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NumericalIntegration.m
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NumericalIntegration.m
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(* ::Package:: *)
(*use 'x' as r/M and \[Omega] to denote 'M\[Omega]'*)
BeginPackage["ReggeWheeler`NumericalIntegration`"];
Begin["`Private`"];
SetAttributes[PsiIn, {NumericFunction}];
PsiIn[s_, l_, \[Omega]_, xmin_, xmax_]:=
Module[{newassoc},
If[\[Omega]==0,
newassoc=<|
"Psi"->Function[y,PsiInStaticOdd[s,l,y]],
"dPsidr"->Function[y,Derivative[0,0,1][PsiInStaticOdd][s,l,y]],
"xmin"->xmin,
"xmax"->xmax
|>;
,
If[s==2,
Module[{boundaryconditions,soln,newAssoc,psiBC,dpsidxBC,xBC},
boundaryconditions = ReggeWheelerInBC[s,l,\[Omega],$MachinePrecision];
psiBC=boundaryconditions["Psi"];
dpsidxBC=boundaryconditions["dPsidx"];
xBC=boundaryconditions["xBC"];
soln = Integrator[s,l,\[Omega],psiBC,dpsidxBC,xBC,xmax,ReggeWheelerPotential,$MachinePrecision];
newassoc=<|
"Psi"->soln[[1,1,2]],
"dPsidr"->soln[[1,2,2]],
"xmin"->xBC,
"xmax"->xmax
|>;
];,
Nothing (*wait for further functionality*)
];
];
newassoc
];
(*Derivative[1][PsiIn[s_Integer, l_Integer, \[Omega]_, rmin_, rmax_]][x_?InexactNumberQ] :=
Module[{},
];
Derivative[n_Integer?Positive][PsiIn[s_Integer, l_Integer, \[Omega]_, rmin_, rmax_]][r0_?InexactNumberQ] :=
Module[{d2R, pderivs, R, r, i},
d2R = -2/(x^2*(1-2/x))*Derivative[1][R][r] -(\[Omega]^2-ReggeWheelerPotential[s,l,x])/(1-2/x)^2*R[r];
pderivs = D[R[r_], {r_, i_}] :> D[d2R, {r, i - 2}] /; i >= 2;
Do[Derivative[i][R][r] = Simplify[D[Derivative[i - 1][R][r], r] /. pderivs];, {i, 2, n}];
Derivative[n][R][r] /. {
R'[r] -> PsiIn[s, l, \[Omega]]'[r0],
R[r] -> PsiIn[s, l, \[Omega]][r0], r -> r0}
];*)
SetAttributes[PsiUp, {NumericFunction}];
PsiUp[s_, l_, \[Omega]_, xmin_, xmax_]:=
Module[{newassoc},
If[\[Omega]==0,
newassoc=<|
"Psi"->Function[y,PsiUpStaticOdd[s,l,y]],
"dPsidr"->Function[y,Derivative[0,0,1][PsiUpStaticOdd][s,l,y]],
"xmin"->xmin,
"xmax"->xmax
|>;
,
If[s==2,
Module[{boundaryconditions,soln,newAssoc,psiBC,dpsidxBC,xBC},
boundaryconditions = ReggeWheelerUpBC[s,l,\[Omega],xmax,$MachinePrecision];
psiBC=boundaryconditions["Psi"];
dpsidxBC=boundaryconditions["dPsidx"];
xBC=boundaryconditions["xBC"];
soln = Integrator[s,l,\[Omega],psiBC,dpsidxBC,xBC,xmin,ReggeWheelerPotential,$MachinePrecision];
newassoc= <|
"Psi"->soln[[1,1,2]],
"dPsidr"->soln[[1,2,2]],
"xmin"->xmin,
"xmax"->xBC
|>
],
newassoc=Nothing (*wait for further functionality*)
];
];
newassoc
];
(*Derivative[1][PsiUp[s_Integer, l_Integer, \[Omega]_, rmin_, rmax_]][x_?InexactNumberQ] :=
Module[{},
];
Derivative[n_Integer?Positive][PsiUp[s_Integer, l_Integer, \[Omega]_, rmin_, rmax_]][r0_?InexactNumberQ] :=
