Stability regions of numerical methods for solving ordinary differential equations

Linear multistep methods

Adams-Bashforth

M=7;
colors={Red,Blue,Green,Orange,Black,Yellow,Brown};
Do[
{α,β}=MyGetLMMCoefficients["Adams-Bashforth",j];
ρ[z_]:=MyPolynomial[α,z];
σ[z_]:=MyPolynomial[β,z];
bb[θ_]=ρ[Exp[I θ]]/σ[Exp[I θ]];
plot[j]=ParametricPlot[{Re[bb[θ]],Im[bb[θ]]},{θ,0,2π},PlotRange->{{-2.5,1.5},{-1.5,1.5}}/1,PlotStyle->{Thick,colors[[j]]}],
{j,1,M}
];
Show[Table[plot[j],{j,1,M}]]

Adams-Moulton

M=6;
colors={Red,Blue,Green,Orange,Black,Yellow,Brown};
Do[
{α,β}=MyGetLMMCoefficients["Adams-Moulton",j];
ρ[z_]:=MyPolynomial[α,z];
σ[z_]:=MyPolynomial[β,z];
bb[θ_]=ρ[Exp[I θ]]/σ[Exp[I θ]];
plot[j]=ParametricPlot[{Re[bb[θ]],Im[bb[θ]]},{θ,0,2π},PlotRange->{{-2.5,0.5},{-1.5,1.5}}/0.4,PlotStyle->{Thick,colors[[j]]}],
{j,1,M}
];
Show[Table[plot[j],{j,1,M}]]

Backward Differentiation

M=6;
colors={Red,Blue,Green,Orange,Black,Yellow,Brown};
Do[
{α,β}=MyGetLMMCoefficients["BD",j];
ρ[z_]:=MyPolynomial[α,z];
σ[z_]:=MyPolynomial[β,z];
bb[θ_]=ρ[Exp[I θ]]/σ[Exp[I θ]];
plot[j]=ParametricPlot[{Re[bb[θ]],Im[bb[θ]]},{θ,0,2π},PlotRange->{{-1.5,2.5},{-1.5,1.5}}/0.2,PlotStyle->{Thick,colors[[j]]}],
{j,1,M}
];
Show[Table[plot[j],{j,1,M}]]

explicit Runge-Kutta methods

M=7;
colors={Red,Blue,Green,Orange,Black,Yellow,Brown};
xmax=5;
Do[
eee[x_,y_]=Abs[Normal[Series[Exp[z],{z,0,j}]]/.{z->x+I y}];
plot[j]=ContourPlot[eee[x,y]==1,{x,-xmax,xmax},{y,-xmax,xmax},ContourStyle->{colors[[j]]}],
{j,1,M}
];
Show[Table[plot[j],{j,1,M}]]