Splitting the wavelet-Galerkin solver

In general, $c$ and $k$ are piecewise constant with the same interfaces. In this case the wavelet-Galerkin equations reduce to a convolutional form except for a region, $R$, of width $m$ centered at a jump point, where $\Omega^{1001}$ and $\Omega^{0000}$ are of size $(m,m,m)$.

For $(p,q)$ not in $R$,

$$\Psi_{tt}(p,q)= c^2k(\left(\Omega_l^{11}\Psi(l+p,q)+\Omega_m^{11}\Psi(p,m+q)\right)$$

For $(p,q)$ in $R$,

$$\Psi_{tt}(p,q)= \left(\Omega_{i,j,k}^{1001}\Omega_{l,m,n}^{0000}+... ...ga_{l,m,n}^{1001}\Omega_{i,j,k}^{0000}\right) C(i+p,l+q)K(j+p,m+q)\Psi(k+p,n+q)$$

If in $R$, the piecewise constant $c$ and $k$ are locally tensor products

$$K(p,q)=a(p)b(q)$$

$$C(p,q)=d(p)e(q)$$

the wavelet-Galerkin equations reduce to the form

$$\Psi_{tt}(p,q)= \left(A(p,l)B(q,m)+B(p,l)A(q,m)\right)\Psi(l+p,m+q).$$

The forms $A(p,l)$ and $B(p,l)$ are deformations of the second derivative and of the delta function through the transition region at a interface. The are represented in the figures. In this transition the wave velocity jumps from one to four.

We split the wavelet-Galerkin algorithm as follows.

• The convolutional part, not in $R$, can use the higher order wavelet systems. Typically, the $D30$ system would be used.
• The locally tensor product form in $R$ is defined by a lower order wavelet system. Typically, the $D10$ wavelet system would be used.