One-step splitting methods for semi-discrete parabolic equations

Abstract

--ZusammenfassungOne-Step Splitting Methods for Semi-Discrete Parabolic Equations. The main purpose of the paper is to discuss splitting methods for parabolic equations via the method of Iines. Firstly, we deal with the formulation of these methods for autonomous semi-discrete equations formeln wird hier definiert welche alle bekannten Split-Methoden entNilt, unter der Bedingung, dab die Funktionen f~ geschickt gew~ihlt sind. Fiir eine Reihe yon Methoden werden ~.tabilit~itsergebnisse angegeben. Ferner wird das alternierende Richtungsverfahren fiir Probleme mit einer wiltkiirlichen nicht-linearen Verbindung zwischen Raumableitungen diskutiert.

Unconditional Convergence of Some Crank-Nicolson Lod Methods for Initial- Boundary Value Problems

“…If F1 and F2 have a more simple structure than F , the computation of Un+I from (1.4) can be done more efficiently than from (1.3). However, the LOD method ( 1.4) will have 2nd order only if F1 , F2 are linear and commuting; in more general situations it will be merely of lst order, owing to lack of symmetry (see [4]). Below we will consider two modifications which restore symmetry and 2nd order.…”

Accuracy and stability of splitting with Stabilizing Corrections

“…This scheme has been called the method of Stabilizing Corrections in [12,18], since the calculation of the vectors v j , j = 1, 2, ..., s, primarily serves to stabilize the forward Euler method that is used to calculate v 0 . When applied with dimensional splitting, the ADI methods of Douglas and Brian with θ = 1 2 , and Douglas-Rachford with θ = 1, see [7,20], are special cases of interest.…”

Splitting methods for three-dimensional bio-chemical transport

“…The order conditions are easy to verify using the relations derived in [7] (see also [6]). For the stability we refer to [14], where, for one particular choice (viz.…”

“…6 reason formulated in property (i) in the Introduction. As a matter of fact, this property of hopscotch type splitting was the motivation for constructing this particular "explicit-in-the-horizontal-implicitin-the-vertical-line-hopscotch-splitting".…”