Wavelet-Galerkin solutions of quasilinear hyperbolic conservation equations

Abstract

The compactly supported orthogonal wavelet bases developed by Daubechies are used in the Galerkin scheme for a class of one-dimensional first-order quasilinear conservation equations with perturbed dissipative terms. We first develop a recursive algorithm to obtain the wavelet coefficients of a dissipative term of the type u n or u 1/n from those of u and then develop the wavelet-Galerkin solutions to several differential equations containing such dissipative terms. It is seen that the wavelet bases are capable of producing stable solutions throughout the domain except in the vicinity of the shock formation, where oscillations are seen to occur. Unlike the Fourier bases, which are known to produce oscillations which are smaller in magnitude but present throughout the domain, the oscillations produced by the wavelet bases are confined to the vicinity of the shocks. Further, we show that these oscillations may be smoothed out by a three-term averaging method.