arXiv subjects

Lukas Einkemmer

Publications and source records attributed to Lukas Einkemmer.

At least 73 records · Page 4Linked to original sources

arXiv2020

Exponential methods for solving hyperbolic problems with application to kinetic equations

The efficient numerical solution of many kinetic models in plasma physics is impeded by the stiffness of these systems. Exponential integrators are attractive in this context as they remove the CFL condition induced by the linear part of the system, which in practice is often the most stringent stability constraint. In the literature, these schemes have been found to perform well, e.g., for drift-kinetic problems. Despite their overall efficiency and their many favorable properties, most of the commonly used exponential integrators behave rather erratically in terms of the allowed time step size in some situations. This severely limits their utility and robustness.Our goal in this paper is to explain the observed behavior and suggest exponential methods that do not suffer from the stated deficiencies. To accomplish this we study the stability of exponential integrators for a linearized problem. This analysis shows that classic exponential integrators exhibit severe deficiencies in that regard. Based on the analysis conducted we propose to use Lawson methods, which can be shown not to suffer from the same stability issues. We confirm these results and demonstrate the efficiency of Lawson methods by performing numerical simulations for both the Vlasov-Poisson system and a drift-kinetic model of a ion temperature gradient instability.

math.NA↗

arXiv2020

An accurate and time-parallel rational exponential integrator for hyperbolic and oscillatory PDEs

Rational exponential integrators (REXI) are a class of numerical methods that are well suited for the time integration of linear partial differential equations with imaginary eigenvalues. Since these methods can be parallelized in time (in addition to the spatial parallelization that is commonly performed) they are well suited to exploit modern high performance computing systems. In this paper, we propose a novel REXI scheme that drastically improves accuracy and efficiency. The chosen approach will also allow us to easily determine how many terms are required in the approximation in order to obtain accurate results. We provide comparative numerical simulations for a shallow water equation that highlight the efficiency of our approach and demonstrate that REXI schemes can be efficiently implemented on graphic processing units.

math.NA↗

arXiv2020

Semi-Lagrangian Vlasov simulation on GPUs

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, which is publicly available under the MIT license abstracts the number of dimensions and uses a shared codebase 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 speedup 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.

physics.comp-ph↗

arXiv2019

Performance optimization and modeling of fine-grained irregular communication in UPC

The UPC programming language offers parallelism via logically partitioned shared memory, which typically spans physically disjoint memory sub-systems. One convenient feature of UPC is its ability to automatically execute between-thread data movement, such that the entire content of a shared data array appears to be freely accessible by all the threads. The programmer friendliness, however, can come at the cost of substantial performance penalties. This is especially true when indirectly indexing the elements of a shared array, for which the induced between-thread data communication can be irregular and have a fine-grained pattern. In this paper we study performance enhancement strategies specifically targeting such fine-grained irregular communication in UPC. Starting from explicit thread privatization, continuing with block-wise communication, and arriving at message condensing and consolidation, we obtained considerable performance improvement of UPC programs that originally require fine-grained irregular communication. Besides the performance enhancement strategies, the main contribution of the present paper is to propose performance models for the different scenarios, in form of quantifiable formulas that hinge on the actual volumes of various data movements plus a small number of easily obtainable hardware characteristic parameters. These performance models help to verify the enhancements obtained, while also providing insightful predictions of similar parallel implementations, not limited to UPC, that also involve between-thread or between-process irregular communication. As a further validation, we also apply our performance modeling methodology and hardware characteristic parameters to an existing UPC code for solving a 2D heat equation on a uniform mesh.

cs.DC↗

arXiv2019

A pseudo-spectral splitting method for linear dispersive problems with transparent boundary conditions

The goal of the present work is to solve a linear dispersive equation with variable coefficient advection on an unbounded domain. In this setting, transparent boundary conditions are vital to allow waves to leave (or even re-enter) the, necessarily finite, computational domain. To obtain an efficient numerical scheme we discretize space using a spectral method. This allows us to drastically reduce the number of grid points required for a given accuracy. Applying a fully implicit time integrator, however, would require us to invert full matrices. This is addressed by performing an operator splitting scheme and only treating the third order differential operator, stemming from the dispersive part, implicitly; this approach can also be interpreted as an implicit-explicit scheme. However, the fact that the transparent boundary conditions are non-homogeneous and depend implicitly on the numerical solution presents a significant obstacle for the splitting/pseudo-spectral approach investigated here. We show how to overcome these difficulties and demonstrate the proposed numerical scheme by performing a number of numerical simulations.

