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

Phil Scadden wrote:
Looking to solve time-independent conservation law.
[f(u)]_x + [g(u)]_y =0

with initial condtions u(x_0(t),y_0(t))=u_0

and boundary conditions u(x_0),y) and u(x,y_0)

snip
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,
Now you can take a bunch of characteristic curves, each having
a different initial value ( y(x_0) ), the value of u(x_0,y(x_0))
is transported constantly along the characteristic curve.

If fp(u) should become zero, you have to switch to a characteristic
curve of the form (x(y),y) which can be computing similarly.


--
Helmut Jarausch

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