Economical Difference Schemes
for Multidimensional Problems
in Mathematical PhysicsOne of the serious develop1nents in con1putational tnathematics owes a debt
to economical difference n1ethods available for solving partial differential
equations of several spatial variables. Recent years have seen the publica-
tions of nun1erous papers on this subject for nmltiple equations of parabolic,
hyperbolic and elliptic types as well as the constructions of various eco-
non1ical schemes. The general stability theory lies in the foundations of the
possible theory of economical methods which will be given special investi-
gation throughout the entire chapter. Two classes of admissible econmnical
schen1es are of great importance: schen1es with a factorized operator on the
upper layer and additive schemes generating a sum1narized approximation
in a certain up-agreed sense. These can depend on the range of situations
to be considered.
9.1 THE ALTERNATING DIRECTION METHOD
(THE LONGITUDINAL-TRANSVERSE SCHEME)
FOR THE HEAT CONDUCTION EQUATION- Some preliminary infonnation on economical sche1nes. One of the tnost
i1nportant issues in numerical tnethocls is the well-founded choice of eco-
nmnical computational algorithrns, the realization of which requires a n1in-
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