Operator splittings and spatial approximations for evolution equations

@article{Btkai2009OperatorSA,
  title={Operator splittings and spatial approximations for evolution equations},
  author={Andr{\'a}s B{\'a}tkai and Petra Csom{\'o}s and Gregor Nickel},
  journal={Journal of Evolution Equations},
  year={2009},
  volume={9},
  pages={613-636}
}
The convergence of various operator splitting procedures, such as the sequential, the Strang and the weighted splitting, is investigated in the presence of a spatial approximation. To this end the relevant notions and results of numerical analysis are presented, a variant of Chernoff’s product formula is proved and the general Trotter–Kato approximation theorem is used. The methods are applied to an abstract partial delay differential equation. 
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20 Citations
Background Citations
5
Methods Citations
6

20 Citations

Operator splitting for non-autonomous evolution equations
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…
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TLDR
The errors between Lie-Trotter splitting and Strang splitting are compared by discretizing the space into N sub-intervals, and the convergence rate is computed.
ON THE ORDER OF OPERATOR SPLITTING METHODS FOR TIME-DEPENDENT LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS
TLDR
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Operator Semigroups for Convergence Analysis
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The paper deals with abstract Cauchy problems and presents how their solutions are approximated by using space and time discretisations and introduces and applies the basic notions of operator semigroup theory.
Fourier-splitting method for solving hyperbolic LQR problems
TLDR
A novel approach based on operator splitting idea combined with Fourier's method to efficiently compute the optimal state of linear quadratic regulator problems for hyperbolic partial differential equations where the dynamics is driven by a strongly continuous semigroup is proposed.
NUMERICAL ERROR ANALYSIS OF A SPLITTING METHOD FOR THE RESOLUTION OF THE ANISOTROPIC SCHRÖDINGER EQUATION
This paper deals with the numerical analysis of three finite difference methods for the resolution of an anisotropic, linear 2D time-dependent Schrödinger equation. The physical context corresponding…
Operator splitting for nonautonomous delay equations
Stability and Convergence of Product Formulas for Operator Matrices
We present easy to verify conditions implying stability estimates for operator matrix splittings which ensure convergence of the associated Trotter, Strang and weighted product formulas. The results…
PDE approximation of large systems of differential equations
A large system of ordinary differential equations is approximated by a parabolic partial differential equation with dynamic boundary condition and a different one with Robin boundary condition. Using…
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