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 L2 norm in space. Numerical results confirm the predicted convergence rates and also exhibit optimal accuracy in the L∞ and H1 norms and superconvergence phenomena.
Original language | British English |
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Pages (from-to) | 344-364 |
Number of pages | 21 |
Journal | SIAM Journal on Numerical Analysis |
Volume | 46 |
Issue number | 1 |
DOIs | |
State | Published - 2007 |
Keywords
- Alternating direction implicit method
- Backward Euler method
- Crank-Nicolson method
- Evolution equation with positive-type memory term
- Orthogonal spline collocation
- Quadrature rules
- Second order backward differentiation formula method