Numerical Methods for 3D Tokamak Simulations Using a Flux‐Surface Independent Grid

Stegmeir, A.; Coster, D.; Maj, O.; Lackner, K.

doi:10.1002/ctpp.201410041

Cited by 9 publications

(10 citation statements)

References 9 publications

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“…For example in a gyro-fluid model we have the nonlinear elliptic equation ∇ · (N∇ ⊥ φ) = n − N, where n is the electron density, N is the ion gyro-center density, φ the electric potential and ∇ ⊥ the gradient in the direction perpendicular to the magnetic field [45]. Current interest also includes the implementation of the flux-coordinate independent approach to discretize derivatives along arbitrary magnetic field lines [27,54,33]. This type of scheme is particularly important if a magnetic field aligned coordinate system is unavailable due to singularities in the coordinate transformations (X-points).…”

Section: Introductionmentioning

confidence: 99%

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

Wiesenberger

Einkemmer

Held

et al. 2019

Computer Physics Communications

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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. We briefly discuss alternative methods to ensure the correctness of results like the convergence of reduced physical quantities of interest, ensemble simulations, invariants or reduced 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 10 −1 and 10 3 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).

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Section: Introductionmentioning

confidence: 99%

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

Wiesenberger

Einkemmer

Held

et al. 2019

Computer Physics Communications

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“…Motivated by previous work from [13,14], a numerical scheme is developed which exhibits very low numerical diffusion. The discussion extends previous work from [15]. Several model problems are also discussed in Appendix A The developed numerical methods are implemented in the new code GRILLIX.…”

Section: Introductionmentioning

confidence: 64%

The field line map approach for simulations of magnetically confined plasmas

Stegmeir

Coster

Maj

et al. 2016

Computer Physics Communications

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In the presented field line map approach the simulation domain of a tokamak is covered with a cylindrical grid, which is Cartesian within poloidal planes. Standard finite-difference methods can be used for the discretisation of perpendicular (w.r.t. magnetic field lines) operators. The characteristic flute mode property k k ⊥ of structures is exploited computationally by a grid sparsification in the toroidal direction. A field line following discretisation of parallel operators is then required, which is achieved via a finite difference along magnetic field lines. This includes field line tracing and interpolation or integration. The main emphasis of this paper is on the discretisation of the parallel diffusion operator. Based on the support operator method a scheme is constructed which exhibits only very low numerical perpendicular diffusion. The schemes are implemented in the new code GRILLIX, and extensive benchmarks are presented which show the validity of the approach in general and GRILLIX in particular. The main advantage of the approach is that it does not rely on field/flux-aligned, which become singular on the separatrix/X-point. Most tokamaks are based on the divertor concept, and the numerical treatment of the separatrix is therefore of importance for simulations of the edge and scrape-off layer.

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“…We use the bump defined in Eq. (20) with (x 0 , y 0 ) = (420, −470) and σ = 50. This is the lower bump in Fig.…”

Section: P=1mentioning

confidence: 99%

“…For completeness let us also mention recent approaches to use non fluxaligned grids for the discretization of model equations [19,20,21]. Like unstructured grids, these avoid numerical issues with the X-point but shift the problem to the question of how to correctly implement a flux-aligned boundary.…”

Section: Introductionmentioning

confidence: 99%

Streamline integration as a method for structured grid generation in X-point geometry

Wiesenberger

Held

Einkemmer

et al. 2018

Journal of Computational Physics

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We investigate structured grids aligned to the contours of a two-dimensional flux-function with an X-point (saddle point). Our theoretical analysis finds that orthogonal grids exist if and only if the Laplacian of the flux-function vanishes at the X-point. In general, this condition is sufficient for the existence of a structured aligned grid with an X-point. With the help of streamline integration we then propose a numerical grid construction algorithm. In a suitably chosen monitor metric the Laplacian of the flux-function vanishes at the X-point such that a grid construction is possible.We study the convergence of the solution to elliptic equations on the proposed grid. The diverging volume element and cell sizes at the X-point reduce the convergence rate. As a consequence, the proposed grid should be used with grid refinement around the X-point in practical applications. We show that grid refinement in the cells neighbouring the X-point restores the expected convergence rate.

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Numerical Methods for 3D Tokamak Simulations Using a Flux‐Surface Independent Grid

Cited by 9 publications

References 9 publications

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

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

The field line map approach for simulations of magnetically confined plasmas

Streamline integration as a method for structured grid generation in X-point geometry

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