Implicit Runge-Kutta Methods and Discontinuous Galerkin Discretizations for Linear Maxwell's Equations

@article{Hochbruck2015ImplicitRM,
  title={Implicit Runge-Kutta Methods and Discontinuous Galerkin Discretizations for Linear Maxwell's Equations},
  author={Marlis Hochbruck and Tomislav Pazur},
  journal={SIAM J. Numer. Anal.},
  year={2015},
  volume={53},
  pages={485-507}
}
In this paper we consider implicit Runge--Kutta methods for the time integration of linear Maxwell's equations. We first present error bounds for the abstract Cauchy problem which respect the unboundedness of the differential operators using energy techniques. The error bounds hold for algebraically stable and coercive methods such as Gauss and Radau collocation methods. The results for the abstract evolution equation are then combined with a discontinuous Galerkin discretization in space using… 

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References

SHOWING 1-10 OF 35 REFERENCES

Interior penalty discontinuous Galerkin method for Maxwell's equations

TLDR
It is shown that the symmetric, interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell's equations in second-order form is also optimally convergent in the L 2 -norm, on tetrahedral meshes and for smooth material coefficients.

Convergence and stability of a discontinuous galerkin time-domain method for the 3D heterogeneous maxwell equations on unstructured meshes

TLDR
Convergence is proved for Discontinuous elements on tetrahedral meshes, as well as a discrete divergence preservation property, and a discrete analog of the electromagnetic energy is conserved for metallic cavities.

Runge-Kutta approximation of quasi-linear parabolic equations

TLDR
It is shown that the convergence properties of implicit Runge-Kutta meth- ods applied to time discretization of parabolic equations with time- or solution- dependent operator depends on the type of boundary conditions.

TEMPORAL CONVERGENCE OF A LOCALLY IMPLICIT DISCONTINUOUS GALERKIN METHOD FOR MAXWELL'S EQUATIONS ∗

TLDR
The temporal convergence of a locally implicit discontinuous Galerkin (DG) method for Maxwell's equations modeling electromagnetic wave propagation is studied to wonder whether the method retains its second-order ordinary differential equation convergence under stable simultaneous space-time grid refinement towards the true partial differential equation solution.

A high-order discontinuous Galerkin method with time-accurate local time stepping for the Maxwell equations

TLDR
The proposed LTS scheme provides high order of accuracy in space and time on unstructured tetrahedral meshes and is applied to a well-acknowledged test case and comparisons with analytical reference solutions confirm the performance of the proposed method.

Explicit local time-stepping methods for Maxwell's equations

Error analysis of a discontinuous Galerkin method for Maxwell equations in dispersive media

Error analysis of implicit and exponential time integration of linear Maxwell's equations

TLDR
The error analysis of time integrators is done both for continuous Maxwell's equations in a semigroup theory framework and for space discrete problem obtained by discretizing Maxwell’s equations in space by using discontinuous Galerkin finite element method.

An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation

We prove Lp stability and error estimates for the discontinuous Galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain. Using finite element analysis

Explicit Runge-Kutta Schemes and Finite Elements with Symmetric Stabilization for First-Order Linear PDE Systems

TLDR
A general set of properties on the space stabilization, encompassing continuous and discontinuous finite elements, is identified, under which stability estimates are proved using energy arguments and quasi-optimal convergence rates for smooth solutions in space and time are established.