A Diagonalization-Based Parareal Algorithm for Dissipative and Wave Propagation Problems

We present a new parareal algorithm based on a diagonalization technique proposed recently. The algorithm uses a single implicit Runge--Kutta method with the same small step-size for both the $\mathcal{F}$ and $\mathcal{G}$ propagators in parareal and would thus converge in one iteration when used directly like this, without, however, any speedup due to the sequential way parareal uses $\mathcal{G}$. We then approximate $\mathcal{G}$ with a head-tail coupled condition such that $\mathcal{G}$ can be parallelized using diagonalization in time. We show that with a suitable choice of the parameter in the head-tail condition, our new parareal algorithm converges very rapidly, both for parabolic and hyperbolic problems, even in the nonlinear case. We illustrate our new algorithm with numerical experiments.