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We introduce some general tools to design exact splitting methods to compute numerically semigroups generated by inhomogeneous quadratic differential operators. More precisely, we factorize these semigroups as products of semigroups that can be approximated efficiently, using, for example, pseudo-spectral methods. We highlight the efficiency of these new methods on the examples of the magnetic linear Schrödinger equations with quadratic potentials, some transport equations and some Fokker–Planck equations. [ABSTRACT FROM AUTHOR]

We consider time discretizations of the Vlasov-HMF (Hamiltonian mean-field) equation based on splitting methods between the linear and nonlinear parts. We consider solutions starting in a small Sobolev neighborhood of a spatially homogeneous state satisfying a linearized stability criterion (Penrose criterion). We prove that the numerical solutions exhibit a scattering behavior to a modified state, which implies a nonlinear Landau damping effect with polynomial rate of damping. Moreover, we prove that the modified state is close to the continuous one and provide error estimates with respect to the time step size. [ABSTRACT FROM AUTHOR]

In this paper, we review recent progress on two related issues. Firstly, the discretisation of partial differential equations of quantum mechanics in a semiclassical regime. Due to the presence of a small parameter, such equations exhibit high oscillations and multiscale behaviour, rendering them difficult to discretise. We describe a methodology, using symmetric Zassenhaus splittings in a free Lie algebra, which allows for their exceedingly fast and accurate numerics. The imperative of preserving the unitarity of the underlying flow takes us to the second theme of this paper, approximation of derivatives by skew-symmetric matrices. Here, we identify a gap in the elementary theory of finite-difference approximations: in the presence of Dirichlet boundary conditions, it is impossible to approximate the derivative to order higher than two on a uniform grid! This motivates the investigation of skew symmetry on non-uniform grids, an endeavour which, although still in its infancy, is already replete with interesting results. We conclude by discussing a number of generalisations and open problems. [ABSTRACT FROM AUTHOR]

A comprehensive linear stability analysis of splitting methods is carried out by means of a 2×2 matrix K( x) with polynomial entries (the stability matrix) and the stability polynomial p( x) (the trace of K( x) divided by two). An algorithm is provided for determining the coefficients of all possible time-reversible splitting schemes for a prescribed stability polynomial. It is shown that p( x) carries essentially all the information needed to construct processed splitting methods for numerically approximating the evolution of linear systems. By conveniently selecting the stability polynomial, new integrators with processing for linear equations are built which are orders of magnitude more efficient than other algorithms previously available. [ABSTRACT FROM AUTHOR]

We present new explicit volume-preserving methods based on splitting for polynomial divergence-free vector fields. The methods can be divided in two classes: methods that distinguish between the diagonal part and the off-diagonal part and methods that do not. For the methods in the first class it is possible to combine different treatments of the diagonal and off-diagonal parts, giving rise to a number of possible combinations. [ABSTRACT FROM AUTHOR]