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We propose efficient and parallel algorithms for the implementation of thehigh-order continuous time Galerkin method for dissipative and wave propagationproblems. By using Legendre polynomials as shape functions, we obtain a specialstructure of the stiffness matrix which allows us to extend the diagonal Pad\'eapproximation to solve ordinary differential equations with source terms. Theunconditional stability, $hp$ error estimates, and $hp$ superconvergence at thenodes of the continuous time Galerkin method are proved. Numerical examplesconfirm our theoretical results.

Efficient and Parallel Solution of High-order Continuous Time Galerkin for Dissipative and Wave Propagation Problems