The Quasi-wavelet Solutions Of A Non-monotone Conservation Law For Dune Problems

Wavelet numerical methods are widely applied to the nonlinear partial equations, of which the quasi-wavelets enjoy both the globally high accuracy and the locally stability. In the thesis we apply the quasi-wavelets numerical method to a class of nonlinear evolution equations that refers to instabilities in landforms of dunes, which occur through the interaction of a turbulent flow with an erodible substrate, and which was shown violating the conservation law. The numerical methods in this thesis are as follows: we apply the quasi-wavelet scheme and the fourth-order Runge-Kutta method respectively to discretize the spatial derivatives and the temporal derivative, while make use of the Newton-Simpson integral method to discretize the derivative of nonlocal operator. The advantage of ours is that due to the Gauss regularizer in the quasi-wavelet basis, the numerical process converges much fast. The numerical results also confirm the violence of the maximum principle in the numerical solutions. In the thesis we also systematically discuss the theory of wavelet analysis and quasi-wavelet method.