas,et al.  Waiting-times and returns in high-frequency financial data: an empirical study , 2002, cond-mat/0203596.

[15]  Chuanju Xu,et al.  Finite difference/spectral approximations for the time-fractional diffusion equation , 2007, J. Comput. Phys..

[16]  Raymond H. Chan,et al.  An Introduction to Iterative Toeplitz Solvers (Fundamentals of Algorithms) , 2007 .

[17]  Raymond H. Chan,et al.  Conjugate Gradient Methods for Toeplitz Systems , 1996, SIAM Rev..

[18]  R. Chan,et al.  An Introduction to Iterative Toeplitz Solvers , 2007 .

[19]  Mark M. Meerschaert,et al.  A second-order accurate numerical approximation for the fractional diffusion equation , 2006, J. Comput. Phys..

[20]  Mechthild Thalhammer,et al.  A second-order Magnus-type integrator for quasi-linear parabolic problems , 2007, Math. Comput..

[21]  Marlis Hochbruck,et al.  Explicit Exponential Runge-Kutta Methods for Semilinear Parabolic Problems , 2005, SIAM J. Numer. Anal..

[22]  Hong Wang,et al.  A direct O(N log2 N) finite difference method for fractional diffusion equations , 2010, J. Comput. Phys..

[23]  Weihua Deng,et al.  Second-order LOD multigrid method for multidimensional Riesz fractional diffusion equation , 2013, BIT Numerical Mathematics.

[24]  Hai-Wei Sun,et al.  Shift-Invert Arnoldi Approximation to the Toeplitz Matrix Exponential , 2010, SIAM J. Sci. Comput..

[25]  Hong Wang,et al.  A fast Galerkin method with efficient matrix assembly and storage for a peridynamic model , 2012, J. Comput. Phys..

[26]  Norbert Heuer,et al.  Numerical Approximation of a Time Dependent, Nonlinear, Space-Fractional Diffusion Equation , 2007, SIAM J. Numer. Anal..

[27]  Mingrong Cui,et al.  Compact finite difference method for the fractional diffusion equation , 2009, J. Comput. Phys..

[28]  Alexander Ostermann,et al.  A second-order Magnus-type integrator for nonautonomous parabolic problems , 2006 .

[29]  Hong Wang,et al.  Fast finite difference methods for space-fractional diffusion equations with fractional derivative boundary conditions , 2015, J. Comput. Phys..

[30]  Vickie E. Lynch,et al.  Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model , 2001 .

[31]  D. Benson,et al.  Application of a fractional advection‐dispersion equation , 2000 .

[32]  Charles Tadjeran,et al.  Finite di$erence approximations for fractional advection-dispersion &ow equations , 2004 .

[33]  Han Zhou,et al.  Quasi-Compact Finite Difference Schemes for Space Fractional Diffusion Equations , 2012, J. Sci. Comput..

[34]  Siu-Long Lei,et al.  Circulant and skew-circulant splitting iteration for fractional advection–diffusion equations , 2014, Int. J. Comput. Math..

[35]  J. Cavendish.  On the Norm of a Matrix Exponential , 1974 .

[36]  Hong Wang,et al.  A Fast Finite Difference Method for Two-Dimensional Space-Fractional Diffusion Equations , 2012, SIAM J. Sci. Comput..

[37]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[38]  Bruce J. West,et al.  Lévy dynamics of enhanced diffusion: Application to turbulence. , 1987, Physical review letters.

[39]  Zhaoxia Yang,et al.  Finite difference approximations for the fractional advection-diffusion equation , 2009 .

[40]  Hong Wang,et al.  AN EFFICIENT COLLOCATION METHOD FOR A NON-LOCAL DIFFUSION MODEL , 2013 .

[41]  Xiao-Qing Jin,et al.  Preconditioned iterative methods for fractional diffusion equation , 2014, J. Comput. Phys..

[42]  Hong Wang,et al.  A fast and faithful collocation method with efficient matrix assembly for a two-dimensional nonlocal diffusion model , 2014 .

[43]  George F. Simmons.  The Existence and Uniqueness of Solutions , 2016 .

[44]  Ercília Sousa,et al.  Finite difference approximations for a fractional advection diffusion problem , 2009, J. Comput. Phys..

[45]  Michael K. Ng,et al.  Preconditioning Techniques for Diagonal-times-Toeplitz Matrices in Fractional Diffusion Equations , 2014, SIAM J. Sci. Comput..

[46]  M. Meerschaert,et al.  Finite difference methods for two-dimensional fractional dispersion equation , 2006 .

[47]  Weihua Deng,et al.  EFFICIENT NUMERICAL ALGORITHMS FOR THREE-DIMENSIONAL FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS , 2014 .

[48]  M. Meerschaert,et al.  Finite difference approximations for two-sided space-fractional partial differential equations , 2006 .

[49]  Hong Wang,et al.  A fast finite difference method for three-dimensional time-dependent space-fractional diffusion equations and its efficient implementation , 2013, J. Comput. Phys..

[50]  Stevens,et al.  Self-similar transport in incomplete chaos. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[51]  Brian Clark,et al.  Physics in Oil Exploration , 2002 .

[52]  David L. Powers.  On the Norm of a Matrix Exponential (J. C. Cavendish) , 1975 .

[53]  Han Zhou,et al.  A class of second order difference approximations for solving space fractional diffusion equations , 2012, Math. Comput..

[54]  Xin Liu,et al.  Fast exponential time integration scheme for option pricing with jumps , 2012, Numer. Linear Algebra Appl..