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

Andras Pataki, Antoine J. Cerfon, Jeffrey P. Freidberg, Leslie Greengard, Michael O'Neil

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)28-45
Number of pages18
JournalJournal of Computational Physics
StatePublished - Jun 5 2013


  • Conformal mapping
  • Grad-Shafranov
  • High-order
  • Kerzman-Stein
  • Plasma physics
  • Poisson solver
  • Spectrally-accurate

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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