A. Arteaga, D. Ruprecht, and R. Krause, A stencil-based implementation of Parareal in the C++ domain specific embedded language STELLA, Applied Mathematics and Computation, 267 (2015), pp. 727–741.
U. M. Ascher, S. J. Ruuth, and B. T. Wetton, Implicit-explicit methods for time-dependent partial differential equations, SIAM Journal on Numerical Analysis, 32 (1995), pp. 797–823.
U. M. Ascher, S. J. Ruuth, and R. J. Spiteri, Implicit-Explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Numer. Math., 25 (1997), pp. 151–167.
E. Aubanel, Scheduling of tasks in the parareal algorithm, Parallel Comput., 37 (2011), pp. 172–182.
G. Bal, On the Convergence and the Stability of the Parareal Algorithm to Solve Partial Differential Equations, in Domain Decomposition Methods in Science and Engineering, R. Kornhuber and et al., eds., vol. 40 of Lecture Notes in Computational Science and Engineering, Berlin, 2005, Springer, pp. 426–432.
L. A. Berry, W. R. Elwasif, J. M. Reynolds-Barredo, D. Samaddar, R. S. Snchez, and D. E. Newman, Event-based parareal: A data-flow based implementation of parareal, Journal of Computational Physics, 231 (2012), pp. 5945–5954.
S. Boscarino and G. Russo, On a class of uniformly accurate IMEX Runge–Kutta schemes and applications to hyperbolic systems with relaxation, SIAM J. Sci. Comput., 31 (2009), pp. 1926–1945.
T. Buvoli, Rogue Waves in optics and water, PhD thesis, Master thesis, University of Colorado at Boulder, 2013.
T. Buvoli, A class of exponential integrators based on spectral deferred correction, SIAM Journal on Scientific Computing, 42 (2020), pp. A1–A27.
T. Buvoli, Codebase for “IMEX Runge-Kutta Parareal for Non- Diffusive Equations”, (2021). https://doi.org/10.5281/zenodo.4513662.
M. P. Calvo, J. De Frutos, and J. Novo, Linearly implicit Runge-Kutta methods for advection-reaction-diffusion equations, Appl. Numer. Math., 37 (2001), pp. 535–549.
A. Cardone, Z. Jackiewicz, H. Zhang, and A. Sandu, Extrapolation-based implicit-explicit general linear methods, (2013).
S. M. Cox and P. C. Matthews, Exponential time differencing for stiff systems, Journal of Computational Physics, 176 (2002), pp. 430–455.
V. A. Dobrev, T. Kolev, N. A. Petersson, and J. B. Schroder, Two-level convergence theory for multigrid reduction in time (mgrit), SIAM Journal on Scientific Computing, 39 (2017), pp. S501–S527.
A. Dutt, L. Greengard, and V. Rokhlin, Spectral deferred correction methods for ordinary differential equations, BIT Numerical Mathematics, 40 (2000), pp. 241–266.
M. J. Gander, Analysis of the Parareal Algorithm Applied to Hyperbolic Problems using Characteristics, Bol. Soc. Esp. Mat. Apl., 42 (2008), pp. 21–35.
M. J. Gander, 50 years of Time Parallel Time Integration, in Multiple Shooting and Time Domain Decomposition, Springer, 2015.
M. J. Gander and S. Vandewalle, Analysis of the Parareal Time-Parallel Time-Integration Method, SIAM Journal on Scientific Computing, 29 (2007), pp. 556–578.
E. Hairer and G. Wanner, Solving Ordinary Differential Equations II : Stiff and Differential-Algebraic Problems, Springer Berlin Heidelberg, 1991.
E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, Math. Comput. Simul., 29 (1987), p. 447.
G. Izzo and Z. Jackiewicz, Highly stable implicit–explicit Runge–Kutta methods, Applied Numerical Mathematics, 113 (2017), pp. 71–92.
C. A. Kennedy and M. H. Carpenter, Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44 (2003), pp. 139–181.
S. Krogstad, Generalized integrating factor methods for stiff PDEs, Journal of Computational Physics, 203 (2005), pp. 72–88.
J.-L. Lions, Y. Maday, and G. Turinici, A “parareal” in time discretization of PDE’s, Comptes Rendus de l’Acadmie des Sciences - Series I - Mathematics, 332 (2001), pp. 661–668.
M. Minion, Semi-implicit spectral deferred correction methods for ordinary differential equations, Communications in Mathematical Sciences, 1 (2003), pp. 471–500.
M. L. Minion, A Hybrid Parareal Spectral Deferred Corrections Method, Communications in Applied Mathematics and Computational Science, 5 (2010), pp. 265–301.
J. Nievergelt, Parallel Methods for Integrating Ordinary Differential Equations, Commun. ACM, 7 (1964), pp. 731–733.
D. Ruprecht, Shared Memory Pipelined Parareal, Springer International Publishing, 2017, pp. 669–681.
D. Ruprecht, Wave Propagation Characteristics of Parareal, Computing and Visualization in Science, 19 (2018), pp. 1–17.
A. Sandu and M. Günther, A generalized-structure approach to additive Runge–Kutta methods, SIAM Journal on Numerical Analysis, 53 (2015), pp. 17–42.
B. S. Southworth, Necessary Conditions and Tight Two-level Convergence Bounds for Parareal and Multigrid Reduction in Time, SIAM J. Matrix Anal. Appl., 40 (2019), pp. 564–608.
B. S. Southworth, W. Mitchell, A. Hessenthaler, and F. Danieli, Tight two-level convergence of linear parareal and mgrit: Extensions and implications in practice, arXiv preprint arXiv:2010.11879, (2020).
G. A. Staff and E. M. Rnquist, Stability of the parareal algorithm, in Domain Decomposition Methods in Science and Engineering, R. Kornhuber and et al., eds., vol. 40 of Lecture Notes in Computational Science and Engineering, Berlin, 2005, Springer, pp. 449–456.
Z. Wang and S.-L. Wu, Parareal algorithms implemented with IMEX runge-kutta methods, Mathematical Problems in Engineering, 2015 (2015).
G. Wanner and E. Hairer, Solving ordinary differential equations II, Springer Berlin Heidelberg, 1996.