Module[{d2R, pderivs, R, r, i},
d2R = -2/(x^2*(1-2/x))*Derivative[1][R][r] -(\[Omega]^2-ReggeWheelerPotential[s,l,x])/(1-2/x)^2*R[r];
pderivs = D[R[r_], {r_, i_}] :> D[d2R, {r, i - 2}] /; i >= 2;
Do[Derivative[i][R][r] = Simplify[D[Derivative[i - 1][R][r], r] /. pderivs];, {i, 2, n}];
Derivative[n][R][r] /. {
R'[r] -> PsiUp[s, l, \[Omega]]'[r0],
R[r] -> PsiUp[s, l, \[Omega]][r0], r -> r0}
];*)
(*should this be in a module for y1 and y2?*)
Integrator[s_,l_,\[Omega]_,y1BC_,y2BC_,xBC_,xend_,potential_,precision_]:=Module[{y1,y2,x},
NDSolve[
{y1'[x]==y2[x],(1-2/x)^2*y2'[x]+2(1-2/x)/x^2*y2[x]+(\[Omega]^2-potential[s,l,x])*y1[x]==0,y1[xBC]==y1BC,y2[xBC]==y2BC},
{y1,y2},
If[xBC<xend,{x,xBC,xend},{x,xend,xBC}],
Method->"StiffnessSwitching",
MaxSteps->Infinity,
WorkingPrecision->precision,
InterpolationOrder->All
]
]
(*boundary conditions for odd-parity Regge-Wheeler eqn.*)
(*currently only working for s=2*)
(*SetAttributes[ReggeWheelerInBC, {NumericFunction}];
SetAttributes[ReggeWheelerUpBC, {NumericFunction}];*)
ReggeWheelerInBC[s_Integer,l_Integer,\[Omega]_,workingprecision_]:=
Module[{rm2M,p,ptrys,expeh,Dexpeh,done=False,delReh,Reh,rstar,drstardr,
nmax,Xn,n,bk,denominator,f1=0,f2=0,f3=0,last,Bkm1=1,Bkm2=0,Bkm3=0,next,psi,dpsidr,om,precision=workingprecision+10,count},
rm2M=4*^-2*\[Omega]^2/(l*(l+1));
om=-\[Omega]; (* <-- this makes the sign of the exponential positive!*)
p=0;
ptrys=6;
nmax=10;
f1=0;
f2=0;
f3=0;
Bkm1=1;
Bkm2=0;
Bkm3=0;
expeh=1;
Dexpeh=0;
Xn=1;
last=1*^33;
count=0;
While[!done && p<ptrys,
delReh=rm2M;
Reh=2+rm2M;
p++;
bk=1;
For[n=1, n<=nmax, n++,
Xn*=delReh;
bk=-(((-3+n) \[Omega] Bkm3)/(2 n (I n+4\[Omega])))+(I (l+l^2-(-2+n) (-3+n-12 I \[Omega])) Bkm2)/(4 n (I n+4 \[Omega]))-((2-l-l^2+2 n^2+s^2+12 I \[Omega]+n (-5-12 I M \[Omega])) Bkm1)/(2 n (n-4 I \[Omega]));
Bkm3=Bkm2;
Bkm2=Bkm1;
Bkm1=bk;
next=bk*Xn;
If[(Abs[next]>Abs[last])&&(n>5),{count+=1,If[count>3,Break[]]}];
If[n>6 && (RealExponent[N[(expeh+next)-expeh,precision]])<-(precision-5),{done=True,Break[]}];
last=next;
expeh+=next;
Dexpeh+=n*next/delReh;
];
If[done==False,
f1=0;
f2=0;
f3=0;
Bkm1=1;
Bkm2=0;
Bkm3=0;
expeh=1;
Dexpeh=0;
Xn=1;
rm2M/=2;
last=1*^33
count=0;
];
];
rstar = Reh+2*Log[rm2M/(2)];
drstardr = Reh/rm2M;
psi=Exp[I*om*rstar]*expeh;
dpsidr=Exp[I*om*rstar]*Dexpeh+I*om*drstardr*psi;
<|"Psi"->N[psi,precision],"dPsidx"->N[dpsidr,precision],"xBC"->N[Reh,precision]|>
]
ReggeWheelerUpBC[s_Integer,l_Integer,\[Omega]_,xmax_,workingprecision_]:=
Module[{An=1,Anm1=0,Anm2=0,Anm3=0,Nmax=75,NNmax=1000,n,nn,rstart,rstar,drstardr,increment=0,
lastincrement=1*^40,S=0,lastS=0,dS=0,lastdS=0,count=0,r,rn,np,continue=True,precision=workingprecision+10,om,BCinc},