math.NA↗

arXiv2019

A low-rank projector-splitting integrator for the Vlasov--Maxwell equations with divergence correction

The Vlasov--Maxwell equations are used for the kinetic description of magnetized plasmas. As they are posed in an up to 3+3 dimensional phase space, solving this problem is extremely expensive from a computational point of view. In this paper, we exploit the low-rank structure in the solution of the Vlasov equation. More specifically, we consider the Vlasov--Maxwell system and propose a dynamic low-rank integrator. The key idea is to approximate the dynamics of the system by constraining it to a low-rank manifold. This is accomplished by a projection onto the tangent space. There, the dynamics is represented by the low-rank factors, which are determined by solving lower-dimensional partial differential equations. The proposed scheme performs well in numerical experiments and succeeds in capturing the main features of the plasma dynamics. We demonstrate this good behavior for a range of test problems. The coupling of the Vlasov equation with the Maxwell system, however, introduces additional challenges. In particular, the divergence of the electric field resulting from Maxwell's equations is not consistent with the charge density computed from the Vlasov equation. We propose a correction based on Lagrange multipliers which enforces Gauss' law up to machine precision.

math.NA↗

arXiv2018

Reproducibility, accuracy and performance of the Feltor code and library on parallel computer architectures

Feltor is a modular and free scientific software package. It allows developing platform independent code that runs on a variety of parallel computer architectures ranging from laptop CPUs to multi-GPU distributed memory systems. Feltor consists of both a numerical library and a collection of application codes built on top of the library. Its main target are two- and three-dimensional drift- and gyro-fluid simulations with discontinuous Galerkin methods as the main numerical discretization technique. We observe that numerical simulations of a recently developed gyro-fluid model produce non-deterministic results in parallel computations. First, we show how we restore accuracy and bitwise reproducibility algorithmically and programmatically. In particular, we adopt an implementation of the exactly rounded dot product based on long accumulators, which avoids accuracy losses especially in parallel applications. However, reproducibility and accuracy alone fail to indicate correct simulation behaviour. In fact, in the physical model slightly different initial conditions lead to vastly different end states. This behaviour translates to its numerical representation. Pointwise convergence, even in principle, becomes impossible for long simulation times. In a second part, we explore important performance tuning considerations. We identify latency and memory bandwidth as the main performance indicators of our routines. Based on these, we propose a parallel performance model that predicts the execution time of algorithms implemented in Feltor and test our model on a selection of parallel hardware architectures. We are able to predict the execution time with a relative error of less than 25% for problem sizes between 0.1 and 1000 MB. Finally, we find that the product of latency and bandwidth gives a minimum array size per compute node to achieve a scaling efficiency above 50% (both strong and weak).

physics.comp-ph↗

arXiv2018

A quasi-conservative dynamical low-rank algorithm for the Vlasov equation

Numerical methods that approximate the solution of the Vlasov-Poisson equation by a low-rank representation have been considered recently. These methods can be extremely effective from a computational point of view, but contrary to most Eulerian Vlasov solvers, they do not conserve mass and momentum, neither globally nor in respecting the corresponding local conservation laws. This can be a significant limitation for intermediate and long time integration. In this paper we propose a numerical algorithm that overcomes some of these difficulties and demonstrate its utility by presenting numerical simulations.

math.NA↗

arXiv2018

A low-rank projector-splitting integrator for the Vlasov--Poisson equation

Many problems encountered in plasma physics require a description by kinetic equations, which are posed in an up to six-dimensional phase space. A direct discretization of this phase space, often called the Eulerian approach, has many advantages but is extremely expensive from a computational point of view. In the present paper we propose a dynamical low-rank approximation to the Vlasov--Poisson equation, with time integration by a particular splitting method. This approximation is derived by constraining the dynamics to a manifold of low-rank functions via a tangent space projection and by splitting this projection into the subprojections from which it is built. This reduces a time step for the six- (or four-) dimensional Vlasov--Poisson equation to solving two systems of three- (or two-) dimensional advection equations over the time step, once in the position variables and once in the velocity variables, where the size of each system of advection equations is equal to the chosen rank. By a hierarchical dynamical low-rank approximation, a time step for the Vlasov--Poisson equation can be further reduced to a set of six (or four) systems of one-dimensional advection equations, where the size of each system of advection equations is still equal to the rank. The resulting systems of advection equations can then be solved by standard techniques such as semi-Lagrangian or spectral methods. Numerical simulations in two and four dimensions for linear Landau damping, for a two-stream instability and for a plasma echo problem highlight the favorable behavior of this numerical method and show that the proposed algorithm is able to drastically reduce the required computational effort.

