1
$\begingroup$

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 $u(x,0)$.

$\endgroup$
  • $\begingroup$ 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
  • 1
    $\begingroup$ 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
0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.