We propose efficient and parallel algorithms for the implementation of the high-order continuous time Galerkin method for dissipative and wave propagation problems. By using Legendre polynomials as shape functions, we obtain a special structure of the stiffness matrix which allows us to extend the diagonal Pad\'e approximation to solve ordinary differential equations with source terms. The unconditional stability, $hp$ error estimates, and $hp$ superconvergence at the nodes of the continuous time Galerkin method are proved. Numerical examples confirm our theoretical results.

翻译：本文针对耗散与波传播问题中高阶连续时间伽辽金方法的实现，提出了高效且并行的算法。通过采用勒让德多项式作为形函数，我们获得了刚度矩阵的特殊结构，从而能够将对角Padé逼近方法拓展至带源项常微分方程的求解。证明了连续时间伽辽金方法的无条件稳定性、$hp$误差估计以及节点处的$hp$超收敛性。数值算例验证了理论结果。