It is well known that wave propagation problems over long time intervals require high order methods, and many such methods have been constructed. However, to keep high order accuracy also in time for problems with variable coefficients, the methods quickly become complicated. In this talk, we will consider acoustic wave propagation, and discuss a few finite difference methods for the wave equation in scalar form as well as in first order system form.
One way to achieve high accuracy in time, is to use the Numerov principle, based on substituting truncation errors in time by space derivatives. It turns out that the first order system form leads to less complicated approximations, while still keeping the conservation properties of the continuous formulation. We will show that the method works well even for discontinuous coefficients without any special procedures across the material interfaces.