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Exponential Rosenbrock-type methods. (English) Zbl 1193.65119

Authors’ 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.

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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