Stability and energy conservation

Stability requires the conservation of energy during the time propagation process. For the elastic system considered here, the conservation of energy is represented by (Auld, 1990):  

$$\begin{array} {ccccccccc} \left( \begin{array} {c} {\bf u} \... ...ay} {c} {\bf u} \\ \dot{\bf u} \end{array} \right).\end{array}$$ (9)

Using the approximation for $\bf P_{+}$ given by equation (8) the righthand side becomes

$$\begin{array} {cccc} \left( \begin{array} {c} {\bf u} \\ {\... ...y} {c} {\bf u} \\ \dot{\bf u} \end{array} \right), \end{array}$$

and equation (9) assumes the form  

$$\begin{array} {ccccccccc} \left( \begin{array} {c} {\bf u} \... ...ay} {c} {\bf u} \\ \dot{\bf u} \end{array} \right).\end{array}$$ (10)

Equation (10) shows that the energy conservation, and thus the stability of the extrapolation process, is controlled by how well the consecutive application of the forward and backward propagation operators can reproduce the original wavefield.