An Orthogonal Spline Collocation Alternating Direction Implicit Crank--Nicolson Method for Linear Parabolic Problems on Rectangles

Abstract

We formulate an alternating direction implicit Crank-Nicolson scheme for solving a general linear variable coefficient parabolic problem in nondivergence form on a rectangle with the solution subject to nonhomogeneous Dirichlet boundary condition. Orthogonal spline collocation with piecewise Hermite bicubics is used for spatial discretization. We show that for sufficiently small time stepsize the scheme is stable and of optimal-order accuracy in time and the H 1 norm in space. We also describe an efficient implementation of the scheme and present numerical results demonstrating the accuracy and convergence rates in various norms.

A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model

“…Let P N (D) be a finite-dimensional subspace of H 1 0 (D; M ) and letψ n N ∈ P N (D) be the solution at time level n of the following fully-discrete Galerkin method 5 :…”

“…However, the method is extremely well suited to implementation on a parallel architecture since the q ∼ -direction solves are completely independent from one another, and similarly the x ∼ -direction solves are decoupled also. We discuss the parallel implementation 5 Here we introduce a backward-Euler temporal discretisation. We will also consider a semi-implicit discretisation in Section 3.…”

“…For example, in the context of collocationbased alternating-direction schemes, Celia and Pinder [8,9] and Bialecki and Fernandes [5], developed methods that could handle equations with general variable coefficients. However, our focus is on developing a Galerkinbased framework, and therefore, again, the work of Douglas and Dupont is the most relevant here.…”

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.…”

Single‐degree of freedom Hermite collocation for multiphase flow and transport in porous media

“…Recently there has been renewed interest in the collocation approach. Bialecky et al [7] formulated a collocation approach for linear parabolic problems on rectangles and Li et al [8] studied the problem of transverse vibrations of a clamped square plate. Elliptic boundary value problems [9], Schrodinger wave equation problems [10] and biharmonic problems [11], as well as techniques to e ciently solve the resulting approximating equations have also been studied by this group of researchers [12].…”