377 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
Entropy and energy conservation for thermal atmospheric dynamics using mixed compatible finite elements
Atmospheric systems incorporating thermal dynamics must be stable with
respect to both energy and entropy. While energy conservation can be enforced
via the preservation of the skew-symmetric structure of the Hamiltonian form of
the equations of motion, entropy conservation is typically derived as an
additional invariant of the Hamiltonian system, and satisfied via the exact
preservation of the chain rule. This is particularly challenging since the
function spaces used to represent the thermodynamic variables in compatible
finite element discretisations are typically discontinuous at element
boundaries. In the present work we negate this problem by constructing our
equations of motion via weighted averages of skew-symmetric formulations using
both flux form and material form advection of thermodynamic variables, which
allow for the necessary cancellations required to conserve entropy without the
chain rule. We show that such formulations allow for stable simulations of both
the thermal shallow water and 3D compressible Euler equations on the sphere
using mixed compatible finite elements without entropy damping
Provably stable numerical method for the anisotropic diffusion equation in toroidally confined magnetic fields
We present a novel numerical method for solving the anisotropic diffusion
equation in toroidally confined magnetic fields which is efficient, accurate
and provably stable. The continuous problem is written in terms of a derivative
operator for the perpendicular transport and a linear operator, obtained
through field line tracing, for the parallel transport. We derive energy
estimates of the solution of the continuous initial boundary value problem. A
discrete formulation is presented using operator splitting in time with the
summation by parts finite difference approximation of spatial derivatives for
the perpendicular diffusion operator. Weak penalty procedures are derived for
implementing both boundary conditions and parallel diffusion operator obtained
by field line tracing. We prove that the fully-discrete approximation is
unconditionally stable and asymptotic preserving. Discrete energy estimates are
shown to match the continuous energy estimate given the correct choice of
penalty parameters. Convergence tests are shown for the perpendicular operator
by itself, and the ``NIMROD benchmark" problem is used as a manufactured
solution to show the full scheme converges even in the case where the
perpendicular diffusion is zero. Finally, we present a magnetic field with
chaotic regions and islands and show the contours of the anisotropic diffusion
equation reproduce key features in the field.Comment: 33 pages, 8 figure
On well-posed boundary conditions and energy stable finite volume method for the linear shallow water wave equation
We derive and analyse well-posed boundary conditions for the linear shallow
water wave equation. The analysis is based on the energy method and it
identifies the number, location and form of the boundary conditions so that the
initial boundary value problem is well-posed. A finite volume method is
developed based on the summation-by-parts framework with the boundary
conditions implemented weakly using penalties. Stability is proven by deriving
a discrete energy estimate analogous to the continuous estimate. The continuous
and discrete analysis covers all flow regimes. Numerical experiments are
presented verifying the analysis.Comment: 23 pages, 4 figure
An efficient method for the anisotropic diffusion equation in magnetic fields
We solve the anisotropic diffusion equation in 2D, where the dominant
direction of diffusion is defined by a vector field which does not conform to a
Cartesian grid. Our method uses operator splitting to separate the diffusion
perpendicular and parallel to the vector field. The slow time scale is solved
using a provably stable finite difference formulation in the perpendicular to
the vector field, and an integral operator for the diffusion parallel to it.
Energy estimates are shown to for the continuous and semi-discrete cases.
Numerical experiments are performed showing convergence of the method, and
examples is given to demonstrate the capabilities of the method
Introduction
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