Difference Methods for Solving
Nonlinear Equations
of Mathematical Physics
In this chapter the new difference schemes are constructed for the quasilin-
ear heat conduction equation and equations of gas dynan1ics with placing a
special emphasis on iterative inethocls available for solving nonlinear differ-
ence equations. Among other things, the convergence of Newton's method
is established for i1nplicit sche1nes of gas clynan1ics.8.1 DIFFERENCE METHODS FOR SOLVING
THE QUASILINEAR HEAT CONDUCTION EQUATION- The stationary probleni. To avoid misunderstanding, we concentrate
primarily on the simplest problem, the state1nent of which is related to the
stationary heat conduction problem with nonlinear sources:
(1) u^11 =-f(u), O<x<l, u(O)=O, u(l)=O.
An excellent start in this direction is to introduce on the segment
0 < x < 1 au equidistant grid wh ={xi= ih, i = 0, 1, ... , N, hN = l} and
proceed to design the difference scheme
(2) Yxx = -f(y), .('. = ih, i=l,2, ... ,N-1,
Yo= YN = U,
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