Question

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.

Answer 1

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).