79 research outputs found
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
A free-energy stable nodal discontinuous Galerkin approximation with summation-by-parts property for the Cahn-Hilliard equation
We present a nodal Discontinuous Galerkin (DG) scheme for the Cahn-Hilliard
equation that satisfies the summation-by-parts simultaneous-approximation-term
(SBP-SAT) property. The latter permits us to show that the discrete free-energy
is bounded, and as a result, the scheme is provably stable. The scheme and the
stability proof are presented for general curvilinear three-dimensional
hexahedral meshes. We use the Bassi-Rebay 1 (BR1) scheme to compute interface
fluxes, and an IMplicit-EXplicit (IMEX) scheme to integrate in time. Lastly, we
test the theoretical findings numerically and present examples for two and
three-dimensional problems
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
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
Entropy Stable Summation-by-Parts Methods for Hyperbolic Conservation Laws on h/p Non-Conforming Meshes
In this work we present high-order primary conservative and entropy stable schemes for hyperbolic systems of conservation laws with geometric (h) and algebraic (p) non-conforming rectangular meshes. Throughout we rely on summation-by-parts (SBP) operators which discretely mimic the integration-by-parts rule to construct stable approximations. Thus, the discrete proofs of primary conservation and entropy stability can be done in a one-to-one fashion to the continuous analysis. Here, we consider different SBP operators based on finite difference as well as discontinuous Galerkin approaches. We derive non-conforming schemes by extending ideas of high-order primary conservative and entropy stable SBP methods on conforming meshes. Here, special attention is given to the coupling between non-conforming elements. The coupling is instructed to entropy stable projection operators. However, these projection operators suffer from a suboptimal degree. Therefore, we develop degree preserving SBP operators where the norm matrix has a higher degree compared to classical SBP operators. With these operators it is possible to construct entropy stable projection operators which have the same degree as the SBP differentiationmatrix. Typically, high-order primary conservative and entropy stable schemes are semi-discrete methods with a discretized spatial domain and assuming continuity in time. Therefore, temporal errors are introduced when integrating the conservation laws in time with standard methods, e.g. Runge-Kutta schemes, for which the entropy can have an unpredictable temporal behaviour. Thus, we extend high-order primary conservative and entropy stable semi-discrete methods to fully-discrete schemes on conforming and non-conforming meshes. This results in an implicit space-time method. We introduce a simple mesh generation strategy to obtain quadrilateral meshes surrounding two dimensional complex geometries. Finally, with the generated meshes we simulate a flow around a NACA0012 airfoil to demonstrate the benefits of considering non-conforming elements for a practical simulation
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