arXiv subjects

T. Warburton

Publications and source records attributed to T. Warburton.

14 recordsLinked to original sources

arXiv2018

Discontinuous Galerkin Discretizations of the Boltzmann Equations in 2D: semi-analytic time stepping and absorbing boundary layers

We present an efficient nodal discontinuous Galerkin method for approximating nearly incompressible flows using the Boltzmann equations. The equations are discretized with Hermite polynomials in velocity space yielding a first order conservation law. A stabilized unsplit perfectly matching layer (PML) formulation is introduced for the resulting nonlinear flow equations. The proposed PML equations exponentially absorb the difference between the nonlinear fluctuation and the prescribed mean flow. We introduce semi-analytic time discretization methods to improve the time step restrictions in small relaxation times. We also introduce a multirate semi-analytic Adams-Bashforth method which preserves efficiency in stiff regimes. Accuracy and performance of the method are tested using distinct cases including isothermal vortex, flow around square cylinder, and wall mounted square cylinder test cases.

math.NA↗

arXiv2017

On the penalty stabilization mechanism for upwind discontinuous Galerkin formulations of first order hyperbolic systems

Penalty fluxes are dissipative numerical fluxes for high order discontinuous Galerkin (DG) methods which depend on a penalization parameter. We investigate the dependence of the spectra of high order DG discretizations on this parameter, and show that as its value increases, the spectra of the DG discretization splits into two disjoint sets of eigenvalues. One set converges to the eigenvalues of a conforming discretization, while the other set corresponds to spurious eigenvalues which are damped proportionally to the parameter. Numerical experiments also demonstrate that undamped spurious modes present in both in the limit of zero and large penalization parameters are damped for moderate values of the upwind parameter.

math.NA↗

arXiv2017

Weight-adjusted discontinuous Galerkin methods: wave propagation in heterogeneous media

Time-domain discontinuous Galerkin (DG) methods for wave propagation require accounting for the inversion of dense elemental mass matrices, where each mass matrix is computed with respect to a parameter-weighted L2 inner product. In applications where the wavespeed varies spatially at a sub-element scale, these matrices are distinct over each element, necessitating additional storage. In this work, we propose a weight-adjusted DG (WADG) method which reduces storage costs by replacing the weighted L2 inner product with a weight-adjusted inner product. This equivalent inner product results in an energy stable method, but does not increase storage costs for locally varying weights. A-priori error estimates are derived, and numerical examples are given illustrating the application of this method to the acoustic wave equation with heterogeneous wavespeed.

math.NA↗

arXiv2016

Reduced storage nodal discontinuous Galerkin methods on semi-structured prismatic meshes

We present a high order time-domain nodal discontinuous Galerkin method for wave problems on hybrid meshes consisting of both wedge and tetrahedral elements. We allow for vertically mapped wedges which can be deformed along the extruded coordinate, and present a simple method for producing quasi-uniform wedge meshes for layered domains. We show that standard mass lumping techniques result in a loss of energy stability on meshes of vertically mapped wedges, and propose an alternative which is both energy stable and efficient. High order convergence is demonstrated, and comparisons are made with existing low-storage methods on wedges. Finally, the computational performance of the method on Graphics Processing Units is evaluated.

math.NA↗

arXiv2016

GPU-accelerated Bernstein-Bezier discontinuous Galerkin methods for wave problems

We evaluate the computational performance of the Bernstein-Bezier basis for discontinuous Galerkin (DG) discretizations and show how to exploit properties of derivative and lift operators specific to Bernstein polynomials for an optimal complexity quadrature-free evaluation of the DG formulation. Issues of efficiency and numerical stability are discussed in the context of a model wave propagation problem. We compare the performance of Bernstein-Bezier kernels to both a straightforward and a block-partitioned implementation of nodal DG kernels in a time-explicit GPU-accelerated DG solver. Computational experiments confirm the advantage of Bernstein-Bezier DG kernels over both straightforward and block-partitioned nodal DG kernels at high orders of approximation.

