112 research outputs found
A massively parallel semi-Lagrangian solver for the six-dimensional Vlasov-Poisson equation
This paper presents an optimized and scalable semi-Lagrangian solver for the
Vlasov-Poisson system in six-dimensional phase space. Grid-based solvers of the
Vlasov equation are known to give accurate results. At the same time, these
solvers are challenged by the curse of dimensionality resulting in very high
memory requirements, and moreover, requiring highly efficient parallelization
schemes. In this paper, we consider the 6d Vlasov-Poisson problem discretized
by a split-step semi-Lagrangian scheme, using successive 1d interpolations on
1d stripes of the 6d domain. Two parallelization paradigms are compared, a
remapping scheme and a classical domain decomposition approach applied to the
full 6d problem. From numerical experiments, the latter approach is found to be
superior in the massively parallel case in various respects. We address the
challenge of artificial time step restrictions due to the decomposition of the
domain by introducing a blocked one-sided communication scheme for the purely
electrostatic case and a rotating mesh for the case with a constant magnetic
field. In addition, we propose a pipelining scheme that enables to hide the
costs for the halo communication between neighbor processes efficiently behind
useful computation. Parallel scalability on up to 65k processes is demonstrated
for benchmark problems on a supercomputer
A mass-conserving sparse grid combination technique with biorthogonal hierarchical basis functions for kinetic simulations
The exact numerical simulation of plasma turbulence is one of the assets and
challenges in fusion research. For grid-based solvers, sufficiently fine
resolutions are often unattainable due to the curse of dimensionality. The
sparse grid combination technique provides the means to alleviate the curse of
dimensionality for kinetic simulations. However, the hierarchical
representation for the combination step with the state-of-the-art hat functions
suffers from poor conservation properties and numerical instability.
The present work introduces two new variants of hierarchical multiscale basis
functions for use with the combination technique: the biorthogonal and full
weighting bases. The new basis functions conserve the total mass and are shown
to significantly increase accuracy for a finite-volume solution of constant
advection. Further numerical experiments based on the combination technique
applied to a semi-Lagrangian Vlasov--Poisson solver show a stabilizing effect
of the new bases on the simulations
Afterlive: A performant code for Vlasov-Hybrid simulations
A parallelized implementation of the Vlasov-Hybrid method [Nunn, 1993] is
presented. This method is a hybrid between a gridded Eulerian description and
Lagrangian meta-particles. Unlike the Particle-in-Cell method [Dawson, 1983]
which simply adds up the contribution of meta-particles, this method does a
reconstruction of the distribution function in every time step for each
species. This interpolation method combines meta-particles with different
weights in such a way that particles with large weight do not drown out
particles that represent small contributions to the phase space density. These
core properties allow the use of a much larger range of macro factors and can
thus represent a much larger dynamic range in phase space density.
The reconstructed phase space density is used to calculate momenta of the
distribution function such as the charge density . The charge density
is also used as input into a spectral solver that calculates the
self-consistent electrostatic field which is used to update the particles for
the next time-step.
Afterlive (A Fourier-based Tool in the Electrostatic limit for the Rapid
Low-noise Integration of the Vlasov Equation) is fully parallelized using MPI
and writes output using parallel HDF5. The input to the simulation is read from
a JSON description that sets the initial particle distributions as well as
domain size and discretization constraints. The implementation presented here
is intentionally limited to one spatial dimension and resolves one or three
dimensions in velocity space. Additional spatial dimensions can be added in a
straight forward way, but make runs computationally even more costly.Comment: Accepted for publication in Computer Physics Communication
Solving the VlasovâMaxwell equations using Hamiltonian splitting
In this paper, the numerical discretizations based on Hamiltonian splitting for solving the VlasovâMaxwell system are constructed. We reformulate the VlasovâMaxwell system in MorrisonâMarsdenâWeinstein Poisson bracket form. Then the Hamiltonian of this system is split into five parts, with which five corresponding Hamiltonian subsystems are obtained. The splitting method in time is derived by composing the solutions to these five subsystems. Combining the splitting method in time with the Fourier spectral method and finite volume method in space gives the full numerical discretizations which possess good conservation for the conserved quantities including energy, momentum, charge, etc. In numerical experiments, we simulate the Landau damping, Weibel instability and Bernstein wave to verify the numerical algorithms
On the error propagation of semi-Lagrange and Fourier methods for advection problems
In this paper we study the error propagation of numerical schemes for the
advection equation in the case where high precision is desired. The numerical
methods considered are based on the fast Fourier transform, polynomial
interpolation (semi-Lagrangian methods using a Lagrange or spline
interpolation), and a discontinuous Galerkin semi-Lagrangian approach (which is
conservative and has to store more than a single value per cell).
We demonstrate, by carrying out numerical experiments, that the worst case
error estimates given in the literature provide a good explanation for the
error propagation of the interpolation-based semi-Lagrangian methods. For the
discontinuous Galerkin semi-Lagrangian method, however, we find that the
characteristic property of semi-Lagrangian error estimates (namely the fact
that the error increases proportionally to the number of time steps) is not
observed. We provide an explanation for this behavior and conduct numerical
simulations that corroborate the different qualitative features of the error in
the two respective types of semi-Lagrangian methods.
The method based on the fast Fourier transform is exact but, due to round-off
errors, susceptible to a linear increase of the error in the number of time
steps. We show how to modify the Cooley--Tukey algorithm in order to obtain an
error growth that is proportional to the square root of the number of time
steps.
Finally, we show, for a simple model, that our conclusions hold true if the
advection solver is used as part of a splitting scheme.Comment: submitted to Computers & Mathematics with Application
Practical 3-splitting beyond Strang
Operator splitting is a popular divide-and-conquer strategy for solving
differential equations. Typically, the right-hand side of the differential
equation is split into a number of parts that can then be integrated
separately. Many methods are known that split the right-hand side into two
parts. This approach is limiting, however, and there are situations when
3-splitting is more natural and ultimately more advantageous. The second-order
Strang operator-splitting method readily generalizes to a right-hand side
splitting into any number of operators. It is arguably the most popular method
for 3-splitting because of its efficiency, ease of implementation, and
intuitive nature. Other 3-splitting methods exist, but they are less
well-known, and evaluations of their performance in practice are scarce. We
demonstrate the effectiveness of some alternative 3-split, second-order methods
to Strang splitting on two problems: the reaction-diffusion Brusselator, which
can be split into three parts that each have closed-form solutions, and the
kinetic Vlasov--Poisson equations that is used in semi-Lagrangian plasma
simulations. We find alternative second-order 3-operator-splitting methods that
realize efficiency gains of 10\%--20\% over traditional Strang splitting
hyper.deal: An efficient, matrix-free finite-element library for high-dimensional partial differential equations
This work presents the efficient, matrix-free finite-element library
hyper.deal for solving partial differential equations in two to six dimensions
with high-order discontinuous Galerkin methods. It builds upon the
low-dimensional finite-element library deal.II to create complex
low-dimensional meshes and to operate on them individually. These meshes are
combined via a tensor product on the fly and the library provides new
special-purpose highly optimized matrix-free functions exploiting domain
decomposition as well as shared memory via MPI-3.0 features. Both node-level
performance analyses and strong/weak-scaling studies on up to 147,456 CPU cores
confirm the efficiency of the implementation. Results of the library hyper.deal
are reported for high-dimensional advection problems and for the solution of
the Vlasov--Poisson equation in up to 6D phase space.Comment: 33 pages, 18 figure
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