math.NA↗

arXiv2018

An adaptive step size controller for iterative implicit methods

The automatic selection of an appropriate time step size has been considered extensively in the literature. However, most of the strategies developed operate under the assumption that the computational cost (per time step) is independent of the step size. This assumption is reasonable for non-stiff ordinary differential equations and for partial differential equations where the linear systems of equations resulting from an implicit integrator are solved by direct methods. It is, however, usually not satisfied if iterative (for example, Krylov) methods are used. In this paper, we propose a step size selection strategy that adaptively reduces the computational cost (per unit time step) as the simulation progresses, constraint by the tolerance specified. We show that the proposed approach yields significant improvements in performance for a range of problems (diffusion-advection equation, Burgers' equation with a reaction term, porous media equation, viscous Burgers' equation, Allen--Cahn equation, and the two-dimensional Brusselator system). While traditional step size controllers have emphasized a smooth sequence of time step sizes, we emphasize the exploration of different step sizes which necessitates relatively rapid changes in the step size.

math.NA↗

arXiv2018

A low-rank algorithm for weakly compressible flow

In this paper, we propose a numerical method for solving weakly compressible fluid flow based on a dynamical low-rank projector splitting. The low-rank splitting scheme is applied to the Boltzmann equation with BGK collision term, which results in a set of constant coefficient advection equations. This procedure is numerically efficient as a small rank is sufficient to obtain the relevant dynamics (described by the Navier--Stokes equations). The resulting method can be combined with a range of different discretization strategies; in particular, it is possible to implement spectral and semi-Lagrangian methods, which allows us to design numerical schemes that are not encumbered by the sonic CFL condition.

math.NA↗

arXiv2018

On the performance of exponential integrators for problems in magnetohydrodynamics

Exponential integrators have been introduced as an efficient alternative to explicit and implicit methods for integrating large stiff systems of differential equations. Over the past decades these methods have been studied theoretically and their performance was evaluated using a range of test problems. While the results of these investigations showed that exponential integrators can provide significant computational savings, the research on validating this hypothesis for large scale systems and understanding what classes of problems can particularly benefit from the use of the new techniques is in its initial stages. Resistive magnetohydrodynamic (MHD) modeling is widely used in studying large scale behavior of laboratory and astrophysical plasmas. In many problems numerical solution of MHD equations is a challenging task due to the temporal stiffness of this system in the parameter regimes of interest. In this paper we evaluate the performance of exponential integrators on large MHD problems and compare them to a state-of-the-art implicit time integrator. Both the variable and constant time step exponential methods of EpiRK-type are used to simulate magnetic reconnection and the Kelvin--Helmholtz instability in plasma. Performance of these methods, which are part of the EPIC software package, is compared to the variable time step variable order BDF scheme included in the CVODE (part of SUNDIALS) library. We study performance of the methods on parallel architectures and with respect to magnitudes of important parameters such as Reynolds, Lundquist, and Prandtl numbers. We find that the exponential integrators provide superior or equal performance in most circumstances and conclude that further development of exponential methods for MHD problems is warranted and can lead to significant computational advantages for large scale stiff systems of differential equations such as MHD.

math.NA↗

arXiv2018

Evaluation of the Intel Xeon Phi 7120 and NVIDIA K80 as accelerators for two-dimensional panel codes

To optimize the geometry of airfoils for a specific application is an important engineering problem. In this context genetic algorithms have enjoyed some success as they are able to explore the search space without getting stuck in local optima. However, these algorithms require the computation of aerodynamic properties for a significant number of airfoil geometries. Consequently, for low-speed aerodynamics, panel methods are most often used as the inner solver. In this paper we evaluate the performance of such an optimization algorithm on modern accelerators (more specifically, the Intel Xeon Phi 7120 and the NVIDIA K80). For that purpose, we have implemented an optimized version of the algorithm on the CPU and Xeon Phi (based on OpenMP, vectorization, and the Intel MKL library) and on the GPU (based on CUDA and the MAGMA library). We present timing results for all codes and discuss the similarities and differences between the three implementations. Overall, we observe a speedup of approximately $2.5$ for adding an Intel Xeon Phi 7120 to a dual socket workstation and a speedup between $3.4$ and $3.8$ for adding a NVIDIA K80 to a dual socket workstation.

