An Adjoint Approach for Stabilizing the Parareal Method

The parareal algorithm seeks to extract parallelism in the time-integration direction of time-dependent differential equations. While it has been applied with success to a wide range of problems, it suffers from some stability issues when applied to non-dissipative problems. We express the method through an iteration matrix and show that the problematic behavior is related to the non-normal structure of the iteration matrix. To enforce monotone convergence we propose an adjoint parareal algorithm, accelerated by the Conjugate Gradient Method. Numerical experiments confirm the stability and suggest directions for further improving the performance.