math.NA↗

arXiv2016

Weight-adjusted discontinuous Galerkin methods: curvilinear meshes

Traditional time-domain discontinuous Galerkin (DG) methods result in large storage costs at high orders of approximation due to the storage of dense elemental matrices. In this work, we propose a weight-adjusted DG (WADG) methods for curvilinear meshes which reduce storage costs while retaining energy stability. A priori error estimates show that high order accuracy is preserved under sufficient conditions on the mesh, which are illustrated through convergence tests with different sequences of meshes. Numerical and computational experiments verify the accuracy and performance of WADG for a model problem on curved domains.

math.NA↗

arXiv2016

Array Program Transformation with Loo.py by Example: High-Order Finite Elements

To concisely and effectively demonstrate the capabilities of our program transformation system Loo.py, we examine a transformation path from two real-world Fortran subroutines as found in a weather model to a single high-performance computational kernel suitable for execution on modern GPU hardware. Along the transformation path, we encounter kernel fusion, vectorization, prefetch- ing, parallelization, and algorithmic changes achieved by mechanized conversion between imperative and functional/substitution- based code, among a number more. We conclude with performance results that demonstrate the effects and support the effectiveness of the applied transformations.

cs.PL↗

arXiv2015

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.

math.NA↗

arXiv2015

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.

math.NA↗

arXiv2015

GPU accelerated spectral finite elements on all-hex meshes

This paper presents a spectral element finite element scheme that efficiently solves elliptic problems on unstructured hexahedral meshes. The discrete equations are solved using a matrix-free preconditioned conjugate gradient algorithm. An additive Schwartz two-scale preconditioner is employed that allows h-independence convergence. An extensible multi-threading programming API is used as a common kernel language that allows runtime selection of different computing devices (GPU and CPU) and different threading interfaces (CUDA, OpenCL and OpenMP). Performance tests demonstrate that problems with over 50 million degrees of freedom can be solved in a few seconds on an off-the-shelf GPU.

cs.CE↗

arXiv2015

Orthogonal bases for vertex-mapped pyramids

Discontinuous Galerkin (DG) methods discretized under the method of lines must handle the inverse of a block diagonal mass matrix at each time step. Efficient implementations of the DG method hinge upon inexpensive and low-memory techniques for the inversion of each dense mass matrix block. We propose an efficient time-explicit DG method on meshes of pyramidal elements based on the construction of a semi-nodal high order basis, which is orthogonal for a class of transformations of the reference pyramid, despite the non-affine nature of the mapping. We give numerical results confirming both expected convergence rates and discuss efficiency of DG methods under such a basis.

math.NA↗

arXiv2014

A Comparison of High Order Interpolation Nodes for the Pyramid

The use of pyramid elements is crucial to the construction of efficient hex-dominant meshes. For conforming nodal finite element methods with mixed element types, it is advantageous for nodal distributions on the faces of the pyramid to match those on the faces and edges of hexahedra and tetrahedra. We adapt existing procedures for constructing optimized tetrahedral nodal sets for high order interpolation to the pyramid with constrained face nodes, including two generalizations of the explicit Warp and Blend construction of nodes on the tetrahedron.

math.NA↗

arXiv2014

Residual based adaptivity and PWDG methods for the Helmholtz equation

We present a study of two residual a posteriori error indicators for the Plane Wave Discontinuous Galerkin (PWDG) method for the Helmholtz equation. In particular we study the h-version of PWDG in which the number of plane wave directions per element is kept fixed. First we use a slight modification of the appropriate a priori analysis to determine a residual indicator. Numerical tests show that this is reliable but pessimistic in that the ratio between the true error and the indicator increases as the mesh is refined. We therefore introduce a new analysis based on the observation that sufficiently many plane waves can approximate piecewise linear functions as the mesh is refined. Numerical results demonstrate an improvement in the efficiency of the indicators.

math.NA↗

arXiv2014

OCCA: A unified approach to multi-threading languages

The inability to predict lasting languages and architectures led us to develop OCCA, a C++ library focused on host-device interaction. Using run-time compilation and macro expansions, the result is a novel single kernel language that expands to multiple threading languages. Currently, OCCA supports device kernel expansions for the OpenMP, OpenCL, and CUDA platforms. Computational results using finite difference, spectral element and discontinuous Galerkin methods show OCCA delivers portable high performance in different architectures and platforms.

cs.DC↗