Runge Kutta for wave equation

Question

Recently I work on mechanical shocks (e.g. impact) in FE (fenics). I've already put together simple timesteppers (Euler, Crank-Nicolson). I use higher order basis, so I think of higher order time integration schemes like RK4. Is it worth the effort in mechanics?

I can't find any appropriate example of this method in elastic waves, so maybe it's pointless. I'm not sure, how to apply it in my case, since it's written in "incremental" form:

$$y_{i+1} = y_i+\frac{1}{6}(k_1+2k_2+2k_3+k_4)h$$

and I'm trying to apply it to system:

$$\begin{eqnarray*} \frac{\partial v}{\partial t} - \nabla u &=& f \\ \frac{\partial u}{\partial t} &=& v \end{eqnarray*}$$

Answer 1

High order RK methods work fine for a large number of wave propagation (an elastic example using a DG discretization). The procedure is usually to discretize the spatial derivatives in the first order form of the wave equation. This is called semi-discretization and it results in a system of ODEs involving a vector of solution coefficients $U$ (for both $u$ and $v$).

$\frac{dU(t)}{dt} = AU(t)$

where $A$ is defined by your choice of discretization and the wave equation you want to solve. You can then apply Runge-Kutta methods directly to the equation - this involves computing RHS evaluations. Since the wave equation is linear, marching from $t_n$ to $t_n + dt$ with RK4 method requires computing four vectors

$\begin{align*} k_1 &= AU(t_n)\\ k_2 &= A(U(t_n) + ({dt/2}) k_1)\\ k_3 &= A(U(t_n) + ({dt/2}) k_2)\\ k_4 &= A(U(t_n) + (dt) k_3) \end{align*}$

then computing the final update $U(t_n + dt) = \frac{dt}{6}(k_1+2k_2+2k_3+k_4)$. Repeat this to march forward in time again.