In this paper, we analyze the Lax-Wendroff discontinuous Galerkin (LWDG) method for solving linear conservation laws. The method was originally proposed by Guo et al., where they applied local discontinuous Galerkin (LDG) techniques to approximate high order spatial derivatives in the Lax-Wendroff time discretization. We show that, under the standard CFL condition $\tau\leq \lambda h$ (where $\tau$ and $h$ are the time step and the maximum element length respectively and $\lambda>0$ is a constant) and uniform or non-increasing time steps, the second order schemes with piecewise linear elements and the third order schemes with arbitrary piecewise polynomial elements are stable in the $L^2$ norm. The specific type of stability may differ with different choices of numerical fluxes. Our stability analysis includes multidimensional problems with divergence-free coefficients. Besides solving the equation itself, the LWDG method also gives approximations to its time derivative simultaneously. We obtain optimal error estimates for both the solution $u$ and its first order time derivative $u_t$ in one dimension, and numerical examples are given to validate our analysis.

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This is the note for a series of lectures that the author gave at the Centre de Recerca Matem´atica (CRM), Bellaterra, Barcelona, Spain on October 19–24, 2009. The aim is to give a comprehensive description of some recent work of the author and his students on generalisations of the Gross-Zagier formula, Euler systems on Shimura curves, and rational points on elliptic curves.

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Computational techniques derived from digital signal processing are playing a significant role in the security and digital copyrights of audio, video, and visual arts. In light of the quantum computing, the corresponding algorithms are becoming a new research direction in today’s high-technology world. The nature of quantum computer guarantees the security of quantum data, so a safe and effective quantum watermarking algorithm is in demand. Quantum watermarking is the technique that embeds the invisible quantum signal into quantum multimedia data for copyright protection. This research presents a quantum watermarking algorithm in discrete M-band wavelet domain. Different from most traditional algorithms, we propose a new algorithm which applies quantum watermarking in M-band wavelet domain. Assured by the Heisenberg uncertainty principle and quantum no-cloning theorem, the security of quantum watermark can reach a very high-level standard. In other words, this watermarking algorithm can defeat nearly all attackers, no matter using classical computer or quantum computer.