**Abstract:** | Energy conservation is an important property for many time dependent PDEs, such as linear hyperbolic systems, linear and nonlinear dispersive wave equations including KdV equations,etc. Discontinuous Galerkin (DG) methods are often used to solve such problems, especially when adaptivity is desired. However,it is difficult to design energy conserving DG methods for such problems with optimal convergence in the $L^2$-norm. In this talk we will describe our recent work in designing such DG schemes, which involves the technique of possible doubling of unknowns. Optimal a priori error estimates of order $k+1$ are obtained for the semi-discrete scheme in one dimension, and in multi-dimensions on Cartesian meshes when tensor-product polynomials of degree $k$ are used, for linear hyperbolic and dispersive wave equations. Computational results for linear and nonlinear problems including those in aeroacoustics, Maxwell equations and KdV equations, on both structured and unstructured meshes, demonstrate the excellent performance of these energy conserving schemes. This is joint workwith Guosheng Fu. Professor Shu received his B.S. degree from the University of Science and Technology of China in 1982. In 1986 he received his Ph.D. degree in UCLA. He then spent a year at IMA in University of Minnesota as a post doctoral fellow. Since 1987 he has been with the Division of Applied Mathematics, Brown University, as an Assistant Professor (1987-91), Associate Professor (1992-96), Professor (1996- ), Chairman (1999-2005), and Theodore B. Stowell University Professor (2008- ). In 1992 he received the NASA Public Service Group Achievement Award. In 1995 he received the first Feng Kang Prize of Scientific Computing.Since 2004 he has been listed as an ISI Highly Cited Author in Mathematics by the ISI Web of Knowledge, Thomson Scientific Company. In 2007 he received the SIAM/ACM Prize in Computational Science and Engineering. In 2009 he was selected as one of the inaugural class of Fellows of the Society for Industrial and Applied Mathematics. In 2012 he was selected as one of the inaugural class of Fellows of the American Mathematical Society. In 2014 he was an invited 45-minute speaker in the International Congress of Mathematicians in Seoul. |