Performance assessment of exponential Rosenbrock methods for large systems of ODEs

Abstract

This article studies time integration methods for stiff systems of ordinary differential equations of large dimension. For such problems, implicit methods generally outperform explicit methods because the step size is usually less restricted by stability constraints. Recently, however, a family of explicit methods, called exponential integrators, have become popular for large stiff problems due to their favourable stability properties and the rapid convergence of non-preconditioned Krylov subspace methods for computing matrix-vector products involving exponential-like functions of the Jacobian matrix. In this article, we implement the so-called exponential Rosenbrock methods using Krylov subspaces. Numerical experiments on a challenging real-world test problem reveal that these methods are a promising preconditioner-free alternative to well-established approaches based on preconditioned Newton--Krylov implementations of the backward differentiation formulas. References E. J. Carr, T. J. Moroney and I. W. Turner. Efficient simulation of unsaturated flow using exponential time integration. Appl. Math. Comput., 217(14): 6587--6596, 2011. doi:10.1016/j.amc.2011.01.041 E. J. Carr, I. W. Turner and M. Ilic. Krylov subspace approximations for the exponential Euler method: error estimates and the harmonic Ritz approximant. ANZIAM Journal, 52: C612--C627, 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3938 E. J. Carr, I. W. Turner and P. Perre. A variable step size Jacobian-free exponential integrator for simulating transport in heterogeneous porous media: application to wood drying. J. Comput. Phys., 233: 66-82, 2013. doi:10.1016/j.jcp.2012.07.024 Sundials user documentation. https://computation.llnl.gov/casc/sundials/documentation/documentation.html M. Hochbruck, C. Lubich and H. Selhofer. Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput., 19(5): 1552--1574, 1998. doi:10.1137/S1064827595295337 M. Hochbruck, A. Ostermann and J. Schweitzer. Exponential Rosenbrock-type methods. SIAM J. Numer. Anal., 47(1): 786--803, 2009. doi:10.1137/080717717 C. T. Kelley. Solving nonlinear equations with Newton's method. SIAM, USA, 2003. D. A. Knoll and D. E. Keyes. Jacobian-free Newton--Krylov methods: a survey of approaches and applications. J. Comput. Phys., 193(2): 357--397, 2004. doi:10.1016/j.jcp.2003.08.010 M. Tokman. Efficient integration of large stiff systems of ODEs with exponential propagation iteration (EPI) methods. J. Comput. Phys., 213: 748--776, 2006. doi:10.1016/j.jcp.2005.08.032