Contour plotting temperature data on a hemisphere or circle

Question

(edited question:) I solved the Laplace equation(PDE) in spherical coordinates (in my problem temperature is steady and it is azimuthal(θ) symmetric, so my data is only a function of r and φ.) Note that I defined the coordinates as Spherical[r, φ, θ].

Now I want to make a contour plot of this data (temperature would be contoured) on the surface of a solid hemisphere(not a hollow one), but I am having problems.

This is the data that should be contoured on a hemisphere of radius 2 (u[r, φ], in which: 0 < r < 2 & 0 < φ < π)

K[n_] := (2 n + 1)/(n*(2^n))Integrate[10 φ LegendreP[n, Cos[φ]] Sin[φ], {φ, 0, π}] // N
u[r_, φ_] := Sum[K[n] r^n LegendreP[n, Cos[φ]], {n, 1, 8}]

After days probing for ans,I tried simpler contour instead of hemisphere and all of them didn't give any answer and were just running..!(except one 2D contourplot which was evaluated after an hour!) But when @WateSoyan answered my question(as below is shown) he solved the "running time" problem,he just put out the "Hue" from code, by using a pure function( #..&) and this accelerated the evaluation amazingly!!but I added Rescale function into the Hue,'coz making the domain (0...1),despite these efforts, YET the color gradients and mesh lines are NOT ideal(as the examples I show here with 2Dcontour or with Maple)" how can the 3Dplot be corrected?

1=> Last edition of my 3D code

plots = RegionPlot3D[(x <= 0) && 0 <= x^2 + y^2 + z^2 < 4, {x, -3, 
 3}, {y, -3, 3}, {z, -3, 3}, 
ColorFunction -> Function[{x, y, z}, #], 
PlotRange -> {{-3, 3}, {-3, 3}, {-3, 3}}, PlotPoints -> 100, 
MaxRecursion -> 10] &@
Hue[Rescale[u[Sqrt[x^2 + y^2 + z^2], ArcCos[z/Sqrt[x^2 + y^2 + z^2]]],
{u[2, 0], u[2, Pi]}]]

result=>

2Dcontour code which this one is somehow a correct answer=

ContourPlot[#1, {x, -2, 2}, {y, -2, 2}, 
RegionFunction -> Function[{x, y, z}, 0 < x^2 + y^2 < 4]] & @ 
u[Sqrt[x^2 + y^2], ArcCos[y/Sqrt[x^2 + y^2]]]

For an analogy to the right answer,I put my code and result of it in Maple,which I'd done last week very easy!=>

K := (2*n+1)*(int(10*phi*LegendreP(n, x)*sin(phi), phi = 0 .. Pi))/n(10)^n
u := (rho, phi) ↦ add(K*simplify(LegendreP(n, cos(phi)))*rho^n, n = 0 .. 10)
#plotting=> #
contourplot([rho, phi, u(rho, phi+(1/2)*Pi)], rho = 0 .. c, phi = -Pi .. Pi,
coords = cylindrical, filledregions = true) 

contour plotting in Maple in less than 30 seconds=>

Answer 1

I think your ColorFunction may be used incorrectly, and my edition is

Clear[k, u]
k[n_] := (2 n + 1)/(n*(2^n)) Integrate[10 \[CurlyPhi] LegendreP[n, 
      Cos[\[CurlyPhi]]] Sin[\[CurlyPhi]], {\[CurlyPhi], 0, \[Pi]}] // N
u[r_, \[CurlyPhi]_] := 
  Evaluate@Sum[k[n] r^n LegendreP[n, Cos[\[CurlyPhi]]], {n, 1, 8}];

plots = RegionPlot3D[(x <= 0) && 0 <= x^2 + y^2 + z^2 < 4, {x, -3, 3}, 
    {y, -3, 3}, {z, -3, 3}, 
    ColorFunction -> Function[{x, y, z}, #],PlotRange -> {{-3, 3}, {-3, 3}, {-3, 3}}, 
    PlotPoints->100,MaxRecursion -> 10] &@
  Hue[u[Sqrt[x^2 + y^2 + z^2], ArcCos[z/Sqrt[x^2 + y^2 + z^2]]]]