Piano String Finite Difference Scheme

The Chaigne and Askenfelt PDE is discretized in space and time to yield a recursive difference scheme with the following space-time ``stencil'':

• 3rd order in time, 4th order in space
• Explicit, enabling recursive time updates

The FDS can be written in matrix form as

$$\begin{eqnarray*} {\bf y}_{n} ={\bf A_{1}y_{n-1}}+{\bf A_{2}y_{n-2}}+{\bf A_{3}y_{n-3}}+{\bf g}f_n({\bf y}_{n-1}[i_h],y^h_{n-1}) \end{eqnarray*}$$

where

• The column vector ${\bf y}_{n}$ contains the displacements of the $N$ nodes at time step $n$ , i.e., ${\bf y}_{n}[i] = y_i^n$

• ${\bf A_{1} A_{2}}$ and ${\bf A_{3}}$ are $N$ by $N$ matrices