1549301742-The_Theory_of_Difference_Schemes__Samarskii

Economical factorized schemes 591

which includes as part the supplen1entary boundary conditions for the

vector-functions w(a)· Cl= l, 2, ... ,p-1, in the case xc, = 0, (,:

p

w(a)= IT (E+r^2 R13)f.-lt, Xa=O,la.

f3=a+l

From here the components of the vector w(a)' cY = 1,2, ... ,p, can

be recovered independently, since the operators Da = E + r^2 Ra possess

diagonal matrices of coefficients with diagonal blocks.

9.3 THE SUMMARIZED* APPROXIMATION METHOD

1. The problem state1nent. First of all, it should be noted that it is impos-
   sible to generalize directly the alternating direction n1ethod for three and
   more measuren1ents as well as for parabolic equations of general form. Sec-
   ond, econon1ical factorized sche1nes which have been under consideration in
   Section 2 of the present chapter are quite applicable under the assumption
   that the argument x = ( x 1 , x 2 , ..• , xp) varies within a parallelepiped.
   Because of this, there is a real need for designing the general method,
   by means of which economical schen1es can be created for equations with
   variable and even discontinuous coefficients as well as fo1· quasilinea1· non-
   stationary equations in complex domains of arbitrary shape and diniension.
   As a inatter of experience, the universal tool in such obstacles is the method
   of smnnicuized approximation, the framework of which will be explained a
   little later on the basis of the heat conduction equation in an arbitrary
   domain G of the din1ension p with the boundary r

(1)

p

Lu= L Lau,

a=l

provided that the conditions hold:

(2) 1tlr=p(x,t), t>O, u(x,0)=11 0 (x), xEC.

*Editor's note: Summarized= summed.