276 research outputs found
An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs
We extend the entropy stable high order nodal discontinuous Galerkin spectral
element approximation for the non-linear two dimensional shallow water
equations presented by Wintermeyer et al. [N. Wintermeyer, A. R. Winters, G. J.
Gassner, and D. A. Kopriva. An entropy stable nodal discontinuous Galerkin
method for the two dimensional shallow water equations on unstructured
curvilinear meshes with discontinuous bathymetry. Journal of Computational
Physics, 340:200-242, 2017] with a shock capturing technique and a positivity
preservation capability to handle dry areas. The scheme preserves the entropy
inequality, is well-balanced and works on unstructured, possibly curved,
quadrilateral meshes. For the shock capturing, we introduce an artificial
viscosity to the equations and prove that the numerical scheme remains entropy
stable. We add a positivity preserving limiter to guarantee non-negative water
heights as long as the mean water height is non-negative. We prove that
non-negative mean water heights are guaranteed under a certain additional time
step restriction for the entropy stable numerical interface flux. We implement
the method on GPU architectures using the abstract language OCCA, a unified
approach to multi-threading languages. We show that the entropy stable scheme
is well suited to GPUs as the necessary extra calculations do not negatively
impact the runtime up to reasonably high polynomial degrees (around ). We
provide numerical examples that challenge the shock capturing and positivity
properties of our scheme to verify our theoretical findings
Flux-corrected transport algorithms for continuous Galerkin methods based on high order Bernstein finite elements
This work extends the flux-corrected transport (FCT) methodology to arbitrary-order continuous finite element discretizations
of scalar conservation laws on simplex meshes. Using Bernstein polynomials as local basis functions, we
constrain the total variation of the numerical solution by imposing local discrete maximum principles on the Bézier
net. The design of accuracy-preserving FCT schemes for high order Bernstein-Bézier finite elements requires the development
of new algorithms and/or generalization of limiting techniques tailored for linear and multilinear Lagrange
elements. In this paper, we propose (i) a new discrete upwinding strategy leading to local extremum bounded low
order approximations with compact stencils, (ii) a high order stabilization operator based on gradient recovery, and
(iii) new localized limiting techniques for antidi usive element contributions. The optional use of a smoothness indicator,
based on a second derivative test, makes it possible to potentially avoid unnecessary limiting at smooth extrema
and achieve optimal convergence rates for problems with smooth solutions. The accuracy of the proposed schemes is
assessed in numerical studies for the linear transport equation in 1D and 2D
A CS guide to the quantum singular value transformation
We present a simplified exposition of some pieces of [Gily\'en, Su, Low, and
Wiebe, STOC'19, arXiv:1806.01838], which introduced a quantum singular value
transformation (QSVT) framework for applying polynomial functions to
block-encoded matrices. The QSVT framework has garnered substantial recent
interest from the quantum algorithms community, as it was demonstrated by
[GSLW19] to encapsulate many existing algorithms naturally phrased as an
application of a matrix function. First, we posit that the lifting of quantum
singular processing (QSP) to QSVT is better viewed not through Jordan's lemma
(as was suggested by [GSLW19]) but as an application of the cosine-sine
decomposition, which can be thought of as a more explicit and stronger version
of Jordan's lemma. Second, we demonstrate that the constructions of bounded
polynomial approximations given in [GSLW19], which use a variety of ad hoc
approaches drawing from Fourier analysis, Chebyshev series, and Taylor series,
can be unified under the framework of truncation of Chebyshev series, and
indeed, can in large part be matched via a bounded variant of a standard
meta-theorem from [Trefethen, 2013]. We hope this work finds use to the
community as a companion guide for understanding and applying the powerful
framework of [GSLW19].Comment: 32 pages; v2 QSVT proofs more self-contained, additional result
separating bounded and unbounded polynomial approximation
Proximal Galerkin: A structure-preserving finite element method for pointwise bound constraints
The proximal Galerkin finite element method is a high-order, low iteration
complexity, nonlinear numerical method that preserves the geometric and
algebraic structure of bound constraints in infinite-dimensional function
spaces. This paper introduces the proximal Galerkin method and applies it to
solve free boundary problems, enforce discrete maximum principles, and develop
scalable, mesh-independent algorithms for optimal design. The paper leads to a
derivation of the latent variable proximal point (LVPP) algorithm: an
unconditionally stable alternative to the interior point method. LVPP is an
infinite-dimensional optimization algorithm that may be viewed as having an
adaptive barrier function that is updated with a new informative prior at each
(outer loop) optimization iteration. One of the main benefits of this algorithm
is witnessed when analyzing the classical obstacle problem. Therein, we find
that the original variational inequality can be replaced by a sequence of
semilinear partial differential equations (PDEs) that are readily discretized
and solved with, e.g., high-order finite elements. Throughout this work, we
arrive at several unexpected contributions that may be of independent interest.
These include (1) a semilinear PDE we refer to as the entropic Poisson
equation; (2) an algebraic/geometric connection between high-order
positivity-preserving discretizations and certain infinite-dimensional Lie
groups; and (3) a gradient-based, bound-preserving algorithm for two-field
density-based topology optimization. The complete latent variable proximal
Galerkin methodology combines ideas from nonlinear programming, functional
analysis, tropical algebra, and differential geometry and can potentially lead
to new synergies among these areas as well as within variational and numerical
analysis
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