RK33[y_,f_,h_]:=Module[{k1,k2,k3,k4,k5,k6,yNew,zNew},
k1=f[y];
k2=f[y+1/3 h k1];
k3=f[y-3/16 h k1+15/16 h k2];
yNew=y+h (1/6 k1+3/10 k2+8/15 k3);
zNew=y+h k1;
{yNew,Norm[yNew-zNew]}];

<<NumericalDifferentialEquationAnalysis`
Eqs=Flatten[
RungeKuttaOrderConditions[3,3,
ButcherRowSum->True,
RungeKuttaMethod->Explicit
]
]

Sols =Solve[Eqs]

Length[Sols]
Sols[[1]]

pErr=ButcherPrincipalError[3,3,
ButcherRowSum->True,
RungeKuttaMethod->Explicit
]/.Sols[[4]]
Goodness[a32_,b2_,b3_,c2_,c3_]=Simplify[
pErr.pErr/.{
(Subscript[a, 3,2])->a32,
(Subscript[b, 2])->b2,
(Subscript[b, 3])->b3,
(Subscript[c, 2])->c2,
(Subscript[c, 3])->c3}
]

ContourPlot[(1/(7776 b3))(80+60 (Sqrt[3]) (Sqrt[b3 (4-6 c2+3 b3 (c2^2))])+3 b3 (54-90 c2+83 (c2^2)-24 (c2^3)+9 (c2^4))-9 (c2^3) (3+(Sqrt[3]) (Sqrt[b3 (4-6 c2+3 b3 (c2^2))]))+6 (c2^2) (9+4 (Sqrt[3]) (Sqrt[b3 (4-6 c2+3 b3 (c2^2))]))-3 c2 (48+19 (Sqrt[3]) (Sqrt[b3 (4-6 c2+3 b3 (c2^2))]))),
{b3,0,1},
{c2,0,1},
Contours->40,
ColorFunction->Hue,
FrameLabel->{"b3","c2"}
]

FindMinimum[{
(1/(7776 b3))(80+60 (Sqrt[3]) (Sqrt[b3 (4-6 c2+3 b3 (c2^2))])+3 b3 (54-90 c2+83 (c2^2)-24 (c2^3)+9 (c2^4))-9 (c2^3) (3+(Sqrt[3]) (Sqrt[b3 (4-6 c2+3 b3 (c2^2))]))+6 (c2^2) (9+4 (Sqrt[3]) (Sqrt[b3 (4-6 c2+3 b3 (c2^2))]))-3 c2 (48+19 (Sqrt[3]) (Sqrt[b3 (4-6 c2+3 b3 (c2^2))]))),
{0≤c2≤1,0<=b3≤1}
},
{b3,0.4},
{c2,0.4}
]

ContourPlot[(1/(7776 b3))(80-60 (Sqrt[3]) (Sqrt[b3 (4-6 c2+3 b3 (c2^2))])+3 b3 (54-90 c2+83 (c2^2)-24 (c2^3)+9 (c2^4))+9 (c2^3) (-3+(Sqrt[3]) (Sqrt[b3 (4-6 c2+3 b3 (c2^2))]))-6 (c2^2) (-9+4 (Sqrt[3]) (Sqrt[b3 (4-6 c2+3 b3 (c2^2))]))+3 c2 (-48+19 (Sqrt[3]) (Sqrt[b3 (4-6 c2+3 b3 (c2^2))]))),
{b3,0,1},
{c2,0,1},
Contours->40,
ColorFunction->Hue,
FrameLabel->{"b3","c2"}
]

FindMinimum[(1/(7776 b3))(80-60 (Sqrt[3]) (Sqrt[b3 (4-6 c2+3 b3 (c2^2))])+3 b3 (54-90 c2+83 (c2^2)-24 (c2^3)+9 (c2^4))+9 (c2^3) (-3+(Sqrt[3]) (Sqrt[b3 (4-6 c2+3 b3 (c2^2))]))-6 (c2^2) (-9+4 (Sqrt[3]) (Sqrt[b3 (4-6 c2+3 b3 (c2^2))]))+3 c2 (-48+19 (Sqrt[3]) (Sqrt[b3 (4-6 c2+3 b3 (c2^2))]))),
{b3,0.4},
{c2,0.4}
]

25/11664.

Simplify[(-(1/4)+(2/27) (3-4 b3)+((8 b3)/27))]

Solve[(-(1/4)+(2/27) (3-4 b3)+((8 b3)/27))==0]

Eqs={
c1==0,
c2==a21,
c3==a31+a32,
b1+b2+b3==1,
b1 c1+b2 c2+b3 c3==(1/2),
b1 (c1^2)+b2 (c2^2)+b3 (c3^2)==(1/3),
b3 a32 c2==(1/6),
(1-c2-b3) a32==(c3-c2)b2
};
{(c1^2)+ (c2^2)a21, a32 c3}/. Solve[Eqs]

NSolve[{c2-a21==0,c2-a31-a32==0,b1+b2+b3-1==0,b2 c2+b3 c3-(1/2)==0,b2 (c2^2)+b3 (c3^2)-(1/3)==0,b3 a32 c2-(1/6)==0,(1-c2-b3) a32-(c3-c2)b2==0},{c2,c3,a21,a31,a32,b1,b2,b3}]

{{c2->0.3619323196694636`,c3->0.3113493515003052`,a21->0.3619323196694636`,a31->0.40952961197220106`,a32->-0.047597292302737404`,b1->0.9706488706232026`,b2->9.704087416563109`,b3->-9.674736287186311`},{c2->2.5471285716870953`,c3->4.226519035267648`,a21->2.5471285716870953`,a31->3.041095473497367`,a32->-0.4939669018102717`,b1->0.7163630072164777`,b2->0.41610165370104973`,b3->-0.1324646609175275`},{c2->-2.7150615050120024`,c3->-0.6906478260904478`,a21->-2.7150615050120024`,a31->-2.765820882811114`,a32->0.05075937779911177`,b1->2.085879294380903`,b2->0.12347287348600042`,b3->-1.2093521678669035`},{c2->-0.1518680561900973`-0.4370573981079545` I,c3->0.69435135084848`-0.5987671572514448` I,a21->-0.1518680561900973`-0.4370573981079545` I,a31->-0.6857980689980456`-0.7387725605122618` I,a32->0.5339300128079483`+0.3017151624043073` I,b1->0.26168783836685394`-0.9832427056485276` I,b2->0.6332031798540179`+0.40537236270352467` I,b3->0.1051089817791282`+0.5778703429450028` I},{c2->-0.1518680561900973`+0.4370573981079545` I,c3->0.69435135084848`+0.5987671572514448` I,a21->-0.1518680561900973`+0.4370573981079545` I,a31->-0.6857980689980456`+0.7387725605122618` I,a32->0.5339300128079483`-0.3017151624043073` I,b1->0.26168783836685394`+0.9832427056485276` I,b2->0.6332031798540179`-0.40537236270352467` I,b3->0.1051089817791282`-0.5778703429450028` I},{c2->0.6040063435997463`,c3->0.7334957903123429`,a21->0.6040063435997463`,a31->-0.23251421080981616`,a32->0.8365205544095625`,b1->0.24291107251745087`,b2->0.4272281842652621`,b3->0.32986074321728703`},{c2->0.2205565849601555`,c3->0.6097330231291715`,a21->0.2205565849601555`,a31->-0.5833450920192567`,a32->0.8039016769794122`,b1->0.3916486806142072`,b2->-0.33164420619479246`,b3->0.9399955255805852`}}