14 research outputs found
von Neumann Stability Analysis of Globally Constraint-Preserving DGTD and PNPM Schemes for the Maxwell Equations using Multidimensional Riemann Solvers
The time-dependent equations of computational electrodynamics (CED) are
evolved consistent with the divergence constraints. As a result, there has been
a recent effort to design finite volume time domain (FVTD) and discontinuous
Galerkin time domain (DGTD) schemes that satisfy the same constraints and,
nevertheless, draw on recent advances in higher order Godunov methods. This
paper catalogues the first step in the design of globally constraint-preserving
DGTD schemes. The algorithms presented here are based on a novel DG-like method
that is applied to a Yee-type staggering of the electromagnetic field variables
in the faces of the mesh. The other two novel building blocks of the method
include constraint-preserving reconstruction of the electromagnetic fields and
multidimensional Riemann solvers; both of which have been developed in recent
years by the first author. We carry out a von Neumann stability analysis of the
entire suite of DGTD schemes for CED at orders of accuracy ranging from second
to fourth. A von Neumann stability analysis gives us the maximal CFL numbers
that can be sustained by the DGTD schemes presented here at all orders. It also
enables us to understand the wave propagation characteristics of the schemes in
various directions on a Cartesian mesh. We find that the CFL of DGTD schemes
decreases with increasing order. To counteract that, we also present
constraint-preserving PNPM schemes for CED. We find that the third and fourth
order constraint-preserving DGTD and P1PM schemes have some extremely
attractive properties when it comes to low-dispersion, low-dissipation
propagation of electromagnetic waves in multidimensions. Numerical accuracy
tests are also provided to support the von Neumann stability analysis
Techniques, Tricks and Algorithms for Efficient GPU-Based Processing of Higher Order Hyperbolic PDEs
GPU computing is expected to play an integral part in all modern Exascale
supercomputers. It is also expected that higher order Godunov schemes will make
up about a significant fraction of the application mix on such supercomputers.
It is, therefore, very important to prepare the community of users of higher
order schemes for hyperbolic PDEs for this emerging opportunity. We focus on
three broad and high-impact areas where higher order Godunov schemes are used.
The first area is computational fluid dynamics (CFD). The second is
computational magnetohydrodynamics (MHD) which has an involution constraint
that has to be mimetically preserved. The third is computational
electrodynamics (CED) which has involution constraints and also extremely stiff
source terms. Together, these three diverse uses of higher order Godunov
methodology, cover many of the most important applications areas. In all three
cases, we show that the optimal use of algorithms, techniques and tricks, along
with the use of OpenACC, yields superlative speedups on GPUs! As a bonus, we
find a most remarkable and desirable result: some higher order schemes, with
their larger operations count per zone, show better speedup than lower order
schemes on GPUs. In other words, the GPU is an optimal stratagem for overcoming
the higher computational complexities of higher order schemes! Several avenues
for future improvement have also been identified. A scalability study is
presented for a real-world application using GPUs and comparable numbers of
high-end multicore CPUs. It is found that GPUs offer a substantial performance
benefit over comparable number of CPUs, especially when all the methods
designed in this paper are used.Comment: 73 pages, 17 figure
Curl Constraint-Preserving Reconstruction and the Guidance it Gives for Mimetic Scheme Design
Several important PDE systems, like magnetohydrodynamics and computational electrodynamics, are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion. Recently, new classes of PDE systems have emerged for hyperelasticity, compressible multiphase flows, so-called first-order reductions of the Einstein field equations, or a novel first-order hyperbolic reformulation of Schrödinger’s equation, to name a few, where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector field. We study the problem of curl constraint-preserving reconstruction as it pertains to the design of mimetic finite volume (FV) WENO-like schemes for PDEs that support a curl-preserving involution. (Some insights into discontinuous Galerkin (DG) schemes are also drawn, though that is not the prime focus of this paper.) This is done for two- and three-dimensional structured mesh problems where we deliver closed form expressions for the reconstruction. The importance of multidimensional Riemann solvers in facilitating the design of such schemes is also documented. In two dimensions, a von Neumann analysis of structure-preserving WENO-like schemes that mimetically satisfy the curl constraints, is also presented. It shows the tremendous value of higher order WENO-like schemes in minimizing dissipation and dispersion for this class of problems. Numerical results are also presented to show that the edge-centered curl-preserving (ECCP) schemes meet their design accuracy. This paper is the first paper that invents non-linearly hybridized curl-preserving reconstruction and integrates it with higher order Godunov philosophy. By its very design, this paper is, therefore, intended to be forward-looking and to set the stage for future work on curl involution-constrained PDEs
Fourth-order finite volume algorithm with adaptive mesh refinement in space and time for multi-fluid plasma modeling, A
2022 Spring.Includes bibliographical references.Improving our fundamental understanding of plasma physics using numerical methods is pivotal to the advancement of science and the continual development of cutting-edge technologies such as nuclear fusion reactions for energy production or the manufacturing of microelectronic devices. An elaborate and accurate approach to modeling plasmas using computational fluid dynamics (CFD) is the multi-fluid method, where the full set of fluid mechanics equations are solved for each species in the plasma simultaneously with Maxwell's equations in a coupled fashion. Nevertheless, multi-fluid plasma modeling is inherently multiscale and multiphysics, presenting significant numerical and mathematical stiffness. This research aims to develop an efficient and accurate multi-fluid plasma model using higher-order, finite-volume, solution-adaptive numerical methods. The algorithm developed herein is verified to be fourth-order accurate for electromagnetic simulations as well as those involving fully-coupled, multi-fluid plasma physics. The solutions to common plasma test problems obtained by the algorithm are validated against exact solutions and results from literature. The algorithm is shown to be robust and stable in the presence of complex solution topology and discontinuities, such as shocks and steep gradients. The optimizations in spatial discretization provided by the fourth-order algorithm and adaptive mesh refinement are demonstrated to improve the solution time by a factor of 10 compared to lower-order methods on fixed-grid meshes. This research produces an advanced, multi-fluid plasma modeling framework which allows for studying complex, realistic plasmas involving collisions and practical geometries
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal