Edge equilibrium code for tokamaks

Li, Xujing; Zakharov, L.

doi:10.1063/1.4861369

Cited by 11 publications

(13 citation statements)

References 29 publications

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“…In order to deal with the semi-linearity of the problem we will resort to an iterative strategy. Due to their simplicity and effectiveness, straightforward fixed-point iterations (also known as a Picard iterations) of the style −∆ * ψ n = F(r, z, ψ n−1 ) have been preferred in many applications [11,12,17,18,19,20,22]. We choose to follow a similar strategy, but enhance it with two simple yet effective acceleration methods.…”

Section: Accelerated Fixed-point Iterationsmentioning

confidence: 99%

“…have been preferred in many applications [11,12,17,18,19,20,22]. We choose to follow a similar strategy, but enhance it with two simple yet effective acceleration methods.…”

Section: Accelerated Fixed-point Iterationsmentioning

confidence: 99%

“…Some other recent alternatives that have attracted attention include the hybrid approach EEC-ESC that couples Hermite elements near the plasma edge with Fourier decomposition methods in the plasma core [20], the use of meshless methods [21,22], the use of approximate particular solutions [23], and the method of fundamental solutions [24].…”

Section: Introductionmentioning

confidence: 99%

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A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation

Sánchez-Vizuet

Solano

2019

Computer Physics Communications

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Section: Accelerated Fixed-point Iterationsmentioning

confidence: 99%

“…have been preferred in many applications [11,12,17,18,19,20,22]. We choose to follow a similar strategy, but enhance it with two simple yet effective acceleration methods.…”

Section: Accelerated Fixed-point Iterationsmentioning

confidence: 99%

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A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation

Sánchez-Vizuet

Solano

2019

Computer Physics Communications

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“…More recently, several other approaches have been proposed and are used by different research groups, e.g. CEDRES++ [32], CHEASE [50,51], CREATE-NL+ [2], ECOM [46,59], EEC [47], ESC [71], HELENA [37], NIMEQ [35], SPIDER [40], etc.…”

Section: Introductionmentioning

confidence: 99%

A mimetic spectral element solver for the Grad–Shafranov equation

Palha

Koren

Felici

2016

Journal of Computational Physics

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In this work we present a robust and accurate arbitrary order solver for the fixed-boundary plasma equilibria in toroidally axisymmetric geometries. To achieve this we apply the mimetic spectral element formulation presented in [56] to the solution of the Grad-Shafranov equation. This approach combines a finite volume discretization with the mixed finite element method. In this way the discrete differential operators (∇, ∇×, ∇·) can be represented exactly and metric and all approximation errors are present in the constitutive relations. The result of this formulation is an arbitrary order method even on highly curved meshes. Additionally, the integral of the toroidal current J φ is exactly equal to the boundary integral of the poloidal field over the plasma boundary. This property can play an important role in the coupling between equilibrium and transport solvers. The proposed solver is tested on a varied set of plasma cross-sections (smooth and with an X-point) and also for a wide range of pressure and toroidal magnetic flux profiles. Equilibria accurate up to machine precision are obtained. Optimal algebraic convergence rates of order p + 1 and geometric convergence rates are shown for Soloviev solutions (including high Shafranov shifts), field-reversed configuration (FRC) solutions and spheromak analytical solutions. The robustness of the method is demonstrated for non-linear test cases, in particular on an equilibrium solution with a pressure pedestal.The Grad-Shafranov equation, (1), is a non-linear elliptic partial differential equation. Its non-linear character stems from the non-linear dependence of P and f on the unknown flux function ψ and from the fact that the plasma domain Ω p is also an unknown, that in general can only be determined once the flux function is known. These two characteristics make the solution of this equation a challenging task.The literature on the numerical solution of MHD equilibria is extensive. A detailed review of different formulations and codes prior to 1991 is presented in [67] and a shorter review up to 1984 is given in [5]. More recently, several other approaches have been proposed and are used by different research groups, e.g. CEDRES++ [32], CHEASE [50, 51], CREATE-NL+ [2], ECOM [46, 59], EEC [47], ESC [71], HELENA [37], NIMEQ [35], SPIDER [40], etc.The solution of MHD equilibria can be grouped into two distinct classes: (i) fixed-boundary (e.g. CHEASE, ECOM, EEC, ESC, HELENA, NIMEQ, SPIDER) and (ii) free-boundary (e.g. CEDRES++, CREATE-NL+). In the fixed-boundary case the plasma domain is prescribed together with ψ = constant at its boundary, ∂Ω p . Equation (1) is then used to find ψ inside the plasma. The free-boundary approach requires the solution of the MHD equilibrium in an infinite domain with homogeneous boundary conditions, ψ = 0, at infinity and taking into consideration the current flowing in a set of external coils and the Grad-Shafranov equation in the plasma region, (1). In this situation, both the plasma domain and the flux function, ψ, are unknowns tha...

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“…Nevertheless, the development of optimized G-S codes remains a topic of active research, for three main reasons. First, G-S solvers must be able to properly resolve complex two-dimensional geometries [10,11], with boundaries that may have a corner, corresponding to a magnetic field X-point [11]. Second, G-S solvers must be fast.…”

Section: Introductionmentioning

confidence: 99%

ECOM: A fast and accurate solver for toroidal axisymmetric MHD equilibria

Lee

Cerfon

2015

Computer Physics Communications

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We present ECOM (Equilibrium solver via COnformal Mapping), a fast and accurate fixed boundary solver for toroidally axisymmetric magnetohydrodynamic equilibria with or without a toroidal flow. ECOM combines conformal mapping and Fourier and integral equation methods on the unit disk to achieve exponential convergence for the poloidal flux function as well as its first and second partial derivatives. As a consequence of its high order accuracy, for dense grids and tokamak-like elongations ECOM computes key quantities such as the safety factor and the magnetic shear with higher accuracy than the finite element based code CHEASE [H. Lütjens et al., Computer physics communications 97, 219 (1996)] at equal run time. ECOM has been developed to provide equilibrium quantities and details of the flux contour geometry as inputs to stability, wave propagation and transport codes.

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Edge equilibrium code for tokamaks

Cited by 11 publications

References 29 publications

A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation

A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation

A mimetic spectral element solver for the Grad–Shafranov equation

ECOM: A fast and accurate solver for toroidal axisymmetric MHD equilibria

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