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Preconditioned Krylov solvers for structure-preserving discretisations
A key consideration in the development of numerical schemes for
time-dependent partial differential equations (PDEs) is the ability to preserve
certain properties of the continuum solution, such as associated conservation
laws or other geometric structures of the solution. There is a long history of
the development and analysis of such structure-preserving discretisation
schemes, including both proofs that standard schemes have structure-preserving
properties and proposals for novel schemes that achieve both high-order
accuracy and exact preservation of certain properties of the continuum
differential equation. When coupled with implicit time-stepping methods, a
major downside to these schemes is that their structure-preserving properties
generally rely on exact solution of the (possibly nonlinear) systems of
equations defining each time-step in the discrete scheme. For small systems,
this is often possible (up to the accuracy of floating-point arithmetic), but
it becomes impractical for the large linear systems that arise when considering
typical discretisation of space-time PDEs. In this paper, we propose a
modification to the standard flexible generalised minimum residual (FGMRES)
iteration that enforces selected constraints on approximate numerical
solutions. We demonstrate its application to both systems of conservation laws
and dissipative systems
GEMPIC: Geometric ElectroMagnetic Particle-In-Cell Methods
We present a novel framework for Finite Element Particle-in-Cell methods
based on the discretization of the underlying Hamiltonian structure of the
Vlasov-Maxwell system. We derive a semi-discrete Poisson bracket, which retains
the defining properties of a bracket, anti-symmetry and the Jacobi identity, as
well as conservation of its Casimir invariants, implying that the semi-discrete
system is still a Hamiltonian system. In order to obtain a fully discrete
Poisson integrator, the semi-discrete bracket is used in conjunction with
Hamiltonian splitting methods for integration in time. Techniques from Finite
Element Exterior Calculus ensure conservation of the divergence of the magnetic
field and Gauss' law as well as stability of the field solver. The resulting
methods are gauge invariant, feature exact charge conservation and show
excellent long-time energy and momentum behaviour. Due to the generality of our
framework, these conservation properties are guaranteed independently of a
particular choice of the Finite Element basis, as long as the corresponding
Finite Element spaces satisfy certain compatibility conditions.Comment: 57 Page
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