2.6. Time stepping

For the semi-discrete system of differential equations of the form

(2.1)\[\begin{split}\begin{aligned} \mathbf M_p\partial_t{\mathbf p}&=\mathbf B\mathbf v,\\ \mathbf M_v\partial_t{\mathbf v}&=-\mathbf B^\top\mathbf p,\\ \mathbf p(0)&=\mathbf p_0,\quad {\mathbf v}(0)=\mathbf v_0, \end{aligned}\end{split}\]

where \(\mathbf M_p,\mathbf M_v\) are the discrete mass matrices and \(\mathbf B\) is the matrix of the discrete differential operator, We use the Leap-Frog scheme given by

(2.2)\[\begin{split}\mathbf v_{1/2}&=\mathbf v_0 -\frac{\tau}{2} \mathbf M_v^{-1}\mathbf B^\top\mathbf p_0,\\ \mathbf p_{j+1}&=\mathbf p_{j} +\tau \mathbf M_p^{-1}\mathbf B\mathbf v_{j+1/2},\\ \mathbf v_{j+1/2}&=\mathbf v_{j-1/2} -\tau \mathbf M_v^{-1}\mathbf B^\top\mathbf p_j.\end{split}\]

It is well-known that this scheme is stable for timesteps \(\tau>0\) fulfilling

\[\tau^2 < \frac{4}{\sigma \left( \mathbf M_p^{-1}\mathbf B\mathbf M_v^{-1}\mathbf B^\top \right)}\]

where \(\sigma\) is the spectral radius.