453 research outputs found
GPU-accelerated discontinuous Galerkin methods on hybrid meshes
We present a time-explicit discontinuous Galerkin (DG) solver for the
time-domain acoustic wave equation on hybrid meshes containing vertex-mapped
hexahedral, wedge, pyramidal and tetrahedral elements. Discretely energy-stable
formulations are presented for both Gauss-Legendre and Gauss-Legendre-Lobatto
(Spectral Element) nodal bases for the hexahedron. Stable timestep restrictions
for hybrid meshes are derived by bounding the spectral radius of the DG
operator using order-dependent constants in trace and Markov inequalities.
Computational efficiency is achieved under a combination of element-specific
kernels (including new quadrature-free operators for the pyramid), multi-rate
timestepping, and acceleration using Graphics Processing Units.Comment: Submitted to CMAM
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 short note on a Bernstein-Bezier basis for the pyramid
We introduce a Bernstein-Bezier basis for the pyramid, whose restriction to
the face reduces to the Bernstein-Bezier basis on the triangle or
quadrilateral. The basis satisfies the standard positivity and partition of
unity properties common to Bernstein polynomials, and spans the same space as
non-polynomial pyramid bases in the literature.Comment: Submitte
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