A second-order accurate numerical method with graded meshes for an evolution equation with a weakly singular kernel

Abstract A second-order accurate numerical method with graded meshes is proposed and analyzed for an evolution equation with a weakly singular kernel. The graded meshes are employed to compensate for the singular behavior of the exact solution at t = 0 . For the time discretization, the product integration rule is used to approximate the Riemann–Liouville fractional integral, a generalized Crank–Nicolson time-stepping is considered and shown that the error is of order k 2 , where k denotes the maximum time step. A fully discrete difference scheme is constructed with space discretization by compact difference method. Numerical experiment is carried out to support the theoretical results. The comparison between the method on uniform grids and graded grids shows the efficiency of our method.

[1]  K. Mustapha.  An implicit finite-difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements , 2011 .

[2]  Da Xu,et al.  A Compact Difference Scheme for an Evolution Equation with a Weakly Singular Kernel , 2012 .

[3]  Jianfei Huang,et al.  Two finite difference schemes for time fractional diffusion-wave equation , 2013, Numerical Algorithms.

[4]  Vidar Thomée,et al.  Numerical solution of an evolution equation with a positive-type memory term , 1993, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[5]  Yasuhiro Fujita,et al.  INTEGRODIFFERENTIAL EQUATION WHICH INTERPOLATES THE HEAT EQUATION AND THE WAVE EQUATION I(Martingales and Related Topics) , 1989 .

[6]  Bangti Jin,et al.  An analysis of the Crank–Nicolson method for subdiffusion , 2016, 1607.06948.

[7]  J. C. López-Marcos.  A difference scheme for a nonlinear partial integrodifferential equation , 1990 .

[8]  Roberto Garrappa,et al.  Trapezoidal methods for fractional differential equations: Theoretical and computational aspects , 2015, Math. Comput. Simul..

[9]  Jingtang Ma,et al.  On a graded mesh method for a class of weakly singular Volterra integral equations , 2009, J. Comput. Appl. Math..

[10]  Hermann Brunner,et al.  The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes , 1985 .

[11]  Raytcho D. Lazarov,et al.  Error Estimates for a Semidiscrete Finite Element Method for Fractional Order Parabolic Equations , 2012, SIAM J. Numer. Anal..

[12]  Santos B. Yuste,et al.  A finite difference method with non-uniform timesteps for fractional diffusion equations , 2011, Comput. Phys. Commun..

[13]  Zhi-Zhong Sun,et al.  Finite difference methods for the time fractional diffusion equation on non-uniform meshes , 2014, J. Comput. Phys..

[14]  Ya-Nan Zhang,et al.  Error Estimates of Crank-Nicolson-Type Difference Schemes for the Subdiffusion Equation , 2011, SIAM J. Numer. Anal..

[15]  Vidar Thomée,et al.  Discretization with variable time steps of an evolution equation with a positive-type memory term , 1996 .

[16]  Jose L. Gracia,et al.  Error Analysis of a Finite Difference Method on Graded Meshes for a Time-Fractional Diffusion Equation , 2017, SIAM J. Numer. Anal..

[17]  Tao Tang,et al.  A finite difference scheme for partial integro-differential equations with a weakly singular kernel , 1993 .

[18]  K. Atkinson.  The Numerical Solution of Integral Equations of the Second Kind , 1997 .

[19]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[20]  Tao Tang,et al.  A note on collocation methods for Volterra integro-differential equations with weakly singular kernels , 1993 .

[21]  Siqing Gan,et al.  A second-order BDF compact difference scheme for fractional-order Volterra equation , 2016, Int. J. Comput. Math..

[22]  William McLean,et al.  A second-order accurate numerical method for a fractional wave equation , 2006, Numerische Mathematik.

[23]  Jason Roberts,et al.  Detailed error analysis for a fractional Adams method with graded meshes , 2018, Numerical Algorithms.

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

[25]  A note on the convergence of product integration and Galerkin method for weakly singular integral equations , 1997 .