Flux-corrected transport algorithms preserving the eigenvalue range of symmetric tensor quantities
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Date
2017-04
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Abstract
This paper presents a new approach to constraining the eigenvalue range of symmetric tensors in numerical advection
schemes based on the flux-corrected transport (FCT) algorithm and a continuous finite element discretization. In the
context of element-based FEM-FCT schemes for scalar conservation laws, the numerical solution is evolved using
local extremum diminishing (LED) antidi usive corrections of a low order approximation which is assumed to satisfy
the relevant inequality constraints. The application of a limiter to antidi usive element contributions guarantees that
the corrected solution remains bounded by the local maxima and minima of the low order predictor.
The FCT algorithm to be presented in this paper guarantees the LED property for the largest and smallest eigenvalues
of the transported tensor at the low order evolution step. At the antidi usive correction step, this property is
preserved by limiting the antidi usive element contributions to all components of the tensor in a synchronized manner.
The definition of the element-based correction factors for FCT is based on perturbation bounds for auxiliary
tensors which are constrained to be positive semidefinite to enforce the generalized LED condition. The derivation of
sharp bounds involves calculating the roots of polynomials of degree up to 3. As inexpensive and numerically stable
alternatives, limiting techniques based on appropriate approximations are considered. The ability of the new limiters
to enforce local bounds for the eigenvalue range is confirmed by numerical results for 2D advection problems.
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Keywords
tensor quantity, continuous Galerkin method, flux-corrected transport, artificial diffusion, local discrete maximum principles
Subjects based on RSWK
Galerkin-Methode, FCT-Verfahren