Matthieu Lecouvez
A parallel-in-time algorithm for variable stepsize multistep methods

Lawrence Livermore National Laboratory
7000 east avenue
L-561
Livermore
CA 94550
matthieu.lecouvez@gmail.com
Rob, D. Falgout
Carol, S. Woodward

As the number of cores increases on current and future architectures, the natural sequential approach to time integration is becoming a more serious bottleneck for achieving high scalability. One alternative to overcome this problem is the use of multigrid-in-time algorithms such as MGRIT [1]. Although first designed for one-step methods, we apply the MGRIT algorithm to multistep BDF methods for the integration of fully implicit Differential Algebraic Equations (DAE) on variable timestep grids. Our step function solves the nonlinear problem

$$F\left(t_n,y_n,\dot{y}_n\right)=0,\quad \dot{y}_n=\sum_{j=0}^s \alpha_{n,j} y_{n-j},$$ (1)

where $s$ is the order of the BDF method used, and the coefficients $\alpha_{n,j}$ depend on the order $s$ and the previous timestep sizes. We will present one approach for implementing variable stepsize BDF methods in a parallel-in-time context based on the XBraid software library. Results on power grid applications will also be given.
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. Lawrence Livermore National Security, LLC. LLNL-ABS-681082.

1

R. D. Falgout, S. Friedhoff, T. V. Kolev, S. P. MacLachlan, and J. B. Schroder, ``Parallel time integration with multigrid'', SIAM J. Sci. Comput., vol. 36, no 6, pp. C635-C661, 2014, ILNL-JRNL-645325.