In this paper, we are concerned with the construction and analysis of high order exponential splitting methods for the time integration of abstract evolution equations which are evolved by analytic semigroups. We derive a new class of splitting methods of orders three to fourteen based on complex coefficients. An optimal convergence analysis is presented for the methods when applied to equations on Banach spaces with unbounded vector fields. These results resolve the open question whether there exist splitting schemes with convergence rates greater then two in the context of semigroups. As a concrete application we consider parabolic equations and their dimension splittings. The sharpness of our theoretical error bounds is further illustrated by numerical experiments.

Abstract. We present a convergence analysis for exponential splitting methods applied to linear evolution equations. Our main result states that the classical order of the splitting method is retained in a setting of unbounded operators, without requiring any additional order condition. This is achieved by basing the analysis on the abstract framework of (semi)groups. The convergence analysis also includes generalizations to splittings consisting of more than two operators, and to variable time steps. We conclude by illustrating that the abstract results are applicable in the context of the Schrödinger equation with an external magnetic field or with an unbounded potential.

In this paper we study the convergence behaviour and geometric properties of Strang splitting applied to semilinear evolution equations. We work in an abstract Banach space setting that allows us to analyse a certain class of parabolic equations and their spatial discretizations. For this class of problems, Strang splitting is shown to be stable and second-order convergent. Moreover, it is shown that exponential operator splitting methods and in particular the method of Strang will preserve positivity in certain situations. A numerical illustration of the convergence behaviour is included.

In this paper, we are concerned with splitting methods for the time integration of abstract evolution equations. We introduce an analytic framework which allows us to prove optimal convergence orders for various splitting methods, including the Lie and Peaceman-Rachford splittings. Our setting is applicable for a wide variety of linear equations and their dimension splittings. In particular, we analyze parabolic problems with Dirichlet boundary conditions, as well as degenerate equations on bounded domains. We further illustrate our theoretical results with a set of numerical experiments.

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Abstract. We consider a splitting-based approximation of the abstract differential Riccati equation in the setting of Hilbert-Schmidt operators. The Riccati equation arises in many different areas and is important within the field of optimal control. In this paper we conduct a temporal error analysis and prove that the splitting method converges with the same order as the implicit Euler scheme, under the same low regularity requirements on the initial values. For a subsequent spatial discretization, the abstract setting also yields uniform temporal error bounds with respect to the spatial discretization parameter. The spatial discretizations commonly lead to large-scale problems, where the use of structural properties of the solution is essential. We therefore conclude by proving that the splitting method preserves low-rank structure in the matrix-valued case. Numerical results demonstrate the validity of the convergence analysis.

The Peaceman-Rachford scheme is a commonly used splitting method for discretizing semilinear evolution equations, where the vector fields are given by the sum of one linear and one nonlinear dissipative operator. Typical examples of such equations are reaction-diffusion systems and the damped wave equation. In this paper we conduct a convergence analysis for the Peaceman-Rachford scheme in the setting of dissipative evolution equations on Hilbert spaces. We do not assume Lipschitz continuity of the nonlinearity, as previously done in the literature. First or second order convergence is derived, depending on the regularity of the solution, and a shortened proof for o(1)-convergence is given when only a mild solution exits. The analysis is also extended to the Lie scheme in a Banach space framework. The convergence results are illustrated by numerical experiments for Caginalp's solidification model and the Gray-Scott pattern formation problem.