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Conference rusure::math

Title:	Mathematics at DEC

Moderator:	RUSURE::EDP

Created:	Mon Feb 03 1986
Last Modified:	Fri Jun 06 1997
Last Successful Update:	Fri Jun 06 1997
Number of topics:	2083
Total number of notes:	14613

1246.0. "References on Numerical Solutions to Non-linear PDE?" by GENRAL::HEINTZE () Wed May 23 1990 18:31

    Hmmph...   I just spent a lot of time discovering that numerical
    methods for linear differential equations (specifically the heat
    equation) don't work well for non-linear equations!
    
    The equation I'm trying to solve is essentially the heat equation (a
    linear differential equation)  with  g(x,y,z,t)/c added to the RHS
    making the equation non-linear where g is the amount of heat being
    absorbed by an element and C is the heat capacity of the material.
    
    Can anyone recommend a reference on numerical solutions to non-linear
    differential equations?
    
    Basically the equation is
    
      u[t] = (K/C) (u[xx] + u[yy] + u[zz])  + g(x,y,z,t)/C
    
    where the subscripts denote partial derivatives and K is the thermal
    conductivity and C is the heat capacity and G is the unit power
    delivered to the medium per unit volume.
    
    
                               Thanks,
    
    				Sieg

T.R	Title	User	Personal Name	Date	Lines
1246.1		HPSTEK::XIA	In my beginning is my end.	`Wed May 23 1990 19:24`	6
	All methods for nonlinear equations are iterative methods where each step of iteration is solving a linear equation. If it is just a one time deal, you could just hack something up on top of your linear equation package. Otherwise, there are packages readily made for this kind of thing on the market. Eugene
1246.2		GUESS::DERAMO	Dan D'Eramo	`Wed May 23 1990 21:17`	5
	There are also likely to be free packages available over the network. Look for topics here with titles containing LIB or NET or even PAC. Dan
1246.3		RDVAX::NG		`Thu May 24 1990 09:57`	4
	It seems to me your PDE IS linear. Nonlinearity in the forcing function g(x,y,z,t) does not make the PDE nonlinear. (Unless g depends on the solution u or its partials, in which case the dependences should be shown explicitly.)
1246.4		HPSTEK::XIA	In my beginning is my end.	`Thu May 24 1990 16:03`	5
	re -1, Hmmm..., you are right. Eugene
1246.5		ALLVAX::JROTH	It's a bush recording...	`Fri May 25 1990 02:08`	39
	� Basically the equation is � � u[t] = (K/C) (u[xx] + u[yy] + u[zz]) + g(x,y,z,t)/C � � where the subscripts denote partial derivatives and K is the thermal � conductivity and C is the heat capacity and G is the unit power � delivered to the medium per unit volume. If the equation is indeed as shown, then it would probably be best to solve it using an iterative method, such as SOR, (Gauss-Seidel), ADI, or something similar, on a finite element or finite difference mesh. This has the advantage that you can prime the iteration for the next value of the inhomogenous forcing term g(t) with the previous solution, leading to swift reconvergence. In fact, you can extrapolate the delta's seen at the nodes from one time step to the next to get even faster convergence. In addition, with an iterative technique you won't have to set up banded matrices, and can do the iteration right on the topology of the problem. I feel that a finite element mesh will be preferable to a finite difference mesh, since it is substantially more accurate for the same number of nodes, particularly if quadratic (or higher) elements are used. At least as long as the solution is expected to be moderatly smooth... There are a large number of books on finite element solutions to PDE's. One introduction that I like is the first volume of the series by J. T. Oden (and someone else) published by Prentiss Hall. I can get you the exact title but the book is at home. It may be worth looking thru NETLIB at ITPACK (iterative solver package) to see if there is something useful. I don't have direct experience with it though. My own experience with this is electromagnetic problems - waveguide and microstrip analysis - rather than heat or elasticity, but the equations are all the same. - Jim