Alternating Direction Implicit Orthogonal Spline Collocation Methods for an Evolution Equation with a Positive-Type Memory Term

Authors:
Amiya Kumar Pani;Graeme Fairweather;Ryan I. Fernandes
Affiliations:
-;-;-
Venue:
SIAM Journal on Numerical Analysis
Year:
2008

Citing 0
Cited 3

Energy-Conserved Splitting Finite-Difference Time-Domain Methods for Maxwell's Equations in Three Dimensions

SIAM Journal on Numerical Analysis
Alternating direction implicit-Euler method for the two-dimensional fractional evolution equation

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Orthogonal spline collocation methods for the subdiffusion equation

Journal of Computational and Applied Mathematics

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Abstract

New numerical techniques are presented for the solution of a class of linear partial integro-differential equations (PIDEs) with a positive-type memory term in the unit square. In these methods, orthogonal spline collocation (OSC) is used for the spatial discretization, and, for the time stepping, new alternating direction implicit (ADI) methods based on the backward Euler, the Crank-Nicolson, and the second order BDF methods combined with judiciously chosen quadrature rules are considered. The ADI OSC methods are proved to be of optimal accuracy in time and in the $L^2$ norm in space. Numerical results confirm the predicted convergence rates and also exhibit optimal accuracy in the $L^{\infty}$ and $H^1$norms and superconvergence phenomena.