Orthogonal spline collocation Laplace-modified and alternating-direction methods for parabolic problems on rectangles

Abstract

Abstract. A complete stability and convergence analysis is given for two-and three-level, piecewise Hermite bicubic orthogonal spline collocation, Laplacemodified and alternating-direction schemes for the approximate solution of linear parabolic problems on rectangles. It is shown that the schemes are unconditionally stable and of optimal-order accuracy in space and time.

Analysis of incompressible miscible displacement in porous media by characteristics collocation method

“…are just like the definitions of [1]. The mathematical controlling model for two-phase incompressible flow in porous media is given by There hold some basic physical conditions: The collocation methods are widely used for solving practice problems in engineering due to its easiness of implementation and high-order accuracy; see, e.g., Douglas and Dupont in [2], Fernandes and Fairweather in [3], and Lu [4][5][6] for linear elliptic and parabolic problems with constant coefficients, Bialecki and Cai in [7] for orthogonal space collocation schemes for elliptic boundary value problems with variable coefficients in two space variable, and Bialecki and Fernandes in [8,9] for orthogonal spline collocation LM, ADI methods, and ADI Crank-Nicolson methods for linear parabolic problems with variable coefficients, respectively. The mathematical controlling model for two-phase incompressible flow in porous media is a strongly nonlinear coupling system of partial differential equations of two different types.…”

An ADI Petrov–Galerkin method with quadrature for parabolic problems

“…This work is motivated by several H 2 trial space ADI (C 1 and C 2 ) spline collocation methods to solve parabolic PDEs, see, for example, the survey paper [3] and references therein. In particular, the C 1 Laplace modified ADI orthogonal spline collocation (OSC) method was investigated for parabolic PDEs in [4]. The C 2 cubic nodal spline collocation (NSC) seems to be gaining popularity in computational physics and engineering over the C 0 and C 1 spline methods [3, p 76].…”

An Alternating-Direction Implicit Orthogonal Spline Collocation Scheme for Nonlinear Parabolic Problems on Rectangular Polygons

“…Section 2 embodies the heart of the paper. While the purpose of this paper is to develop the ADI OSC scheme for any rectangular polygon, in the interest of giving a simple presentation of the scheme and its implementation, in section 2.1 we first formulate the ADI OSC scheme for a rectangle and then comment on its comparison with other schemes in [1,4,5,11] and show how to extend it to solve (1.4). In section 2.2, we present an efficient implementation and give the computational cost of the scheme.…”