2,038 research outputs found
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
A Polynomial Spectral Calculus for Analysis of DG Spectral Element Methods
We introduce a polynomial spectral calculus that follows from the summation
by parts property of the Legendre-Gauss-Lobatto quadrature. We use the calculus
to simplify the analysis of two multidimensional discontinuous Galerkin
spectral element approximations
An Entropy Stable Nodal Discontinuous Galerkin Method for the Two Dimensional Shallow Water Equations on Unstructured Curvilinear Meshes with Discontinuous Bathymetry
We design an arbitrary high-order accurate nodal discontinuous Galerkin
spectral element approximation for the nonlinear two dimensional shallow water
equations with non-constant, possibly discontinuous, bathymetry on
unstructured, possibly curved, quadrilateral meshes. The scheme is derived from
an equivalent flux differencing formulation of the split form of the equations.
We prove that this discretisation exactly preserves the local mass and
momentum. Furthermore, combined with a special numerical interface flux
function, the method exactly preserves the mathematical entropy, which is the
total energy for the shallow water equations. By adding a specific form of
interface dissipation to the baseline entropy conserving scheme we create a
provably entropy stable scheme. That is, the numerical scheme discretely
satisfies the second law of thermodynamics. Finally, with a particular
discretisation of the bathymetry source term we prove that the numerical
approximation is well-balanced. We provide numerical examples that verify the
theoretical findings and furthermore provide an application of the scheme for a
partial break of a curved dam test problem
Spectral/hp element methods: recent developments, applications, and perspectives
The spectral/hp element method combines the geometric flexibility of the
classical h-type finite element technique with the desirable numerical
properties of spectral methods, employing high-degree piecewise polynomial
basis functions on coarse finite element-type meshes. The spatial approximation
is based upon orthogonal polynomials, such as Legendre or Chebychev
polynomials, modified to accommodate C0-continuous expansions. Computationally
and theoretically, by increasing the polynomial order p, high-precision
solutions and fast convergence can be obtained and, in particular, under
certain regularity assumptions an exponential reduction in approximation error
between numerical and exact solutions can be achieved. This method has now been
applied in many simulation studies of both fundamental and practical
engineering flows. This paper briefly describes the formulation of the
spectral/hp element method and provides an overview of its application to
computational fluid dynamics. In particular, it focuses on the use the
spectral/hp element method in transitional flows and ocean engineering.
Finally, some of the major challenges to be overcome in order to use the
spectral/hp element method in more complex science and engineering applications
are discussed
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
A discontinuous Galerkin method for a new class of Green-Naghdi equations on simplicial unstructured meshes
In this paper, we introduce a discontinuous Finite Element formulation on
simplicial unstructured meshes for the study of free surface flows based on the
fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a
new class of asymptotically equivalent equations, which have a simplified
analytical structure, we consider a decoupling strategy: we approximate the
solutions of the classical shallow water equations supplemented with a source
term globally accounting for the non-hydrostatic effects and we show that this
source term can be computed through the resolution of scalar elliptic
second-order sub-problems. The assets of the proposed discrete formulation are:
(i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary
order of approximation in space, (iii) the exact preservation of the motionless
steady states, (iv) the preservation of the water height positivity, (v) a
simple way to enhance any numerical code based on the nonlinear shallow water
equations. The resulting numerical model is validated through several
benchmarks involving nonlinear wave transformations and run-up over complex
topographies
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