17 research outputs found
Extension of the Finite Integration Technique including dynamic mesh refinement and its application to self-consistent beam dynamics simulations
An extension of the framework of the Finite Integration Technique (FIT)
including dynamic and adaptive mesh refinement is presented. After recalling
the standard formulation of the FIT, the proposed mesh adaptation procedure is
described. Besides the linear interpolation approach, a novel interpolation
technique based on specialized spline functions for approximating the discrete
electromagnetic field solution during mesh adaptation is introduced. The
standard FIT on a fixed mesh and the new adaptive approach are applied to a
simulation test case with known analytical solution. The numerical accuracy of
the two methods are shown to be comparable. The dynamic mesh approach is,
however, much more efficient. This is also demonstrated for the full scale
modeling of the complete RF gun at the Photo Injector Test Facility DESY
Zeuthen (PITZ) on a single computer. Results of a detailed design study
addressing the effects of individual components of the gun onto the beam
emittance using a fully self-consistent approach are presented.Comment: 33 pages, 14 figures, 4 table
A Space-Time Discontinuous Galerkin Trefftz Method for time dependent Maxwell's equations
We consider the discretization of electromagnetic wave propagation problems
by a discontinuous Galerkin Method based on Trefftz polynomials. This method
fits into an abstract framework for space-time discontinuous Galerkin methods
for which we can prove consistency, stability, and energy dissipation without
the need to completely specify the approximation spaces in detail. Any method
of such a general form results in an implicit time-stepping scheme with some
basic stability properties. For the local approximation on each space-time
element, we then consider Trefftz polynomials, i.e., the subspace of
polynomials that satisfy Maxwell's equations exactly on the respective element.
We present an explicit construction of a basis for the local Trefftz spaces in
two and three dimensions and summarize some of their basic properties. Using
local properties of the Trefftz polynomials, we can establish the
well-posedness of the resulting discontinuous Galerkin Trefftz method.
Consistency, stability, and energy dissipation then follow immediately from the
results about the abstract framework. The method proposed in this paper
therefore shares many of the advantages of more standard discontinuous Galerkin
methods, while at the same time, it yields a substantial reduction in the
number of degrees of freedom and the cost for assembling. These benefits and
the spectral convergence of the scheme are demonstrated in numerical tests
Symmetric Interior Penalty Discontinuous Galerkin Discretisations and Block Preconditioning for Heterogeneous Stokes Flow
Provable stable arbitrary order symmetric interior penalty (SIP) discontinuous Galerkin discretizations of heterogeneous, incompressible Stokes flow utilizing -- elements and hierarchical Legendre basis polynomials are developed and investigated. For solving the resulting linear system, a block preconditioned iterative method is proposed. The nested viscous problem is solved by a -multilevel preconditioned Krylov subspace method. For the -coarsening, a two-level method utilizing element-block Jacobi preconditioned iterations as a smoother is employed. Piecewise bilinear () and piecewise constant () -coarse spaces are considered. Finally, Galerkin -coarsening is proposed and investigated for the two -coarse spaces considered. Through a number of numerical experiments, we demonstrate that utilizing the coarse space results in the most robust -multigrid method for heterogeneous Stokes flow. Using this coarse space we observe that the convergence of the overall Stokes solver appears to be robust with respect to the jump in the viscosity and only mildly depending on the polynomial order . It is demonstrated and supported by theoretical results that the convergence of the SIP discretizations and the iterative methods rely on a sharp choice of the penalty parameter based on local values of the viscosity
A priori error analysis of space–time Trefftz discontinuous Galerkin methods for wave problems
We present and analyse a space–time discontinuous Galerkin method for wave propagation problems. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space–time) mesh. The method considered is a modification of the discontinuous Galerkin schemes of Kretzschmar et al. (2014) and of Monk & Richter (2005). For Maxwell’s equations in one space dimension, we prove stability of the method, quasi-optimality, best approximation estimates for polynomial Trefftz spaces and (fully explicit) error bounds with high order in the meshwidth and in the polynomial degree.
The analysis framework also applies to scalar wave problems and Maxwell’s equations in higher space dimensions. Some numerical experiments demonstrate the theoretical results proved and the faster convergence compared to the non-Trefftz version of the scheme
Trefftz Absorbing Boundary Conditions in Analytical, Discontinuous Galerkin and Finite Difference Form
We explore a novel avenue for generating absorbing boundary conditions for wave problems. The key part of our approach is Trefftz approximations of the solution, i.e. approximations by functions satisfying locally the underlying wave equation. Trefftz functions include outgoing waves only (and possibly evanescent waves), but no incoming waves. We show how this idea can be applied in three different contexts: analytical, Discontinuous Galerkin, and finite difference