The symmetric wave equation

$$\psi_{tt}=(c\bigtriangledown \cdot k\bigtriangledown c)\psi$$

The natural approximation would use five term connection coefficients. Since five term coefficients are not currently available, an algorithm was developed that uses four term connection coefficients. The results, although an approximation, are close to the four term wave equation results.

The term that requires five term connection coefficients is:

$$\lambda=c\cdot\bigtriangledown (k\bigtriangledown c).$$

The form:

$$\lambda=\Delta(\frac{kc^2}{2})-\bigtriangledown c\cdot\bigtriangledown (kc)-\frac{c^2}{2}\Delta k$$

can be evaluated using four term connection coefficients. In the first two terms we expand the functions $(kc,c)$, and in the last term expand the functions $(k,c^2)$.

The wavelet-Galerkin method based on five term connections will preserve the invariant:

$$\frac{\partial}{\partial t}\int\left(\frac{\theta_t^2}{2}+k\bigtriangledown (c\theta) \cdot \bigtriangledown (c\theta)\right)=0.$$

Therefore, it would be of interest to develop the five term wavelet-Galerkin algorithm for this equation.