### 13.3PASS: Poisson problem with a pure spherical harmonics solution

Author
Sébastien Delaux
Command
sh harmonic.sh harmonic.gfs
Version
110208
Required files
harmonic.sh gerris.gfv res-7.ref error.ref order.ref
Running time
4 minutes 22 seconds

The spherical harmonics are by definition a family of functions which satisfy the Laplace equation on the sphere

2 f =
 1 r2

 ∂ ∂ r

r2
 ∂ f ∂ r

 1 r2 sinθ
 ∂ ∂ θ

sinθ
 ∂ f ∂ θ

 1 r2 sin2 θ

 ∂2 f ∂ ϕ2
= 0.

If we look for solutions of the form f (r, θ, ϕ) = R (r) Y (θ, ϕ), then two differential equations result. If we consider only the equation for the angles, we get

 1 Y

 1 sinθ

 ∂ ∂ θ

sinθ
 ∂ Y ∂ θ

 1 Y
 1 sin2 θ

 ∂2 Y ∂ ϕ2
= − λ

were λ is a constant. This equation has a whole range of solutions noted Ylm (θ, ϕ) so that

 ∇ Ylm (θ, ϕ) = − l (l + 1) Ylm (θ, ϕ) .

The Ylm can be expressed as functions of the Legendre polynomials as

Ylm (θ, ϕ) =
 √
 2 l + 1 4 π

 (l − m) ! (l + m) !
Plm (cosθ) cos(m ϕ)

where Plm is the associated Legendre polynomial.

For this test we used l = 4 and m = 2.

Figure 166 illustrates the evolution of the maximum residual as a function of CPU time. Figure 167 illustrates the average residual reduction factor (per V-cycle). The evolution of the norms of the error of the final solution as a function of resolution is illustrated on Figure 168. The corresponding order of convergence is given on Figure 169.

#### 13.3.1PASS: Spherical harmonics with longitude-latitude coordinates

Author
Sébastien Delaux
Command
sh ../harmonic.sh lonlat.gfs
Version
110208
Required files