Re: euler without d/dt? time-independent conservation law.

Phil Scadden wrote:
let's rewrite your equation to
(*) fp(u)*u_x + gp(u)*u_y=0 where fp(u) is the derivate of f at
u(x,y).
Now make the ansatz (characteristic) y= y(x), then

(**) d/dx (u(x,y(x))= u_x + u_y*yp where yp= d/dx y .
Assume we can divide (*) by fp(u), then we get by plugging (*)
into (**) :

d/dx (u(x,y(x))= (-gp(u)/fp(u) + yp) * u_y = (**)

Now, if we solve the ODE yp(x)= gp(u(x,y(x))) / fp(u(x,y(x)),
then (***) = 0, or u(x,y(x)) = constant.

I.e., along the so-called characteristic curve (x,y(x))
the solution u is constant,

Thanks very much for taking the time to comment.
The method of characteristics works well for a system of 2 PDEs and 2
unknown functions where it is easy to identify the so called Riemann
Invariants but what if you have 4 of each? This is this issue that
struggling with here.


This wasn't clear from your original post.
Yes, for systems and more than 1 space dimension,
it gets very much harder.

You might have to study e.g. Randall Leveque's book
"Finite Volume Methods for Hyperbolic Problems",
Cambridge Univ. Press 2002.
He has authored the CLAWPACK software package, see
http://www.amath.washington.eud/~claw

Helmut.


--
Helmut Jarausch

Lehrstuhl fuer Numerische Mathematik
RWTH - Aachen University
D 52056 Aachen, Germany
.

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