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-   -   Exact solution of Burgers equation (http://www.cfd-online.com/Forums/main/81270-exact-solution-burgers-equation.html)

mcaro October 21, 2010 10:47

Exact solution of Burgers equation
 
Hi,
well the doubt that i have is the following:
Let an inviscid Burgers equation u_t + (1/2u^2)_x = 0, with initial conditions u_0(x,0) = f(x), e.g. f(x) = 0.5 + sin(pi * x/2).
This problem can be seen like an initial value problem(IVP), i mean too that an exact solution may be found.
For example the exact solution can be written like this:
u(x,t)= f(x-ut);

My question is:
assume that i want to find the solution at time t=0.16, the exact solution would be
u(x,t) = 0.5 + sin(pi * (x-u*t) / 2)
From where will i get the values u in the expression ..(x-u*t)/2.
Sincerely i have so many doubt on the implementation the exact solution, I have to compare the exact solution with numerical solutions implementing new upwind schemes.
Really, i need your help!!
Thank you!
P.S> I have travelede a long time on Google, searching details from this solutions and nothing...
Another thing , on the Euler equations it is possible to find and exact solution analyzing the problem like an IVP. (If i resolve this problem for Burgers equation(inviscid) i think i could resolve for Euler eq.s) On this article FINITE DIFFERENCE WENO SCHEMES WITH LAX–WENDROFF-TYPE TIME DISCRETIZATIONS(Jianxian Qiu-2003) , you can see the comparison between exact and numerical solutions for Burgers equations and Euler equations.

bjohn October 21, 2010 20:35

It is an implicit equation.
You just need to solve it for u `numerically' at a grid point.
For example, iterate by

u(xj,t)^{n+1} = 0.5 + sin(pi * (xj-u(xj,t)^{n}*t) / 2)

with some initial solution, u(xj,t)^{0}.

bjohn

mcaro October 21, 2010 21:08

Thanks bjohn
 
Thanks bjohn, certainly i will try !!!

Thanks man!!!

mcaro January 25, 2011 07:46

Analitycal solution of Buckley_Leveret equation
 
Hi everybody,

How would be an anlytical solution of a Buckley_Leverett equation with initial condition :
u=1 when -1/2<=x <=0 and u=0 elsewhere, computed at t=0.4.

any help i really appreciate.

Thank you all.
Miguel.


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