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.



Is is preferable to use a difference formula of higher order of accuracy for spatial derivatives to solve this IVP problem ?
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).


