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In this talk I will describe two elements from my research program on high order accurate methods for time dependent problems: arbitrary order Hermite methods and energy based discontinuous Galerkin methods.Ìý

Hermite methods are general purpose nodal based methods that use the solution and its derivatives asÌýdegrees of freedom. The methods have exceptional resolving power, excellent explicit time stepping propertiesÌýand very high computation to communication ratio, making them ideally suited for parallel implementation. The first part of the talk will discuss the basic elements of Hermite methods for time-dependent PDE. Time permitting I will show some extensions of the basic algorithms such as h and p-adaptivity,Ìýpropagation of discontinuities and shocks.Ìý

In the second part of the talk I will present a new general theory for spatial discontinuous Galerkin discretization of time dependent partial differential equations in first and second order form. Common to the problems that we consider is that they have a Hamiltonian that is conserved in time, and that the evolution equations governing the solution can be derived by Hamilton's principle. The goals of our discretization strategy is to conserve the Hamiltonian, evaluated for the semi-discretization, as well as to obtain a spectrally accurate approximation to the evolution equations. By using a non-standard test quantity combined with an appropriate choice of numerical fluxes we show that these goals can be achieved.