The Princeton spectral equilibrium code: PSEC

Ling, K.M.; Jardin, S.C.

doi:10.1016/0021-9991(85)90165-2

Cited by 32 publications

(11 citation statements)

References 5 publications

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“…A second respect in which the present procedure departs from existing methods is through the use of a spectral Green's function approach for computing the scalar magnetic potential $ and hence the self-consistent vacuum magnetic field, which is normal to S p . Green's functions have been previously used [18] in plasmas with toroidal symmetry to compute the axisymmetric component of the vector potential, V> = -RA^. Compared with a relaxation scheme for computing the vacuum potential [4], there are at least two advantages to a Green's function approach when the vacuum field is coupled with the moving plasma boundary:…”

Section: Free Boundary Equationsmentioning

confidence: 99%

Three-dimensional free boundary calculations using a spectral Green's function method

Hirshman

Rij

Merkel

1986

Computer Physics Communications

635

584

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The plasma energy W p = j [\B 2-h p)dV is minimized over a toroidal domain ft p using an inverse representation for the cylindrical coordinates R = ^ R mn (s) cos(m6-nf) and Z = VJ Z mn (s) sin(m#-n<;) , where (.s,0,f) are radial, poloidal, and toroidal flux coordinates, respectively. The radial resolution of the MHD equations is significantly improved by separating R and Z into contributions from even and odd poloidal harmonics which are individually analytic near the magnetic axis. A free boundary equilibrium results when tt p is varied to make the total pressure-B 2 + p continuous at the plasma surface E p and when tie vacuum magnetic field B^ satisfies the Neumann condition B w • dS p = 0. The vacuum field is decomposed as B v = Bo I V$, where Bo is the field arising from plasma currents and external coils and $ is a single-valued potential necessary to satisfy B v • d*£, p = 0 when p == 0. A Green's function method is used to obtain an integral equation over S p for the scalar magnetic potential $ = YJ$ mrl sin(mfl-n<;). A linear matrix equation is solved for $ mn to determine ^B% on the boundary. Real experimental conditions are simulated by keeping the external and net plasma currents constant during the iteration. Applications to I-2 stellarator equilibria are presented.

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Section: Free Boundary Equationsmentioning

confidence: 99%

Three-dimensional free boundary calculations using a spectral Green's function method

Hirshman

Rij

Merkel

1986

Computer Physics Communications

635

584

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“…These schemes generally fall into two categories. Eulerian or "direct" solvers use a prescribed mesh to calculate the unknown function [13,17,19,21,29], while Lagrangian or "inverse" solvers find the mapping of the plasma geometry in terms of magnetic coordinates [7,16,25,27,42]. The advantages and disadvantages of one formulation as compared to the other depend on the application of interest [43], on the plasma geometry, and on the type of inputs that plasma stability and transport codes require.…”

Section: Introductionmentioning

confidence: 99%

A fast, high-order solver for the Grad–Shafranov equation

Pataki

Cerfon

Freidberg

et al. 2013

Journal of Computational Physics

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We present a new fast solver to calculate fixed-boundary plasma equilibria in toroidally axisymmetric geometries. By combining conformal mapping with Fourier and integral equation methods on the unit disk, we show that high-order accuracy can be achieved for the solution of the equilibrium equation and its first and second derivatives. Smooth arbitrary plasma cross-sections as well as arbitrary pressure and poloidal current profiles are used as initial data for the solver. Equilibria with large Shafranov shifts can be computed without difficulty. Spectral convergence is demonstrated by comparing the numerical solution with a known exact analytic solution. A fusion-relevant example of an equilibrium with a pressure pedestal is also presented.

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“…Nevertheless, when the source term has a simple form, we can use Green's functions to solve the GS equation. However, except for a small number of simple forms for the source function, numerical methods such as finite differences (Johnson et al 1979), spectral methods (Ling & Jardin 1985), spectral elements (Howell & Sovinec 2014), and linear finite elements (Gruber et al 1987) should be used. In applications to NSs, numerical solvers such as the HSCF method (Lander & Jones 2009), Gauss-Seidel method (Gourgouliatos et al 2013) or the generalised Newton's method (Armaza et al 2015) have been used.…”

Section: Introductionmentioning

confidence: 99%

The impact of superconductivity and the Hall effect in models of magnetised neutron stars

Sur

Haskell

2021

Publ. Astron. Soc. Aust.

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Equilibrium configurations of the internal magnetic field of a pulsar play a key role in modelling astrophysical phenomena from glitches to gravitational wave emission. In this paper, we present a numerical scheme for solving the Grad–Shafranov equation and calculating equilibrium configurations of pulsars, accounting for superconductivity in the core of the neutron star, and for the Hall effect in the crust of the star. Our numerical code uses a finite difference method in which the source term appearing in the Grad–Shafranov equation, which is used to model the magnetic equilibrium is non-linear. We obtain solutions by linearising the source and applying an under-relaxation scheme at each step of computation to improve the solver’s convergence. We have developed our code in both C++ and Python, and our numerical algorithm can further be adapted to solve any non-linear PDEs appearing in other areas of computational astrophysics. We produce mixed toroidal–poloidal field configurations, and extend the portion of parameter space that can be investigated with respect to previous studies. We find that in even in the more extreme cases, the magnetic energy in the toroidal component does not exceed approximately 5% of the total. We also find that if the core of the star is superconducting, the toroidal component is entirely confined to the crust of the star, which has important implications for pulsar glitch models which rely on the presence of a strong toroidal field region in the core of the star, where superfluid vortices pin to superconducting fluxtubes.

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The Princeton spectral equilibrium code: PSEC

Cited by 32 publications

References 5 publications

Three-dimensional free boundary calculations using a spectral Green's function method

Three-dimensional free boundary calculations using a spectral Green's function method

A fast, high-order solver for the Grad–Shafranov equation

The impact of superconductivity and the Hall effect in models of magnetised neutron stars

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