Ecole doctorale de Physique de la région Parisienne (ED107)

182 Spectral methods and vectorial equations
correct analytical properties at r = 0. Indeed, they did not vanish as r l . Consequently
high spatial frequency terms were still generated that did not have the good analytical
properties and a runaway toward the instability was once again generated. To avoid this
phenomenon, we project, at each time step, all the scalar quantities (div W )lm[r], Po,lm [r]
and Th,lm [r] on a Legendre space P l n[r] in such a way to satisfy the analytical conditions.
Note that the necessity of this procedure is due to the fact that in the EE, solved with
spectral methods, no dissipative term (numerical or physical) is present. Consequently
the numerical instability are not damped.
Solving the vectorial heat equation
Now we give more details concerning the algorithm to solve vectorial PDE with spectral
methods. In solving NSE, we need to solve a vectorial heat equation that can be reduced
to an equation of the type
∂ ˆ B
∂t + ∇ ∧ (µ∇ ∧ ˆ B) = ˆJ (B.10)
where ˆ J is the divergence-free term of the source. We remember that it is obtained by the
decomposition of the source term J in a potential part ∇φ and a divergence-free part ˆ J :
J = ˆ J + ∇φ.
The Equation (B.10) can be written in the following form :
∂ B
∂t = µ △ B + (∇ ∧ B) ∧ ∇µ + ˆ J. (B.11)
From a numerical point of view, the term S = (∇ ∧ ˆ B) ∧ ∇µ is considered as a source
and is computed by a second order scheme using its values at the times tj−1 and tj−2.
Therefore, we have to solve the equation
with the condition ∇ · ˆ B = 0.
∂ B
∂t = µ∆ B + ˆ J + S (B.12)
The technique generally used was already described in Bonazzola et al. (1999). Nevertheless,
we adopted here a slightly different approach which gives more accurate results.
To solve the Equation (B.12) it is convenient to introduce the poloidal and toroidal
potentials Po and Th as defined in Appendix B.1.2. Here we will only consider the case
of µ = 1, as the semi-implicit scheme which must be used in the more general case is