cs.DC↗

arXiv2018

A comparison of boundary correction methods for Strang splitting

In this paper we consider splitting methods in the presence of non-homogeneous boundary conditions. In particular, we consider the corrections that have been described and analyzed in Einkemmer, Ostermann 2015 and Alonso-Mallo, Cano, Reguera 2016. The latter method is extended to the non-linear case, and a rigorous convergence analysis is provided. We perform numerical simulations for diffusion-reaction, advection-reaction, and dispersion-reaction equations in order to evaluate the relative performance of these two corrections. Furthermore, we introduce an extension of both methods to obtain order three locally and evaluate under what circumstances this is beneficial.

math.NA↗

arXiv2018

A comparison of semi-Lagrangian discontinuous Galerkin and spline based Vlasov solvers in four dimensions

The purpose of the present paper is to compare two semi-Lagrangian methods in the context of the four-dimensional Vlasov--Poisson equation. More specifically, our goal is to compare the performance of the more recently developed semi-Lagrangian discontinuous Galerkin scheme with the de facto standard in Eulerian Vlasov simulation (i.e. using cubic spline interpolation). To that end, we perform simulations for nonlinear Landau damping and a two-stream instability and provide benchmarks for the SeLaLib and sldg codes (both on a workstation and using MPI on a cluster). We find that the semi-Lagrangian discontinuous Galerkin scheme shows a moderate improvement in run time for nonlinear Landau damping and a substantial improvement for the two-stream instability. It should be emphasized that these results are markedly different from results obtained in the asymptotic regime (which favor spline interpolation). Thus, we conclude that the traditional approach of evaluating numerical methods is misleading, even for short time simulations. In addition, the absence of any All-to-All communication in the semi-Lagrangian discontinuous Galerkin method gives it a decisive advantage for scaling to more than 256 cores.

math.NA↗

arXiv2018

A split step Fourier/discontinuous Galerkin scheme for the Kadomtsev--Petviashvili equation

In this paper we propose a method to solve the Kadomtsev--Petviashvili equation based on splitting the linear part of the equation from the nonlinear part. The linear part is treated using FFTs, while the nonlinear part is approximated using a semi-Lagrangian discontinuous Galerkin approach of arbitrary order. We demonstrate the efficiency and accuracy of the numerical method by providing a range of numerical simulations. In particular, we find that our approach can outperform the numerical methods considered in the literature by up to a factor of five. Although we focus on the Kadomtsev--Petviashvili equation in this paper, the proposed numerical scheme can be extended to a range of related models as well.

math.NA↗

arXiv2017

Efficient boundary corrected Strang splitting

Strang splitting is a well established tool for the numerical integration of evolution equations. It allows the application of tailored integrators for different parts of the vector field. However, it is also prone to order reduction in the case of non-trivial boundary conditions. This order reduction can be remedied by correcting the boundary values of the intermediate splitting step. In this paper, three different approaches for constructing such a correction in the case of inhomogeneous Dirichlet, Neumann, and mixed boundary conditions are presented. Numerical examples that illustrate the effectivity and benefits of these corrections are included.

math.NA↗

arXiv2017

An exponential integrator for the drift-kinetic model

We propose an exponential integrator for the drift-kinetic equation in cylindrical geometry. This approach removes the CFL condition from the linear part of the system (which is often the most stringent requirement in practice) and treats the remainder explicitly using Arakawa's finite difference scheme. The present approach is mass conservative, up to machine precision, and significantly reduces the computational effort per time step. In addition, we demonstrate the efficiency of our method by performing numerical simulations in the context of the ion temperature gradient instability. In particular, we find that our numerical method can take time steps comparable to what has been reported in the literature for the (predominantly used) splitting approach. In addition, the proposed numerical method has significant advantages with respect to conservation of energy and efficient higher order methods can be obtained easily. We demonstrate this by investigating the performance of a fourth order implementation.

physics.comp-ph↗