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I want to numerically integrate the equation $\partial_t u= a(t) \partial_xu+b\partial_{xxx}u+c$ to get $u(t)$. Is is preferable to use a difference formula of higher order of accuracy for spatial derivatives? Thanks. I am using a Semi-Discretization method called Method of Lines (vertical) to treat this problem.

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Could you tell us more about the problem - what is the domain, etc.? –  Tom Dickens Oct 3 '12 at 1:40
    
hi, @tom-dickens it is a equation resulting from fluid dynamics. –  gstar2002 Oct 4 '12 at 15:25
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1 Answer

up vote 2 down vote accepted

You could use the method of lines to solve this PDE. If you use an explicit finite difference method, you will need to take a rather small time step (${\mathcal O(\Delta x^3))}$ due to the $u_{xxx}$ term.

Given that you don't specify any boundary conditions, I will assume that you are solving the Cauchy problem. In that case (or in case of periodic boundary conditions), there is a much more efficient approach. Since your PDE is linear and the coefficients don't vary in space, you can solve in terms of Fourier modes. Decompose the initial data as usual:

$$u(x,0) = \sum_k c_k\exp{ikx}.$$

Then use the ansatz

$$u(x,0) = \sum_k g_k(t)\exp{ikx}$$

with $g_k(0) = c_k$. Substituting this in your PDE gives

$$g_k'(t) = ik a(t) - ik^3b + c.$$

This can be solved very easily by numerical quadrature (or exactly, if $a(t)$ can be integrated symbolically).

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Thanks David. I decided to use MOL to solve the problem. But I will try to use the Fourier method, if I have one more chance. –  gstar2002 Oct 29 '12 at 20:00
    
The acture system is more complicated, but I think the Fourier method is applicable. –  gstar2002 Oct 29 '12 at 20:28
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