19,692 research outputs found
The role of numerical boundary procedures in the stability of perfectly matched layers
In this paper we address the temporal energy growth associated with numerical
approximations of the perfectly matched layer (PML) for Maxwell's equations in
first order form. In the literature, several studies have shown that a
numerical method which is stable in the absence of the PML can become unstable
when the PML is introduced. We demonstrate in this paper that this instability
can be directly related to numerical treatment of boundary conditions in the
PML. First, at the continuous level, we establish the stability of the constant
coefficient initial boundary value problem for the PML. To enable the
construction of stable numerical boundary procedures, we derive energy
estimates for the variable coefficient PML. Second, we develop a high order
accurate and stable numerical approximation for the PML using
summation--by--parts finite difference operators to approximate spatial
derivatives and weak enforcement of boundary conditions using penalties. By
constructing analogous discrete energy estimates we show discrete stability and
convergence of the numerical method. Numerical experiments verify the
theoretical result
A Provably Stable Discontinuous Galerkin Spectral Element Approximation for Moving Hexahedral Meshes
We design a novel provably stable discontinuous Galerkin spectral element
(DGSEM) approximation to solve systems of conservation laws on moving domains.
To incorporate the motion of the domain, we use an arbitrary
Lagrangian-Eulerian formulation to map the governing equations to a fixed
reference domain. The approximation is made stable by a discretization of a
skew-symmetric formulation of the problem. We prove that the discrete
approximation is stable, conservative and, for constant coefficient problems,
maintains the free-stream preservation property. We also provide details on how
to add the new skew-symmetric ALE approximation to an existing discontinuous
Galerkin spectral element code. Lastly, we provide numerical support of the
theoretical results
Review of Summation-by-parts schemes for initial-boundary-value problems
High-order finite difference methods are efficient, easy to program, scales
well in multiple dimensions and can be modified locally for various reasons
(such as shock treatment for example). The main drawback have been the
complicated and sometimes even mysterious stability treatment at boundaries and
interfaces required for a stable scheme. The research on summation-by-parts
operators and weak boundary conditions during the last 20 years have removed
this drawback and now reached a mature state. It is now possible to construct
stable and high order accurate multi-block finite difference schemes in a
systematic building-block-like manner. In this paper we will review this
development, point out the main contributions and speculate about the next
lines of research in this area
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