Convergence in $\ell_2$ and $\ell_\infty$ Norm of One-Stage AMF-W-Methods for Parabolic Problems

Convergence in the maximum norm of ADI-type methods for parabolic problems

“…The first goal of this section is to show unconditional convergence of order two in the maximum norm for semilinear parabolic problems with constant diffusion coefficients (and a time dependent source term) and time-independent Dirichlet boundary conditions ( 1)-( 2), when the one-step AMF-W-method (henceforth denoted as AMF-W1) in [4,6] is considered with the parameter choice θ = 1/2…”

“…where v(t) stands for the derivative of a function v(t) regarding t. The following discussion can be applied in similar terms to the Douglas method [10, p. 373]. We use the same notations as in [4]. The global error at the time step…”

“…was introduced in [4]. For this method, second order convergence in the ℓ ∞ −norm for time independent boundary conditions can be shown as in Theorem 2 above considering that its local error can be expressed as (see [4, formula (5.4)] )…”

“…For the choice θ = 1/2, this is the stability matrix of the Peaceman-Rachford method (when m = 2), also the one of the Douglas scheme ([2], [7], [10, p. 373]) and the one of the one-stage AMF-W-method [4]. Furthermore, the stability matrix of the so-called Hundsdorfer-Verwer scheme [10, Section IV.5.2], which is a 2-stage W-method of order 2 in general, and of order 3 for θ = (3 + √ 3)/6, is given by…”

Boundary corrections on multi-dimensional PDEs

“…In this work our focus is on the time integration of the differential systems that come from some spatial discretization of (1)-( 2)-(3) by using Finite Differences. In particular, we are interested in AMF-type W-methods (based on Approximate Matrix Factorization) which avoid the solution of non-linear equations and only require the solution of linear systems with tridiagonal matrices, as it can be seen in [7], [6], [5]. They have also been successful in applications to the case of variable coefficients in [9] and multi-dimensional problems (N > 3) in [13].…”