Consider the Burgers' equation $$\partial_t u = \alpha u\partial_xu$$ Intend to solve this using Fourier Galerkin method. So When I convert this into $N$th Fourier partial sum, I get a system of ODE's as follows $$\partial_t u_m = \sum_{k=-N}^{k=N}iku_ku_{m-k} $$

How do I solve this system of ODE numerically or analytically?, given we know the first $N$ Fourier coefficients of the initial condition $u(x,0)$.

  • $\begingroup$ From what I googled, it seemed to be a coupled ODE. Wonder any ready made solution available? $\endgroup$ – Rajesh Dachiraju Jan 27 '16 at 12:14
  • $\begingroup$ Usually, you do the linear part of the time evolution in frequency space and the non-linear part in state space. Cf. the examples in stackoverflow.com/questions/29803342/… and stackoverflow.com/questions/29617089/… $\endgroup$ – LutzL Jan 27 '16 at 12:37
  • $\begingroup$ @LutzL : I am trying to solve in Fourier-Galerkin method, where it is done like this. But the references I read do not tell how the ODE is solved, perhaps thats a well known thing in the field of numerical PDE or CFD, I am trying hard to get references. $\endgroup$ – Rajesh Dachiraju Jan 27 '16 at 12:56
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    $\begingroup$ Use any of the standard solvers. Since quadratic systems tend to be stiff, preferably use an implicit solver. Many are based on the fortran code lsoda (? or lsode?). $\endgroup$ – LutzL Jan 27 '16 at 13:01

Analytic solving of the PDE : $$\partial_t u - \alpha u\partial_xu=0$$ Thanks to the method of characteristics : $$\frac{dt}{1}=\frac{dx}{-\alpha u}=\frac{du}{0}$$ On the characteristic curves : \begin{cases} u=c_1\\ \alpha c_1 dt +dx=0\:\:\rightarrow\:\: \alpha u t +x=c_2\\ \end{cases} The general solution can be expressed on the form : $$\Phi\left(u\:,\: \alpha t u +x\right)=0$$ where $\Phi$ is any derivable function of two variables. This is equivalent of the implicit relationship : $$u=F\left(\alpha \;t\; u +x\right)$$ where $F$ is any derivable function.

Nothing can be done further until the boundary conditions be defined, in order to find the convenient function $F$ and then $u(x,t)$ when it is possible to explicit $u$ from the implicit equation.

  • $\begingroup$ Hi JJacquelin : Thnaks for the answer. My question is (what I want to learn) the solution of the coupled ODE system I have mentioned, and not the Burger's equation itself. Why I am interested is that I can build a computational model which can compute numerically, for any given initial data $u(x,0)$. $\endgroup$ – Rajesh Dachiraju Jan 27 '16 at 15:06

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