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Abstract: We consider the approximation of trigonometric operator functions that arise in the numerical solution of wave equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behavior if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we propose and analyze a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators. In contrast to standard Krylov methods, the convergence will be independent of the norm of the operator and thus of its spatial discretization. We will discuss efficient implementations for finite element discretizations and illustrate our analysis with numerical experiments.

Abstract: We consider the approximation of trigonometric operator functions that arise in the numerical solution of wave equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behavior if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we propose and analyze a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators. In contrast to standard Krylov methods, the convergence will be independent of the norm of the operator and thus of its spatial discretization. We will discuss efficient implementations for finite element discretizations and illustrate our analysis with numerical experiments.

In this paper we present a new one-dimensional full electromagnetic relativistic hybrid plasma model. The full kinetic particle-in-cell (PIC) and hydrodynamic model have been combined in the single hybrid plasma code H-VLPL (hybrid virtual laser plasma laboratory). The semi-implicit algorithm allows to simulate plasmas of arbitrary densities via automatic reduction of the highest plasma frequencies down to the numerically stable range. At the same time, the model keeps the correct spatial scales like the plasma skin depth. We discuss the numerically efficient implementation of this model. Further, we carefully test the hybrid model validity by applying it to a series of physical examples. The new mathematical method allows to overcome the typical time step restrictions of explicit PIC codes. [Copyright &y& Elsevier]