(*rstarstart=xmax+2*Log[xmax/2-1]+5*Pi/Abs[om];*)
rstart=xmax;
om=-\[Omega];
nn=1;
r=rstart;
rn=1;
BCinc=10;
While[continue&&nn<NNmax,
For[n=0,n<=Nmax,n++,
increment=An/rn;
If[(Abs[increment]>Abs[lastincrement])&&(n>5),{count+=1,If[count>3,Break[]]}];
lastS=S;
lastdS=dS;
S=S+increment;
dS=dS-n*increment/r;
If[n>4 && (RealExponent[N[S-lastS,precision]]<-(precision-5))&&(RealExponent[N[dS-(dS-n*increment/r),precision]]<-(precision-5)),{continue=False,Break[]}];
np=n+1;
rn=rn*r;
lastincrement=Abs[increment];
Anm3=Anm2;
Anm2=Anm1;
Anm1=An;
An=N[(I(-1+np-s) (-1+np+s) Anm2)/(np \[Omega])+(I (l+l^2+np-np^2) Anm1)/(2 np \[Omega]),precision];
];
If[continue==True,
rstart+=BCinc;
nn++;
r=rstart;
rn=1;
An=1;
Anm1=0;
Anm2=0;
Anm3=0;
increment=0;
lastincrement=1*^40;
S=0;
lastS=0;
dS=0;
lastdS=0;
count=0;
];
];
If[nn>=NNmax,{Print["The UP boundary condition loop ran out of steps!"],Abort[]}];
rstar=rstart+2*Log[rstart/2-1];
drstardr = rstart/(rstart-2);
<|"Psi"->N[S*Exp[-I*om*rstart],precision],
"dPsidx"->N[(-I*om*S*drstardr+dS)*Exp[-I*om*rstart],precision],
"xBC"->N[rstart,precision]|>
]
(*radial potentials for ODE*)
(*SetAttributes[ReggeWheelerPotential,{NumericFunction}];*)
SetAttributes[ZerilliPotential,{NumericFunction}];
ReggeWheelerPotential[s_Integer,l_Integer,x_]:=(1-2/x)*(2(1-s^2)+l*(l+1)*x)/x^3;
(*only for spin s=2*)
ZerilliPotential[l_Integer,x_Numeric]:=
Module[{n},
n=(l-1)*(l+2)/2;
2*(1-2/x)*(9+9*n*x+n^2*x^2*(3+x)+n^3*x^3)/(x^3(3+n*x)^2)
];
(*analytic solutions to the homogeneous static ReggeWheeler Eq. (see e.g. Field, Hesthaven, and Lau, PRD81, 124030 (2010)*)
SetAttributes[PsiUpStaticOdd,{NumericFunction}]
SetAttributes[PsiInStaticOdd,{NumericFunction}]
PsiUpStaticOdd[s_Integer,l_Integer,x_]:=(x/2)^(-l)*Hypergeometric2F1[l+s+1,l-s+1,2*(l+1),2/x];
PsiInStaticOdd[s_Integer,l_Integer,x_]:=(x/2)^(-l)*Hypergeometric2F1[l+s+1,l-s+1,1,(x-2)/x];
(*SetAttributes[PsiUpStaticEven,{NumericFunction}]
SetAttributes[PsiInStaticEven,{NumericFunction}]*)
(*even-parity static solutions for s=2 given by the intertwining operatior*)
PsiUpStaticEven[s_Integer,l_Integer,x_]:=
If[s==2,(*handle spin-2 case*)
Module[{n},
n=(l-1)*(l+2)/2;
2*(1-2/x)*Derivative[0,0,1][PsiUpStaticOdd][s,l,x]+(2*n*(n+1)/3+6*(x-2)/(x^2*(3+n*x)))*PsiUpStaticOdd[s,l,x]
]
,
Nothing
];
PsiInStaticEven[s_Integer,l_Integer,x_]:=
If[s==2,(*handle spin-2 case*)
Module[{n},
n=(l-1)*(l+2)/2;
2*(1-2/x)*Derivative[0,0,1][PsiInStaticOdd][s,l,x]+(2*n*(n+1)/3+6*(x-2)/(x^2*(3+n*x)))*PsiInStaticOdd[s,l,x]
]
,
Nothing
];
(*useful functions?*)
rfromrstar=Function[rstar,2 (1+ProductLog[Sqrt[E^(-2+rstar)]])];
End[]
EndPackage[];