64 research outputs found
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
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
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
Stability of Discontinuous Galerkin Spectral Element Schemes for Wave Propagation when the Coefficient Matrices have Jumps
We use the behavior of the L2 norm of the solutions of linear hyperbolic equations with discontinuous coefficient matrices as a surrogate to infer stability of discontinuous Galerkin spectral element methods (DGSEM). Although the L2 norm is not bounded by the initial data for homogeneous and dissipative boundary conditions for such systems, the L2 norm is easier to work with than a norm that discounts growth due to the discontinuities. We show that the DGSEM with an upwind numerical flux that satisfies the Rankine-Hugoniot (or conservation) condition has the same energy bound as the partial differential equation does in the L2 norm, plus an added dissipation that depends on how much the approximate solution fails to satisfy the Rankine-Hugoniot jump
Entropy-stable discontinuous Galerkin approximation with summation-by-parts property for the incompressible Navier-Stokes equations with variable density and artificial compressibility
We present a provably stable discontinuous Galerkin spectral element method
for the incompressible Navier-Stokes equations with artificial compressibility
and variable density. Stability proofs, which include boundary conditions, that
follow a continuous entropy analysis are provided. We define a mathematical
entropy function that combines the traditional kinetic energy and an additional
energy term for the artificial compressiblity, and derive its associated
entropy conservation law. The latter allows us to construct a provably stable
split-form nodal Discontinuous Galerkin (DG) approximation that satisfies the
summation-by-parts simultaneous-approximation-term (SBP-SAT) property. The
scheme and the stability proof are presented for general curvilinear
three-dimensional hexahedral meshes. We use the exact Riemann solver and the
Bassi-Rebay 1 (BR1) scheme at the inter-element boundaries for inviscid and
viscous fluxes respectively, and an explicit low storage Runge-Kutta RK3 scheme
to integrate in time. We assess the accuracy and robustness of the method by
solving the Kovasznay flow, the inviscid Taylor-Green vortex, and the
Rayleigh-Taylor instability
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