Journal of Computational and Applied Mathematics
On Krylov Subspace Approximations to the Matrix Exponential Operator
SIAM Journal on Numerical Analysis
Exponential Integrators for Large Systems of Differential Equations
SIAM Journal on Scientific Computing
Exponential time differencing for stiff systems
Journal of Computational Physics
Fourth-Order Time-Stepping for Stiff PDEs
SIAM Journal on Scientific Computing
Generalized integrating factor methods for stiff PDEs
Journal of Computational Physics
Explicit Exponential Runge-Kutta Methods for Semilinear Parabolic Problems
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Implementation of exponential Rosenbrock-type integrators
Applied Numerical Mathematics
Exponential Runge--Kutta methods for parabolic problems
Applied Numerical Mathematics
Exponential Rosenbrock-Type Methods
SIAM Journal on Numerical Analysis
A new class of exponential propagation iterative methods of Runge-Kutta type (EPIRK)
Journal of Computational Physics
Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators
SIAM Journal on Scientific Computing
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The aim of this paper is to construct high-order exponential Rosenbrock methods and to analyze their convergence properties for the time discretization of large-scale systems of stiff differential equations. We present a new and simple approach for deriving the stiff order conditions. These conditions allow us to construct new pairs of embedded methods of high order. As an example, we present a fifth-order method with five stages. For particular problems the order conditions can be simplified. It is then even possible to construct a method of order 5 with three stages only. The error analysis is performed in an abstract framework of strongly continuous semigroups that allows us to treat semilinear evolution equations in Banach spaces. Convergence results are proved for variable step size implementations. To demonstrate the efficiency of the new integrators, we give some numerical experiments in MATLAB. In particular, numerical comparisons for semilinear parabolic PDEs in one and two space dimensions are presented.