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Consider the following PDE: $$ \begin{align} &u_t+uu_x=0.1u_{xx},\qquad 0<x<1,t>0\\ &u(x+1,t)=u(x,t),\qquad t\geq 0 \\ &u(x,0) = \sin 2\pi x,\qquad 0\leq x\leq 1 \end{align} $$

I used two different numerical schemes(Finite difference and Spectral method) and implemented it by MATLAB to plot $u$ at $t = 0.21$. The results are very different: Using Finite Difference Using Spectral Method

Here are my questions:

  • Is there a name for this nonlinear PDE?
  • Does the solution to the PDE decay in time?
  • [EDITED:]Which one of the figures above is close to the true solution?
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I think this is the viscous Burger's equation, in wiki you might get what you want.

BTW, by decay you mean the $C^0$norm decay?

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Thanks. The reference for Burger's equation helps. – Jack Dec 5 '11 at 3:22

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