

Subject:
1d Nonlinera Burger's Equation
Category: Science > Math Asked by: javier5440ga List Price: $10.00 
Posted:
15 Feb 2005 06:46 PST
Expires: 17 Mar 2005 06:46 PST Question ID: 474879 
I want to know the "analytical" solution for the 1d nonlinear Burger's equation, given by : with a computational domain of (0,1) and periodic boundary conditions : u(x=0)=u(x=1), the initial condition is a sine wave given by: u= 2 + (1/(4*pi))*sin(2*pi*x); thanks, I am looking for the mathematics of it, is for a CFD study and I am looking at the wavenumber of 3 numerical schemes, best regards, Javier  
 
 
 
 


Subject:
Re: 1d Nonlinera Burger's Equation
Answered By: mathtalkga on 16 Feb 2005 17:08 PST Rated: 
Hi, Javier: Your question certainly brought back memories of graduate classes in numerical PDE's and fluid dynamics. Here's a recent paper that details the approach for analytically expressing solutions to the inviscid Burgers equation in 1D, for smooth periodic initial data defined on the entire xaxis: [A Simple Illustration Of A Weak Spectral Cascade  David J. Muraki] http://www.math.sfu.ca/~dmuraki/research/cascade.pdf Given the inviscid Burgers equation: u_t + u*u_x = 0 and smooth initial data: u(x,0) = f(x) that is periodic in x, one can give an implicit "solution" for Burgers equation by: u(x,t) = f(x  u*t) It is easily verified that for t = 0, such a function u must satisfy the initial condition. Furthermore implicit differentiation with respect to x,t will show that the PDE is satisfied as well. For fixed time t > 0, the periodicity of u follows from the periodicity of f. For consistency with the notation in that paper, we will assume that f(x) is of period 2pi, rather than of period 1 as assumed in your problem. It is easy to show that a common change of scale for both x,t preserves the solutions of Burgers equation while adjusting the period of initial values f. As the above linked paper shows, Platzman's earlier analysis of the Fourier series for u(x,t): Platzman, G.W. An exact integral of complete spectral equations for unsteady onedimensional flow, Tellus, XVI (1964), pp. 422?431 can be generalized to give (up to the time of singularity): a_0 +oo u(x,t) =  + SUM [ a_n(t)*cos(nx) + b_n(t)*sin(nx) ] 2 n=1 where for n > 0: a_0 = (1/pi) INTEGRAL f(z) dz OVER [pi,+pi] 1 a_n(t) =  INTEGRAL sin(nz + nt*f(z)) dz OVER [pi,+pi] pi*nt 1 b_n(t) =  INTEGRAL cos(nz + nt*f(z)) dz OVER [pi,+pi] pi*nt Note that the "constant term" a_0 is truly a constant, independent of time t, corresponding to the conservation of the mean exhibited by Burgers equation. This Fourier series expansion of u(x,t) determines a singlevalued function for all time t, but it only satisfies Burgers equation up to the point in time when a singularity develops. The author of the paper above, David J. Muraki, explains the distinction between the singlevalued function defined by the Fourier series, the multivalued function defined by the method of characteristics, and the "entropy" solution to which one ascribes "physical reality". Please feel free to post a Request for Clarification if I can be of further assistance. regards, mathtalkga 
javier5440ga
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Thanks for your time, I greatly appreciate your answer and compliment you on the refernce you attached, good work, best regards, Javier5440 

Subject:
Re: 1d Nonlinera Burger's Equation
From: mathtalkga on 16 Feb 2005 07:32 PST 
The PDE which Javier has asked about is a nonlinear evolution equation. Without loss of generality we may view the initial conditions as extended periodically to the entire xaxis. The particular initial data consists of a constant plus a superimposed "fundamental" sine wave: u(x,0) = 2 + (1/(4pi))*sin(2pi*x) so that the period in x is length 1. As the solution evolves all the higher harmonics are "excited". In 1964 G.W. Platzmann published a paper in an obscure meterology journal Tellus which gives the exact solution (up to singularity) for the case of initial data which is purely sinusoidal. The "time" coefficients of the various harmonics are then given explicitly in terms of Bessel functions. This approach has recently been generalized to all smooth periodic initial conditions, but it would require some effort to work out the details for the particular initial condition posed here. Other approaches to obtaining exact solutions consider the extension of the domain to complex or even hypercomplex numbers. regards, mathtalkga 
Subject:
Re: 1d Nonlinera Burger's Equation
From: javier5440ga on 16 Feb 2005 08:20 PST 
this information is enough for now, i greatly appreciate it and you CAN go ahead and bill me the amount I had agreed when I posted the response ($10) your service is great and I am quite impressed, I will try to find your reference, best regards, javier5440 
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