Spectral collocation method for collisional power absorption in radially inhomogeneous helicon plasmas

A spectral collocation method using Chebyshev polynomials is applied to compute the electromagnetic fields and the collisional power absorption in radially inhomogeneous helicon plasmas. The governing equation has a singularity at the cylindrical axis, which has been resolved by imposing the appropriate boundary conditions. The results are compared with those of the finite difference method. The present work not only shows the superior accuracy of the spectral collocation method with a proper treatment of the singularity, but also discusses the advantages of using Chebyshev nodes for helicon plasmas in particular. It is possible to extend the range of parameters efficiently in cases for which the finite difference method fails to obtain reliable solutions without drastically decreasing the grid size. The spectral collocation method is shown to be especially useful near the lower hybrid resonance condition.

Figure 1

The normalized perpendicular wave numbers given as functions of $N_{\parallel }$ for a uniform plasma. The real and imaginary parts are indicated by the solid and dashed lines, respectively.

Figure 2

Relative errors of the numerical solutions obtained by the finite difference and spectral collocation methods with respect to the analytic solution. These are denoted by FDM and SCM, respectively.

Figure 3

Relative errors of power conservation for the numerical solutions obtained by the finite difference and spectral methods. The plasma is not uniform with $\mu =2.2$, but other parameters are the same as in Fig. 2.

Figure 4

(a) Plasma resistance from the spectral method and (b) the relative errors of power conservation from both methods given as functions of the magnetic field.

Figure 5

Comparisons of the power absorption profiles between the finite difference and spectral methods at $B_0=300$, 1600, and 2000 G, respectively. The profiles are normalized to their respective maximum values.

Figure 6

Plasma resistance and its accuracy of the FDM compared with those of the spectral method. The numbers denote the number of subintervals used in the FDM. For the spectral method, the number of subintervals is 100.

Figure 7

Plasma resistance from the spectral method and its accuracy given as functions of the magnetic field for different numbers of subintervals.

Figure 8

Relative time and error versus the number of subintervals when $B_0=1000$ G and $N_{\parallel }=100$. The time is normalized with respect to that of the spectral method with 100 subintervals.

Figure 9

Plasma resistance from the spectral method and relative errors of the spectral and the finite difference methods when the electromagnetic fields are excited by a Nagoya Type III antenna. The $m=\pm 1$ modes are included, and the other parameters are the same as in Fig. 4.