A Sparse Grid Discontinuous Galerkin Method for High-Dimensional Transport Equations and Its Application to Kinetic Simulations

Guo, Wei; Cheng, Yingda

doi:10.1137/16m1060017

Cited by 56 publications

(91 citation statements)

References 43 publications

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“…Now, we are ready to incorporate the sparse finite element space defined above into the CDG framework. The approximation properties for the sparse finite element space have been established in previous work [33,12]. By using a lemma in [12], we can have estimates for L 2 projection operator onto the spaceŝ V k N,P ,V k N,D .…”

Section: Periodic Problemsmentioning

confidence: 96%

“…The approximation properties for the sparse finite element space have been established in previous work [33,12]. By using a lemma in [12], we can have estimates for L 2 projection operator onto the spaceŝ V k N,P ,V k N,D . To facilitate the discussion, below we introduce some notations about norms and semi-norms.…”

Section: Periodic Problemsmentioning

confidence: 96%

“…Based on the scheme constructed in [12], the goal of the present paper is to design and analyze the sparse grid CDG method. The CDG schemes [18,20,21] are a class of DG schemes on overlapping cells that combine the idea of the central schemes [24,16,19] with the DG weak formulation.…”

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confidence: 99%

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Sparse Grid Central Discontinuous Galerkin Method for Linear Hyperbolic Systems in High Dimensions

Tao¹,

Chen²,

Zhang³

et al. 2019

SIAM J. Sci. Comput.

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In this paper, we develop sparse grid central discontinuous Galerkin (CDG) scheme for linear hyperbolic systems with variable coefficients in high dimensions. The scheme combines the CDG framework with the sparse grid approach, with the aim of breaking the curse of dimensionality. A new hierarchical representation of piecewise polynomials on the dual mesh is introduced and analyzed, resulting in a sparse finite element space that can be used for non-periodic problems. Theoretical results, such as L 2 stability and error estimates are obtained for scalar problems. CFL conditions are studied numerically comparing discontinuous Galerkin (DG), CDG, sparse grid DG and sparse grid CDG methods. Numerical results including scalar linear equations, acoustic and elastic waves are provided.

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Section: Periodic Problemsmentioning

confidence: 96%

Section: Periodic Problemsmentioning

confidence: 96%

See 1 more Smart Citation

Sparse Grid Central Discontinuous Galerkin Method for Linear Hyperbolic Systems in High Dimensions

Tao¹,

Chen²,

Zhang³

et al. 2019

SIAM J. Sci. Comput.

Self Cite

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“…We will also discuss some aspects that are pertinent to the present work in the next section. Let us also mention that recently the use of dimension reduction techniques, such as low-rank approximations, have been explored [22,23,13,12,18,15].…”

Section: Introductionmentioning

confidence: 99%

Semi-Lagrangian Vlasov simulation on GPUs

Einkemmer

2020

Computer Physics Communications

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In this paper, our goal is to efficiently solve the Vlasov equation on GPUs. A semi-Lagrangian discontinuous Galerkin scheme is used for the discretization. Such kinetic computations are extremely expensive due to the high-dimensional phase space. The SLDG code abstracts the number of dimensions and uses a shared code base for both GPU and CPU based simulations. We investigate the performance of the implementation on a range of both Tesla (V100, Titan V, K80) and consumer (GTX 1080 Ti) GPUs. Our implementation is typically able to achieve a performance of approximately 470 GB/s on a single GPU and 1600 GB/s on four V100 GPUs connected via NVLink. This results in a speed up of about a factor of ten (comparing a single GPU with a dual socket Intel Xeon Gold node) and approximately a factor of 35 (comparing a single node with and without GPUs). In addition, we investigate the effect of single precision computation on the performance of the SLDG code and demonstrate that a template based dimension independent implementation can achieve good performance regardless of the dimensionality of the problem.

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“…It is similar to the hyperbolic -anisotropic-hierarchical base. It allows approximation of highdimensional problems for which there is no medium or high frequency oscillation [10,37]. Tensorization consists of expressing a multi-dimensional function as the product of matrices of one-dimensional functions, i.e.…”

mentioning

confidence: 99%

Six-Dimensional Adaptive Simulation of the Vlasov Equations Using a Hierarchical Basis

Deriaz¹,

Peirani²

2018

Multiscale Model. Simul.

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Abstract. We present an original adaptive scheme using a dynamically refined grid for the simulation of the six-dimensional Vlasov-Poisson equations. The distribution function is represented in a hierarchical basis that retains only the most significant coefficients. This allows considerable savings in terms of computational time and memory usage. The proposed scheme involves the mathematical formalism of Multiresolution Analysis and computer implementation of Adaptive Mesh Refinement. We apply a finite difference method to approximate the Vlasov-Poisson equations although other numerical methods could be considered. Numerical experiments are presented for the d-dimensional Vlasov-Poisson equations in the full 2d-dimensional phase space for d = 1, 2 or 3. The six-dimensional case is compared to a Gadget N-body simulation.keywords: adaptive mesh refinement, hierarchical basis, wavelet, multiresolution analysis, finite differences, phase-space simulations, Vlasov-Poisson equations 1. Introduction. The general idea of Adaptivity in Numerical Analysis lies in the representation of a complex system by a reduced number of elements. It aims to decrease the computational time and memory resources needed to simulate the system. Often, the situation boils down to a trade-off between a numerical volume (computer memory, number of elementary operations etc.) and an increase in the implementation complexity (data structures or algorithms).The most common adaptive technique used in Mechanics [24,40,45,52,53], Physics [7,42] and Astrophysics [1,59,60] is called the Adaptive Mesh Refinement (AMR). In AMR, a Cartesian grid refines locally into finer Cartesian grids. Such refinement is often associated with finite volume methods [40,52,60]. We can distinguish two different approaches among AMR methods: in patch-based AMR the data structure is a collection of grids of varying sizes and accuracies, which are embedded within each other [42,56], whereas in the fully-threaded tree AMR in which the points or cells, or groups thereof, are stored in the nodes or leaves of a large tree structure [11,36,52,53,60]. We use the latter method.In the present study, we are particularly interested in using adaptive methods to tackle the simulation of large systems. Most of these rely on reduced functional basis discretization. This implicates finite elements where the underlying mesh does not have to be Cartesian [41,55]; wavelets associated with a vast non-linear approximation theory [17]; spectral methods that show a good numerical accuracy [31,51]. The hierarchical bases [7] are special kinds of wavelet bases for which the primal scaling function and the primal wavelet are identical, although their duals remain different. A sparse grid method itself [7,50] is a sub-class of hierarchical bases. It is similar to the hyperbolic -anisotropic-hierarchical base. It allows approximation of highdimensional problems for which there is no medium or high frequency oscillation [10,37]. Tensorization consists of expressing a multi-dimensional function as t...

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A Sparse Grid Discontinuous Galerkin Method for High-Dimensional Transport Equations and Its Application to Kinetic Simulations

Cited by 56 publications

References 43 publications

Sparse Grid Central Discontinuous Galerkin Method for Linear Hyperbolic Systems in High Dimensions

Sparse Grid Central Discontinuous Galerkin Method for Linear Hyperbolic Systems in High Dimensions

Semi-Lagrangian Vlasov simulation on GPUs

Six-Dimensional Adaptive Simulation of the Vlasov Equations Using a Hierarchical Basis

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