Consider a differential equation y ′ = A(t, y)y, y(0) = y0 with y0 ∈ G and A : R+ × G → g, where g is a Lie algebra of the matricial Lie group G. Every B ∈ g can be mapped to G by the matrix exponential map exp (tB) with t ∈ R. Most numerical methods for solving ordinary differential equations (ODEs) on Lie groups are based on the idea of representing the approximation yn of the exact solution ...

Error bounds for the Strang splitting in the presence of unbounded operators are derived in a general setting and are applied to evolutionary Schrödinger equations and their pseudo-spectral space discretization. AMS subject classification: 65M15, 65L05, 65M70.

In this work, the error behavior of operator splitting methods is analyzed for highly-oscillatory differential equations. The scope of applications includes time-dependent nonlinear Schrödinger equations, where the evolution operator associated with the principal linear part is highly-oscillatory and periodic in time. In a first step, a known convergence result for the second-order Strang split...

We propose and analyze a numerical method for time-dependent linear Schrödinger equations with uncertain parameters in both the potential initial data. The random are discretized by stochastic collocation on sparse grid, sample solutions nodes approximated Strang splitting method. computational work is reduced multilevel strategy, i.e., combining information obtained from computed different ref...

Using periodic Strang{Fix conditions, we can give an approach to error estimates for periodic interpolation on equidistant and sparse grids for functions from certain Besov spaces.

Consider a diierential equation y 0 = A(t; y)y; y(0) = y0 with y0 2 G and A : R + G ! g, where g is a Lie algebra of the matricial Lie group G. Every B 2 g can be mapped to G by the matrix exponential map exp (tB) with t 2 R. Most numerical methods for solving ordinary diierential equations (ODEs) on Lie groups are based on the idea of representing the approximation yn of the exact solution y(t...

Jing-Mei Qiu and Andrew Christlieb 3 Abstract In this paper, we propose a novel Vlasov solver based on a semi-Lagrangian method which combines Strang splitting in time with high order WENO (weighted essentially nonoscillatory) reconstruction in space. A key insight in this work is that the spatial interpolation matrices, used in the reconstruction process of a semi-Lagrangian approach to linear...

Usually transport-chemistry models are solved by operator-splitting methods where the chemistry and the vertical transport processes are integrated in a coupled way by implicit methods of order two or higher with automatic error control. In mesoscale applications the integration intervals are relatively short and small initial transients occur for the chemistry-diffusion operator which are an a...

We first recall the linear approximation of Strang and Fix [1], in the context of multiresolution analysis and wavelets. The original theorem is concerned with general finite element approximations. Theorem 1 (Fix-Strang) Let p ∈ N, φ a (bi)orthogonal scaling function, and ψ * its conjugate wavelet. The following three conditions are equivalent • for any 0 ≤ k < p, there exists a polynomial θ k...