ADI orthogonal spline collocation methods for parabolic partial integro–differential equations

Alternating direction implicit (ADI) orthogonal spline collocation schemes are formulated and analysed for a class of partial integro-differential equations of parabolic type. These techniques are based on the θ-method, for θ ∈ [1/2, 1], (where θ = 1 yields the backward Euler method and θ = 1/2 yields the Crank-Nicolson method) and the second-order backward differentiation formula (BDF) method. For each method, optimal estimates in various norms at each time step are derived and confirmed by results of numerical experiments.

[1]  Akiyoshi Wakatani.  A parallel and scalable algorithm for ADI method with pre-propagation and message vectorization , 2004, Parallel Comput..

[2]  Weidong Zhao,et al.  A modified alternating‐direction finite volume method for modeling secondary hydrocarbon migration and accumulation processes , 2003 .

[3]  Hong Cheng,et al.  Solving Degenerate Reaction-Diffusion Equations via Variable Step Peaceman-Rachford Splitting , 2004, SIAM J. Sci. Comput..

[4]  Chuan Miao Chen,et al.  Finite Element Methods for Integrodifferential Equations , 1998 .

[5]  Jr. J. E. Dendy.  AN ANALYSIS OF SOME GALERKIN SCHEMES FOR THE SOLUTION OF NONLINEAR TIME-DEPENDENT PROBLEMS* , 1975 .

[6]  G. Fairweather,et al.  Orthogonal spline collocation methods for partial di erential equations , 2001 .

[7]  J. C. Jorge,et al.  A generalization of Peaceman-Rachford fractional step method , 2006 .

[8]  Kassem Mustapha,et al.  A Crank‐Nicolson and ADI Galerkin method with quadrature for hyperbolic problems , 2005 .

[9]  K. Teo,et al.  On application of an alternating direction method to Hamilton-Jacobin-Bellman equations , 2004 .

[10]  Bernard Bialecki.  Superconvergence of the orthogonal spline collocation solution of Poisson's equation , 1999 .

[11]  Samir Karaa,et al.  A high-order ADI method for parabolic problems with variable coefficients , 2009, Int. J. Comput. Math..

[12]  Bernard Bialecki.  Convergence Analysis of Orthogonal Spline Collocation for Elliptic Boundary Value Problems , 1998 .

[13]  Graeme Fairweather,et al.  Discrete-time Orthogonal Spline Collocation Methods for Schrödinger Equations in Two Space Variables , 1998 .

[15]  Jim Douglas,et al.  Collocation Methods for Parabolic Equations in a Single Space Variable , 1974 .

[16]  Graeme Fairweather,et al.  Algorithms for Almost Block Diagonal Linear Systems , 2004, SIAM Rev..

[17]  Tongke Wang.  Alternating direction finite volume element methods for 2D parabolic partial differential equations , 2008 .

[18]  G. Fairweather.  Finite Element Galerkin Methods for Differential Equations , 1978 .

[19]  Jim Douglas,et al.  ALTERNATING-DIRECTION GALERKIN METHODS ON RECTANGLES , 1971 .

[20]  Carmelo Clavero,et al.  A high order uniformly convergent alternating direction scheme for time dependent reaction–diffusion singularly perturbed problems , 2007, Numerische Mathematik.

[21]  L. J. Hayes.  Galerkin Alternating-Direction Methods for Nonrectangular Regions Using Patch Approximations , 1981 .

[22]  Donghyun You.  A high-order Padé ADI method for unsteady convection-diffusion equations , 2006, J. Comput. Phys..

[23]  Graeme Fairweather,et al.  Alternating-Direction Galerkin Methods for Parabolic and Hyperbolic Problems on Rectangular Polygons , 1975 .

[24]  Bernard Bialecki,et al.  An Alternating-Direction Implicit Orthogonal Spline Collocation Scheme for Nonlinear Parabolic Problems on Rectangular Polygons , 2006, SIAM J. Sci. Comput..

[25]  José Luis Gracia Lozano,et al.  A uniformly convergent alternating direction HODIE finite difference scheme for 2D time dependent convection-diffusion problems , 2005 .

[26]  Graeme Fairweather,et al.  Analysis of alternating direction collocation methods for parabolic and hyperbolic problems in two space variables , 1993 .

[27]  Yanping Lin,et al.  A priori L 2 error estimates for finite-element methods for nonlinear diffusion equations with memory , 1990 .

[28]  Jun Zhang,et al.  High order ADI method for solving unsteady convection-diffusion problems , 2004 .

[29]  Graeme Fairweather,et al.  Alternating Direction Implicit Orthogonal Spline Collocation Methods for an Evolution Equation with a Positive-Type Memory Term , 2007, SIAM J. Numer. Anal..

[30]  G. Fairweather,et al.  Almost Block Diagonal Linear Systems: Sequential and Parallel Solution Techniques, and Applications , 2000 .