Discretization errors at free boundaries of the Grad ... - Springer Link

Numer. Math. 59, 683-710 (1991)

Numerische Mathematik

9 Springer-Verlag1991

Discretization errors at free boundaries of the Grad-Sehliiter-Shafranov equation Rita Meyer-Spasche 1 and Bengt Fornberg 2 1 Max-Planck-lnstitut ffir Plasmaphysik, IPP-EURATOM Association, W-8046 Garching, Federal Republic of Germany 2 Exxon Research & Engineering Co., Annandale, NJ 08801, USA Received September 13, 1990

Dedicated to the memory of Professor Lothar Collatz

Summary. The numerical error of s t a n d a r d finite-difference schemes is analyzed at free b o u n d a r i e s of the G r a d - S c h l i i t e r - S h a f r a n o v e q u a t i o n of plasma physics. A simple correction strategy is devised to eliminate (to leading order) the errors which arise as the free b o u n d a r y crosses the rectangular grid at irregular locations. The resulting scheme can be solved by G a u s s - N e w t o n or Inverse iterations, or by multigrid iterations. E x t r a p o l a t i o n (from 2 "d to 3 rd order of accuracy) is possible for the new scheme.

Mathematics Subject Classification (1991)." 65N12, 65N06, 35J60

Summary of contents 1 Introduction 1.1 The model equations . . . . . . . . . . 1.2 Example . . . . . . . . . . . . . . . 1.3 Numerical model . . . . . . . . . . . 1.4 Implementation . . . . . . . . . . . . 1.4.1 Discretization of L~ 1.4.2 Normal derivatives at the boundary 1.4.3 Newton's method 1.5 Explicitly known solutions . . . . . . . 1.5.1 Example 1 (1 D, e=0) 1.5.2 Example 2 (2D, e.=0) 1.5.3 Example 3 (1 D, e+0, tip= 1) 1.5.4 Example 4 (1D, e+0, tip=0) 2 The discretization errors 2.1 Previous results . . . . . . . . . . . . 2.2 Some test cases . . . . . . . . . . . . 2.3 Analysis of the test cases . . . . . . . .

Offprint requests to: R. Meyer-Spasche

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

684 687 688 689

. . . . . . . . . . . . . . . . .

691

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

695 696 698

684

R. Meyer-Spasche and B. Fornberg

2.4 Further test cases: multigrid computations . . . . . . . . . . . . . . . . . . 3 A correction strategy 3.1 The simpliest case (1 D, e=0) . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 The new scheme 3.1.2 Derivation of the new scheme 3.1.3 Implementation of the new scheme 3.2 The general 1 D case, e 4=0 . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The new scheme in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Derivation of the scheme 3.3.2 Implementation and test results References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

701 703

706 707 710

1 Introduction In magnetohydrodynamic equilibrium computations, the location of the plasma - vacuum interface is not known in advance. It is a free surface, to be determined while solving the equations. Several ways of doing this have been employed. In some of the codes equations for the forces at the interface are added. A convenient approach for axisymmetric equilibria is to "ignore" the free boundary when defining the grid. This strategy was used both with finite difference schemes and with finite elements (see, for example, I-4, 5, 7, 9, 11, 15]). The discretization error of this approach is investigated in Sect. 2. For 2nd order centered differences (O(h z) convergence on problems without free boundary), the residual is O(h) in a neighborhood of the free boundary. But this neighborhood is of order h. Thus O(h 2) convergence is obtained globally, with less favorable constants than in the regular cases. The size of the m a x i m u m error depends on the position of the grid points with respect to the free boundary. Higher order methods reduce to O(h 2) methods unless there is special treatment of the free boundary. Extrapolation h ~ 0 is of doubtful value. As a consequence, a naive application of multigrid methods might fail to converge in h. This is demonstrated in Fig. 2.6 of Sect. 2.4. A modification of the standard 2 D difference scheme is devised in Sect. 3 such that the residual becomes O (h 2) locally. The global error is reduced and becomes independent of the location of the grid points with respect to the free boundary.

I.I The model equations An axisymmetric plasma equilibrium can be modeled by the Grad-SchliiterShafranov equation (see, for instance, [Chap. 4 in 2; 12, or 16]J: (1.1.1)

A*~I+II'+R2p'=O, a2 A*=

OR 2

1

0 02 - - ~-

R dR

Oz 2'

for the flux function ~b in a plasma region f2p. Here, R, tk and z are cylindrical coordinates, p(~b) is the pressure, B~,=I(~k)/R is the toroidal magnetic field, and

Discretization errors at free boundaries

685

primes indicate differentiation with respect to ~k. The poloidal field Bp is related to ~ by (1.1.2)

Bp =Vq~ x V~k.

We consider the case where a plasma is contained in a conducting shell, but separated from it by a vacuum region f2v. With this shell coinciding with a magnetic surface, ~ becomes constant there: (1.1.3)

~b=a 0

r

y)