62 research outputs found
A semi-Lagrangian Vlasov solver in tensor train format
In this article, we derive a semi-Lagrangian scheme for the solution of the
Vlasov equation represented as a low-parametric tensor. Grid-based methods for
the Vlasov equation have been shown to give accurate results but their use has
mostly been limited to simulations in two dimensional phase space due to
extensive memory requirements in higher dimensions. Compression of the solution
via high-order singular value decomposition can help in reducing the storage
requirements and the tensor train (TT) format provides efficient basic linear
algebra routines for low-rank representations of tensors. In this paper, we
develop interpolation formulas for a semi-Lagrangian solver in TT format. In
order to efficiently implement the method, we propose a compression of the
matrix representing the interpolation step and an efficient implementation of
the Hadamard product. We show numerical simulations for standard test cases in
two, four and six dimensional phase space. Depending on the test case, the
memory requirements reduce by a factor in four and a factor
in six dimensions compared to the full-grid method
A "metric" semi-Lagrangian Vlasov-Poisson solver
We propose a new semi-Lagrangian Vlasov-Poisson solver. It employs elements
of metric to follow locally the flow and its deformation, allowing one to find
quickly and accurately the initial phase-space position of any test
particle , by expanding at second order the geometry of the motion in the
vicinity of the closest element. It is thus possible to reconstruct accurately
the phase-space distribution function at any time and position by
proper interpolation of initial conditions, following Liouville theorem. When
distorsion of the elements of metric becomes too large, it is necessary to
create new initial conditions along with isotropic elements and repeat the
procedure again until next resampling. To speed up the process, interpolation
of the phase-space distribution is performed at second order during the
transport phase, while third order splines are used at the moments of
remapping. We also show how to compute accurately the region of influence of
each element of metric with the proper percolation scheme. The algorithm is
tested here in the framework of one-dimensional gravitational dynamics but is
implemented in such a way that it can be extended easily to four or
six-dimensional phase-space. It can also be trivially generalised to plasmas.Comment: 32 pages, 14 figures, accepted for publication in Journal of Plasma
Physics, Special issue: The Vlasov equation, from space to laboratory plasma
High order and energy preserving discontinuous Galerkin methods for the Vlasov-Poisson system
We present a computational study for a family of discontinuous Galerkin
methods for the one dimensional Vlasov-Poisson system that has been recently
introduced. We introduce a slight modification of the methods to allow for
feasible computations while preserving the properties of the original methods.
We study numerically the verification of the theoretical and convergence
analysis, discussing also the conservation properties of the schemes. The
methods are validated through their application to some of the benchmarks in
the simulation of plasma physics.Comment: 44 pages, 28 figure
Suppressing Instability in a Vlasov-Poisson System by an External Electric Field Through Constrained Optimization
Fusion energy offers the potential for the generation of clean, safe, and
nearly inexhaustible energy. While notable progress has been made in recent
years, significant challenges persist in achieving net energy gain. Improving
plasma confinement and stability stands as a crucial task in this regard and
requires optimization and control of the plasma system. In this work, we deploy
a PDE-constrained optimization formulation that uses a kinetic description for
plasma dynamics as the constraint. This is to optimize, over all possible
controllable external electric fields, the stability of the plasma dynamics
under the condition that the Vlasov--Poisson (VP) equation is satisfied. For
computing the functional derivative with respect to the external field in the
optimization updates, the adjoint equation is derived. Furthermore, in the
discrete setting, where we employ the semi-Lagrangian method as the forward
solver, we also explicitly formulate the corresponding adjoint solver and the
gradient as the discrete analogy to the adjoint equation and the Frechet
derivative. A distinct feature we observed of this constrained optimization is
the complex landscape of the objective function and the existence of numerous
local minima, largely due to the hyperbolic nature of the VP system. To
overcome this issue, we utilize a gradient-accelerated genetic algorithm,
leveraging the advantages of the genetic algorithm's exploration feature to
cover a broader search of the solution space and the fast local convergence
aided by the gradient information. We show that our algorithm obtains good
electric fields that are able to maintain a prescribed profile in a beam
shaping problem and uses nonlinear effects to suppress plasma instability in a
two-stream configuration
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
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