From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Subject: Re: Method of lines
Date: 3 Jan 2000 17:03:56 GMT
Newsgroups: sci.math.num-analysis

In article <3865FD0B.D6BAC49E@central.ntua.gr>,
 mc96118@central.ntua.gr writes:
|> Hello ,
|> could someone please tell me what the method of lines is
|> Many thanks in advance
|> 
given a partial differential equation in a form like 
    u_t = F(u,u_x,u_y,u_xx ,.....)
or 
    u_tt = F(....) 
such that F depends only on the function u and partial derivatives of u with
respect to other variables than the variable t:
replace the partial derivatives with respect to the other variables by 
finite differences on a grid and consider now the equation only
on the grid points 
or
 make an "ansatz" 
u(t,...)=sum_i alpha_i(t)*phi_i(x,..) 
with finite element basis functions phi_i on a grid (in x-y-.. space),
 mulptiply by phi_j
and integrate (over the domain of x,y,..) 

then you obtain in both cases a (large) system of ordinary differential equations
in the variable t which you can integrate by the appropriate methods
(taking stiffness, highly oscillatory solutions etc) into account.

this is the basic idea of the methods of lines. 
hope that helps
peter