# Exact solution of Burgers equation

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 October 21, 2010, 10:47 Exact solution of Burgers equation #1 New Member   Miguel Caro Join Date: Apr 2010 Posts: 26 Rep Power: 16 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.

 October 21, 2010, 20:35 #2 New Member   Join Date: Jun 2010 Posts: 16 Rep Power: 15 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

 October 21, 2010, 21:08 Thanks bjohn #3 New Member   Miguel Caro Join Date: Apr 2010 Posts: 26 Rep Power: 16 Thanks bjohn, certainly i will try !!! Thanks man!!!

 January 25, 2011, 06:46 Analitycal solution of Buckley_Leveret equation #4 New Member   Miguel Caro Join Date: Apr 2010 Posts: 26 Rep Power: 16 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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