Re: Numerical solution for nonlinear PDE
garcia22@llnl.gov (Manuel Garcia)
Fri, 27 Jun 1997 11:12:37 +0800
In article <33A502D0.41C6@iatfrisc2.fzk.de>, Michael Bunk
wrote:
> In the context of viscous gravity currents I have to solve an equation
>
> dH/dtau - d/dX(H^3 dH/dX) == 0
>
> with
>
> Integrate[H[X,tau] ,{X,0,A[tau]}] == CV tau^alpha
>
> H[A[tau],X] == 0 .
>
> I can get a similarity solution, but how to get a full numerical
> solution?
>
> Thanks for all comments!!!
>
> Michael
>
> --
> Michael Bunk
>
> Forschungszentrum Karlsruhe
> Institut fuer Angewandte Thermo- und Fluiddynamik
> Postfach 3640
> D-76021 Karlsruhe
> Tel.: 07247/82-2528
****************************************************************************
Your equation is a nonlinear diffusion equation, quite interesting. I
cannot decode the "integrate" line of your message, and the last condition
seems to have the parameters reversed (?). In any case my comments are
fairly general.
CONVENTIONS***********************************************
In order to transmit math via ascii I will use the conventions below:
1) partial derivative: dx, dy, deta ("d" "eta"), d2x (2nd derivative)
2) differential operators: (d/dx)[...], (d2/d2x)[...]
3) integral:
Integral function I, from lower limit "a," to upper limit "b,"
with integrand f(..,..,x,..,..), and integration over "x":
I{a,b,f(..,..,x,..,..)}dx
4) double integral:
inner integral over dx, from a to y,
outer integral over dy, from b to z:
I{b,z,I{a,y,f(..,x,..)}dx}dy
5) a) exponential: ^2, ^58,...
b) multiplication: *
c) add, sub, divide: +, -, /
d) precedence (binding): ^#, *, /, + and -
(exponentials "bind" most tightly, then products, ratios, and
sums)
EQUATION**************************************************
The problem at hand:
(d/dt)H(x,t) = (d/dx)[ H(x,t)^3 * (d/dx)[ H(x,t) ]]
initial condition: H(x,0) (known)
boundary condition #1: H(0,t) (known)
boundary condition #2: (d/dx)[H(0,t)] (known)
ANALYSIS**************************************************
08 steps follow:
01) (d/dt)[H(x,t)] = (d2/d2x)[(1/4)*(H(x,t)^4]
02) integrate time ("t") from 0 to t:
H(x,t) - H(x,0) = (d2/d2x)[ I{0,t,(1/4)*H(x,nu)^4}dnu ]
03) integrate space ("x") from 0 to x:
I{0,x,(H(eta,t) - H(eta,0))}deta =
(d/dx)[I{0,t,(1/4)*(H(x,nu)^4)}dnu ] + ...
-(d/dx)[ I{0,t,(1/4)*(H(0,nu)^4)}dnu ]
04) integrate space again:
I{0,x, I{0,eta, (H(zeta,t) - H(zeta,0))}dzeta }deta = ...
I{0,t,(1/4)*(H(x,nu)^4)}dnu - I{0,t,(1/4)*(H(0,nu)^4)}dnu
+...
- x*( (d/dx)[ I{0,t,(1/4)*(H(0,nu)^4)}dnu ] )
05) reverse the order of double integration (easily shown graphically):
I{0,x,I{0,eta,f(..,zeta,..)}dzeta}deta =>
I{0,x,I{zeta,x,f(..,zeta,..)}deta}dzeta
in this case find:
I{0,x,I{zeta,x, (H(zeta,t) - H(zeta,0)) }deta}dzeta = ...
(1/4)*I{0,t,H(x,nu)^4}dnu - (1/4)*I{0,t,H(0,nu)^4}dnu + ...
- x*( (d/dx)[ I{0,t,(1/4)*(H(0,nu)^4)}dnu ] )
06) double integral becomes single integral:
I{0,x,(x-zeta)*(H(zeta,t) - H(zeta,0))}dzeta = ...
(1/4)*I{0,t,(H(x,nu)^4 - H(0,nu)^4)}dnu +...
- x*( (d/dx)[ I{0,t,(1/4)*(H(0,nu)^4)}dnu ] )
07) group "unknowns" on left, "knowns" on right:
I{0,x,(x-zeta)*H(zeta,t)}dzeta - (1/4)*I{0,t,H(x,nu)^4}dnu = ...
I{0,x,(x-zeta)*H(zeta,0)}dzeta - (1/4)*I{0,t,H(0,nu)^4}dnu +
...
- x*( (d/dx)[ I{0,t,(1/4)*(H(0,nu)^4)}dnu ] )
08) iterate for a solution:
H(x,t) into => I{0,x,(x-zeta)*H(zeta,t)}dzeta -
(1/4)*I{0,t,H(x,nu)^4}dnu
must equal => F(x,t) = known = (right-hand-side of 07))
Note the (left-hand-side) => integral(dx@t) + integral(dt@x)
Note that time integral involves H^4, so it is always positive
(H assumed real), the space integral can have either sign.
SUMMARY
This approach reduces the problem to single integrals which sum to a know
field,
F(x,t), which is determined from the initial and boundary conditions.
Iteration is used because of the nonlinearity, H(x,t)^4, (linear equation
has matrix solution). The advantage here is an integral form (stable
numerically), and single integration (single loops). Variants for different
"edge" (boundary and initial) conditions affect integral limits and
resulting F(x,t), not general approach. I have written this out clearly by
hand on a single side of a sheet of paper, including two figures. If you
would like a copy please e-mail me your address (for response via mail) or
fax number (for faster and fuzzier). If it works, let me know. Have fun. MG
--
Manuel Garcia
LLNL L-153
POB 808, Livermore, CA 94550
(510) 422-6017
garcia22@llnl.gov
garcia22@popgun.llnl.gov
