The two-dimensional wave equation

For the wavelet-Galerkin method we expand $K$, $C^2$, and $\psi$ in a scaling function expansion:

$$k=\sum\sum K_{i,j}\phi(x-i)\phi(y-j)$$

$$c^2=\sum\sum C_{i,j}\phi(x-i)\phi(y-j)$$

$$\psi=\sum\sum \Psi_{j,k}\phi(x-j)\phi(y-k)$$

and apply the Galerkin procedure to determine the coefficients $(\Psi_{j,k})$.

Substituting into the equation and projecting the result onto the subspace spanned by $\{\phi (x-j)\phi (y-k): \;\; j=1,\cdots,N ; \; k=1,\cdots,N\}$ requires evaluating terms of the form

$$\int\phi (x)\phi (x-k)\phi (x-m)\phi (x-n)dx.$$

$$\int\phi _{x}(x)\phi (x-k)\phi (x-m)\phi _x(x-n)dx.$$

$$\int\phi (x)\phi _x(x-k)\phi (x-m)\phi _x(x-n)dx.$$

This uniquely determines the $\psi_{j,k}$ as solutions of the Wavelet-Galerkin ordinary differential equations.

Define the four term connection coefficients

$$\Omega_{jkl}^{1001} = \int\phi _{x}(x)\phi (x-j)\phi (x-k)\phi _{x}(x-l)dx$$

$$\Omega_{jkl}^{0101} = \int\phi (x)\phi _x(x-j)\phi (x-k)\phi _{x}(x-l)dx$$

$$\Omega_{jkl}^{0000} = \int\phi (x)\phi (x-j)\phi (x-k)\phi (x-l)dx$$

With the summation convention on the indices $(i,j,k,l,m,n,)$ the Wavelet-Galerkin equations are

$$\Psi_{tt}(p,q)=$$

$$\left(\left(\Omega_{i,j,k}^{1001}+\Omega_{i,j,k}^{0101}\right)\Om... ...,n}^{0101}\right)\Omega_{i,j,k}^{000}\right) C(i+p,l+q)K(j+p,m+q)\Psi(k+p,n+q).$$

It can be shown that the wavelet-Galerkin equations for the symmetric form of the wave equation which are defined by five term connection coefficients will preserve the quadratic integral. The quadratic integral is not preserved by the wavelet-Galerkin for the usual form of the wave equation (which uses four term connection coefficients).