13 research outputs found
PyFR: An Open Source Framework for Solving Advection-Diffusion Type Problems on Streaming Architectures using the Flux Reconstruction Approach
High-order numerical methods for unstructured grids combine the superior
accuracy of high-order spectral or finite difference methods with the geometric
flexibility of low-order finite volume or finite element schemes. The Flux
Reconstruction (FR) approach unifies various high-order schemes for
unstructured grids within a single framework. Additionally, the FR approach
exhibits a significant degree of element locality, and is thus able to run
efficiently on modern streaming architectures, such as Graphical Processing
Units (GPUs). The aforementioned properties of FR mean it offers a promising
route to performing affordable, and hence industrially relevant,
scale-resolving simulations of hitherto intractable unsteady flows within the
vicinity of real-world engineering geometries. In this paper we present PyFR,
an open-source Python based framework for solving advection-diffusion type
problems on streaming architectures using the FR approach. The framework is
designed to solve a range of governing systems on mixed unstructured grids
containing various element types. It is also designed to target a range of
hardware platforms via use of an in-built domain specific language based on the
Mako templating engine. The current release of PyFR is able to solve the
compressible Euler and Navier-Stokes equations on grids of quadrilateral and
triangular elements in two dimensions, and hexahedral elements in three
dimensions, targeting clusters of CPUs, and NVIDIA GPUs. Results are presented
for various benchmark flow problems, single-node performance is discussed, and
scalability of the code is demonstrated on up to 104 NVIDIA M2090 GPUs. The
software is freely available under a 3-Clause New Style BSD license (see
www.pyfr.org)
Positivity-preserving entropy filtering for the ideal magnetohydrodynamics equations
In this work, we present a positivity-preserving adaptive filtering approach
for discontinuous spectral element approximations of the ideal
magnetohydrodynamics equations. This approach combines the entropy filtering
method (Dzanic and Witherden, J. Comput. Phys., 468, 2022) for shock capturing
in gas dynamics along with the eight-wave method for enforcing a
divergence-free magnetic field. Due to the inclusion of non-conservative source
terms, an operator-splitting approach is introduced to ensure that the
positivity and entropy constraints remain satisfied by the discrete solution.
Furthermore, a computationally efficient algorithm for solving the optimization
process for this nonlinear filtering approach is presented. The resulting
scheme can robustly resolve strong discontinuities on general unstructured
grids without tunable parameters while recovering high-order accuracy for
smooth solutions. The efficacy of the scheme is shown in numerical experiments
on various problems including extremely magnetized blast waves and
three-dimensional magnetohydrodynamic instabilities.Comment: 24 pages, 17 figure
A positivity-preserving and conservative high-order flux reconstruction method for the polyatomic Boltzmann--BGK equation
In this work, we present a positivity-preserving high-order flux
reconstruction method for the polyatomic Boltzmann--BGK equation augmented with
a discrete velocity model that ensures the scheme is discretely conservative.
Through modeling the internal degrees of freedom, the approach is further
extended to polyatomic molecules and can encompass arbitrary constitutive laws.
The approach is validated on a series of large-scale complex numerical
experiments, ranging from shock-dominated flows computed on unstructured grids
to direct numerical simulation of three-dimensional compressible turbulent
flows, the latter of which is the first instance of such a flow computed by
directly solving the Boltzmann equation. The results show the ability of the
scheme to directly resolve shock structures without any ad hoc numerical shock
capturing method and correctly approximate turbulent flow phenomena in a
consistent manner with the hydrodynamic equations.Comment: 31 pages, 20 figure
Foundations of space-time finite element methods: polytopes, interpolation, and integration
The main purpose of this article is to facilitate the implementation of
space-time finite element methods in four-dimensional space. In order to
develop a finite element method in this setting, it is necessary to create a
numerical foundation, or equivalently a numerical infrastructure. This
foundation should include a collection of suitable elements (usually
hypercubes, simplices, or closely related polytopes), numerical interpolation
procedures (usually orthonormal polynomial bases), and numerical integration
procedures (usually quadrature rules). It is well known that each of these
areas has yet to be fully explored, and in the present article, we attempt to
directly address this issue. We begin by developing a concrete, sequential
procedure for constructing generic four-dimensional elements (4-polytopes).
Thereafter, we review the key numerical properties of several canonical
elements: the tesseract, tetrahedral prism, and pentatope. Here, we provide
explicit expressions for orthonormal polynomial bases on these elements. Next,
we construct fully symmetric quadrature rules with positive weights that are
capable of exactly integrating high-degree polynomials, e.g. up to degree 17 on
the tesseract. Finally, the quadrature rules are successfully tested using a
set of canonical numerical experiments on polynomial and transcendental
functions.Comment: 34 pages, 18 figure