Exponential Rosenbrock-Type Methods
Top Cited Papers
- 1 January 2009
- journal article
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Numerical Analysis
- Vol. 47 (1), 786-803
- https://doi.org/10.1137/080717717
Abstract
We introduce a new class of exponential integrators for the numerical integration of large-scale systems of stiff differential equations. These so-called Rosenbrock-type methods linearize the flow in each time step and make use of the matrix exponential and related functions of the Jacobian. In contrast to standard integrators, the methods are fully explicit and do not require the numerical solution of linear systems. We analyze the convergence properties of these integrators in a semigroup framework of semilinear evolution equations in Banach spaces. In particular, we derive an abstract stability and convergence result for variable step sizes. This analysis further provides the required order conditions and thus allows us to construct pairs of embedded methods. We present a third-order method with two stages, and a fourth-order method with three stages, respectively. The application of the required matrix functions to vectors are computed by Krylov subspace approximations. We briefly discuss these implementation issues, and we give numerical examples that demonstrate the efficiency of the new integrators.Keywords
This publication has 20 references indexed in Scilit:
- Approximation of matrix operators applied to multiple vectorsMathematics and Computers in Simulation, 2008
- A class of explicit multistep exponential integrators for semilinear problemsNumerische Mathematik, 2005
- Fourth-Order Time-Stepping for Stiff PDEsSIAM Journal on Scientific Computing, 2005
- Explicit Exponential Runge--Kutta Methods for Semilinear Parabolic ProblemsSIAM Journal on Numerical Analysis, 2005
- Interpolating discrete advection–diffusion propagators at Leja sequencesJournal of Computational and Applied Mathematics, 2004
- Exponential Time Differencing for Stiff SystemsJournal of Computational Physics, 2002
- Exponential Integrators for Large Systems of Differential EquationsSIAM Journal on Scientific Computing, 1998
- On Krylov Subspace Approximations to the Matrix Exponential OperatorSIAM Journal on Numerical Analysis, 1997
- Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmeticNumerical Linear Algebra with Applications, 1995
- VODE: A Variable-Coefficient ODE SolverSIAM Journal on Scientific and Statistical Computing, 1989