28 research outputs found
A full space-time convergence order analysis of operator splittings for linear dissipative evolution equations
The Douglas--Rachford and Peaceman--Rachford splitting methods are common
choices for temporal discretizations of evolution equations. In this paper we
combine these methods with spatial discretizations fulfilling some easily
verifiable criteria. In the setting of linear dissipative evolution equations
we prove optimal convergence orders, simultaneously in time and space. We apply
our abstract results to dimension splitting of a 2D diffusion problem, where a
finite element method is used for spatial discretization. To conclude, the
convergence results are illustrated with numerical experiments
Operator splitting for dissipative delay equations
We investigate Lie-Trotter product formulae for abstract nonlinear evolution
equations with delay. Using results from the theory of nonlinear contraction
semigroups in Hilbert spaces, we explain the convergence of the splitting
procedure. The order of convergence is also investigated in detail, and some
numerical illustrations are presented.Comment: to appear in Semigroup Foru
New versions of iterative splitting methods for the momentum equation
[EN] In this paper we propose some modifications in the schemes for the iterative splitting techniques defined in Geiser (2009) for partial differential equations and introduce the parallel version of these modified algorithms. Theoretical results related to the order of the iterative splitting for these schemes are obtained. In the numerical experiments we compare the obtained results by applying iterative methods to approximate the solutions of the nonlinear systems obtained from the discretization of the splitting techniques to the mixed convection-diffusion Burgers' equation and a momentum equation that models a viscous flow. The differential equations in each splitting interval are solved by the back-Euler-Newton algorithm using sparse matrices. (C) 2016 Elsevier B.V. All rights reserved.This work has been supported by Ministerio de Economía y Competitividad de España MTM2014-52016-02-2-PGeiser, J.; Hueso Pagoaga, JL.; Martínez Molada, E. (2017). New versions of iterative splitting methods for the momentum equation. Journal of Computational and Applied Mathematics. 309:359-370. https://doi.org/10.1016/j.cam.2016.06.002S35937030
Convergence analysis of Strang splitting for Vlasov-type equations
A rigorous convergence analysis of the Strang splitting algorithm for
Vlasov-type equations in the setting of abstract evolution equations is
provided. It is shown that under suitable assumptions the convergence is of
second order in the time step \tau. As an example, it is verified that the
Vlasov-Poisson equation in 1+1 dimensions fits into the framework of this
analysis. Also, numerical experiments for the latter case are presented.Comment: submitted to the SIAM Journal on Numerical Analysi