3. Wavelet-Galerkin method.

In this study we utilize the Daubechies wavelets. They are compactly-supported and form an orthonormal basis for square integrable functions [1,2]. The wavelet function can be derived from the scaling function $\varphi$ and it is this function, together with its dilations and translations, that we use for our numerical analysis. The scaling function $\varphi$ satisfies the scaling relation

$$\varphi(x)=\sum a_k\varphi(2x-k),$$

and can be made to have desired properties by suitable choice of the coefficients $\{a_k\}$. The scaling function $\varphi$ will have compact support if and only if only finitely many $a_k$ are nonzero. The orthonormality of the set of functions $\varphi(x-k)$ can be obtained from the conditions $\sum a_ka_{k-2m}=2\delta_{0m}$ and $\sum (-1)^k k^m a_k = 0.$ See [4] for a relaxation of these conditions preserving orthonormality. We use the Daubechies scaling function defined by six nonzero coefficients, or D6. It has a continuous first derivative and the vector space spanned by $\varphi$ and its translates contains all polynomials of degree two. We will numerically solve Burgers' equation by projecting the solution of that PDE onto the space spanned by the integer translates of D6. That is, we assume the solution to be approximated by its projection into the subspace defined by the scaling function and a finite number of its translates

$$u = \sum_{k= - M}^N u_k \varphi(x-k).$$

Substitution of this into Burgers' equation and projection of the result onto the subspace $\Omega$ spanned by $\{ \varphi(x-k) : k = -M, -M+1, \ldots, N \}$ defines a set of ordinary differential equations for the coefficients $u_k(t)$. The good approximation properties of $\Omega$ are a consequence of both the orthogonality and compact-support of the basis elements. In $\Omega$ the orthogonal basis is the most compactly-supported basis possible since linear combinations of translates of the scaling function cannot have a smaller support than the scaling function itself. The orthogonality of the basis diagonalizes the $u_t$ term, preserving the finite bandwidth of the coefficient matrix. The asymmetry of the scaling function does not introduce a significant asymmetry in the numerical solution. Use of the mirror image of the scaling function does not change, to a tolerance, the numerical solution.

In this note we utilize the semi-implicit time differencing scheme for solving Burgers' equation. That is, we solve for $u_{n+1}$ from

$$(u_{n+1} - u_n)/\Delta t - u_nu_{n+1,x} = \sigma u_{n+1,xx}$$

where $u_0 = u(x,0)$. In the Galerkin approach, the coefficients in the expansion of $u_{n+1}$ are chosen to be the unique set of coefficients that make the above equation hold when both sides are projected back to the space spanned by $\varphi$ and its translates.

To apply the Galerkin method we require the definite integrals of products of scaling functions and their derivatives. The direct numerical evaluation of these terms can have inaccuracies associated with the oscillations of the derivatives. We have therefore developed exact methods for the evaluation of these terms. These methods will be described elsewhere.