Interpolating Wavelet Adaptive Solving Hyperbolic Partial Differential Equations

In order to avoid waste of computational resources and loss of precision because of irregularity of solution, an adaptive method usually is needed for numerically solving PDEs. Taking advantage of wavelet抯 ability, We can construct ideal adaptive algorithms. But, duo to interactions between scales, nonlinear operators such as mutiplication and differentiation are too computationally expensive when done directly in a wavelet basis. how to find a efficient method to treatment nonlinear operators of PDEs is very important. In this paper, a wavelet-like methods ?adaptive interpolation wavelets methods is introduced for solving hyperbolic PDEs. By using polynomial interpolation on dyadic grids, an interpolating wavelet transform is constructed. In the interpolating wavelet basis, each wavelet coefficients of function corresponds to function value at a grid point. The adaptability is performed automatically by thresholding the wavelet coefficients. When ignoring all wavelet coefficients with magnitude smaller than some threshold value, we get the sparse wavelet representation(SWR). To avoid the difficulties of applying nonlinear operators in the wavelet basis, we do the inverse transform to SWR and get a sparse point representation(SPR).The SPR is the set of discrete function values. In this set, the nonlinear operations such as differentiation and multiplication are fast and simple. In this paper, we apply the method for solving linear and nonlinear hyperbolic PDEs. Algorithms are achieved in MATLAB language. The numerical experiments show the method抯 efficiency when it is applied to the problems that solutions are well compressed in a wavelet basis, e.g. The function is smooth in most of the domain with small areas of sharp variation.