<< Graphics`

(* 
      Euler explicit *) 
qabs[θ_,  ν_]  =  1 + (ν^2) Sin[θ]^2;
φ  =  -ArcTan[ν Sin[θ]];
φexact  =  - ν θ;
φnorm[θ_,  ν_]  =  (φ/φ)exact;
PolarPlot[{qabs[θ, 0.25], qabs[θ, 0.5], 
        qabs[θ, 0.75], qabs[θ, 1.0]}, {θ, 0, π}, 
      AspectRatio -> 1]
PolarPlot[{φnorm[θ, 0.25], φnorm[θ, 
          0.5], φnorm[θ, 0.75], φnorm[θ, 
          1.0]}, {θ, 0, π}, AspectRatio -> 1]

(* 
      Lax *) 
qabs[θ_,  ν_]  =  (((Cos[θ]^2 + (ν^2) Sin[θ]^2))^(1/2));
φ  =  -(ArcTan[ν Tan[θ]] + π IntegerPart[((2  θ)/π)]);
φexact  =  - ν θ;
φnorm[θ_,  ν_]  = (φ/φ)exact;
ϵ = 0.01;
PolarPlot[{qabs[θ, 0.25], qabs[θ, 0.5], 
        qabs[θ, 0.75], qabs[θ, 1.0]}, {θ, 
        0, π - ϵ}, AspectRatio -> 1]
PolarPlot[{φnorm[θ, 0.25], φnorm[θ, 
          0.5], φnorm[θ, 0.75], φnorm[θ, 
          1.0]}, {θ, 0, π}, AspectRatio -> 1]

(* Lax - 
        Wendroff *) 
qabs[θ_,  ν_]  =  ((((((1 - (ν^2) (1 - Cos[θ])))^2) + (ν^2) Sin[θ]^2))^(1/2));
φ  =  -ArcTan[((ν Sin[θ])/(1 - (ν^2) (1 - Cos[θ])))] - (π/2) (1 - 
                Sign[((ν Sin[θ])/(1 - (ν^2) (1 - Cos[θ])))]);
φexact  =  - ν θ;
φnorm[θ_,  ν_]  =  (φ/φ)exact;
PolarPlot[{qabs[θ, 0.25], qabs[θ, 0.5], 
        qabs[θ, 0.75], qabs[θ, 1.0]}, {θ, 
        0, π}]
PolarPlot[{φnorm[θ, 0.25], φnorm[θ, 
          0.5], φnorm[θ, 0.75], φnorm[θ, 
          1.0]}, {θ, 0, π}]

(* 1  
        st order upwind *) 
qabs[θ_,  ν_]  =  ((((((1 - ν (1 - Cos[θ])))^2) + (ν^2) Sin[θ]^2))^(1/2));
φ  =  -ArcTan[((ν Sin[θ])/(1 - ν (1 - Cos[θ])))] - (π/2) (1 - 
                Sign[((ν Sin[θ])/(1 - ν (1 - Cos[θ])))]);
φexact  =  - ν θ;
φnorm[θ_,  ν_]  =  (φ/φ)exact;
PolarPlot[{qabs[θ, 0.25], qabs[θ, 0.5], 
        qabs[θ, 0.75], qabs[θ, 1.0]}, {θ, 
        0, π}]
PolarPlot[{φnorm[θ, 0.25], φnorm[θ, 
          0.5], φnorm[θ, 0.75], φnorm[θ, 
          1.0]}, {θ, 0, π}]