Efficient solvers for time-dependent problems: a review of IMEX, LATIN, PARAEXP and PARAREAL algorithms for heat-type problems with potential use of approximate exponential integrators and reduced-order models

Antoulas AC. An overview of approximation methods for large-scale dynamical systems. Annu Rev Control. 2005;29:181–90.

Audouze C, De Vuyst F, Nair PB. Nonintrusive reduced-order modeling of parametrized time-dependent partial differential equations. Numeri Methods Partial Differen Equ. 2013;29(5):1587–628.

Baffico L, Bernard S, Maday Y, Turinici G, Zrah G. Parallel-in-time molecular dynamics simulations. Phys Rev E. 2002;66(5):057701.

Bal G, Maday Y. A “parareal” time discretization for nonlinear PDEs with application to the pricing of an American put. Recent developments in domain decomposition methods. Berlin: Springer; 2002. p. 189–202.

Chaturantabut S, Sorensen DC. Nonlinear model reduction via discrete empirical interpolation. SIAM J Sci Comput. 2010;32(5):2737–64.

Chinesta F, Ammar A, Lemarchand F, Beauchène P, Boust F. Parallel time integration and high resolution homogenization. CMAME. 2008;197(5):400–13.

Cortial J, Fahrat C, Guibas LJ, Rajashekhar M. Compressed sensing and time-parallel reduced-order modeling of structural health monitoring using DDDAS. Computational science-ICCS 2007. Berlin: Springer; 2007. p. 1171–9.

Filbet F, Negulescu C, Yang C. Numerical study of a nonlinear heat equation for plasma Physics. Int J Comp Math. 2012;89(8):1060–82.

Gander MJ, Hairer E. Nonlinear convergence analysis for the PARAREAL algorithm. In: Domain decomposition methods in Science and Engineering. 2008. vol. 60, p. 4556.

Gander MJ, Güttel S. PARAEXP: a parallel integrator for linear initial-value problems. SIAM J Sci Comput. 2013;35(2):C123–42.

Gander MJ. 50 years of time parallel time integration. Multiple shooting and time domain decomposition. Berlin: Springer; 2015.

Hochbruck M, Lubich C. On Krylov subspace approximations to the matrix exponential operator. SIAM J Num Anal. 1997;34(5):1911–25.

Kalashnikova I, Barone MF. On the stability of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far-field boundary treatment. IJNME. 2010;83:1345–75.

Kalashnikova I, Barone MF, Arunajatesan S, von Bloemen Waanders BG. Construction of energy-stable projection-based reduced order models. Appl Math Comp. 2014;249:569–96.

Ladevèze P. Non linear computational structural mechanics new approaches and non-incremental methods of calculation. New York: Springer-Verlag; 1999.

Ladevèze P, Passieux JC, Néron D. The LATIN multiscale computational method and the proper generalized decomposition. Comput Methods Appl Mech Eng. 2010;199(21):1287–96.

Lions J, Maday Y, Turinici G. A “parareal” in time discretization of PDE’s. Comptes Rendus de l’Académie des Sciences, Séries I, Mathematics. 2001;332(7):661–8.

Maday Y, Nguyen NC, Patera AT, Pau GSH. A general multipurpose interpolation procedure: the magic points. Commun Pure Appl Anal. 2009;81:383–404.

Minion M. A hybrid parareal spectral deferred corrections method. Commun Appl Math Comput Sci. 2010;5(2):265–301.

Nievergelt J. Parallel methods for integrating ordinary differential equations. Comm ACM. 1964;7:731–3.

Prud’homme C, Rovas D, Veroy K, Maday Y, Patera AT, Turinici G. Reliable real time solution of parametrized partial differential equations: reduced-basis output bound methods. J Fluids Eng. 2002;124(1):70–80.

Willcox K, Peraire J. Balanced model reduction via the proper orthogonal decomposition. AIAA J. 2002;40(11):2323–30.