Biorthogonal wavelets and approximations of operators

We present an explicit scheme, based on wavelets, for the approximation of evolution equations of type partial derivative U/partial derivative t + partial derivative/partial derivative x(f(U)) = 0. It is constructed using a Taylor scheme for the time approximation and a wavelet method for spatial approximation. This scheme is shown to be stable and convergent under a specific stability condition and can be implemented fast on c-structured wavelet spaces of approximation thanks to the introduction of scheme dependant biorthogonal multiresolution analyses. A scale dependant time step version is presented that allows a strong reduction of complexity. For discontinuous solutions, a non linear version has been proposed and numerically tested.