The summarized approximation method

The problem statement. First of all, it should be noted that it is impossible to generalize directly the alternating direction method for three and more measurements as well as for parabolic equations of general form. Second, economical factorized schemes which have been under consideration in Section 2 of the present chapter are quite applicable under the assumption that the argument x = (xq, x, ., Xp) varies within a parallelepiped. 

Because of this, there is a real need for designing the general method, by means of which economical schemes can be created for equations with variable and even discontinuous coefhcients as well as for quasilinear non-stationary equations in complex domains of arbitrary shape and dimension. As a matter of experience, the universal tool in such obstacles is the method of summarized 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 dimension p with the boundary F 

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The quasilinear heat conduction equation reproduces the case when = k x,t,u) and / = f(x,t,u). 

Of course, the words arbitrary domain cannot be understood in a literal sense. Before giving further motivations, it is preassumed that the boundary F is smooth enough to ensure the existence of a smooth solution u = u x,t) of the original problem (l)-(2). In the accurate account of the approximation error and accuracy we always take for granted that the solution of the original problem associated w ith the governing differential equation exists and possesses all necessary derivatives which do arise in the further development. 

However, throughout this book, the classification of difference methods is mostly based on the origin of difference schemes rather than on a possible way of constructing them and a perfect tool for solving this or that difference scheme (equation). 

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