init

.gitignore

@@ -0,0 +1,4 @@
+.venv
+parfiles/
+*.csv
+*.log

.vscode/settings.json

@@ -0,0 +1,4 @@
+{
+	"editor.rulers": [80],
+	"editor.tabSize": 2
+}

Teuk_psi4.wl

@@ -0,0 +1,189 @@
+(*
+  This script is used to get a analytical formula of \psi_4,
+  and save to a file.
+*)
+
+(*
+	Usage: Please set the fowllowing variables from outside as
+    command line arguments, like this:
+		  wolframscript -file Teuk_psi4.wl m a L n in|out [filename]
+    for example:
+      wolframscript -file Teuk_psi4.wl 0 1e-8 1 1 out
+    if filename is not given, it's "./psi4_analytical.txt" by default.
+*)
+
+(* variable name : description *)
+
+(* Begin of variables *)
+(* m : the Teukolsky multiple order *)
+(* a : wave amplitude *)
+(* L : wave width *)
+(* n : the order N in F formula *)
+(* End of variables *)
+
+
+(* Parsing arguments *)
+
+argv = Rest@$ScriptCommandLine;
+Check[
+  {m, a, L, n} = Interpreter["Number"]/@argv[[;;4]];
+  type = argv[[5]];
+  If[Length@argv >= 6,
+    filename = argv[[6]],
+    filename = "./psi4_analytical.txt"
+  ],
+  Print["Error: Check your command line arguments!"];Abort[]
+];
+
+(* Set Assumptions *)
+$Assumptions = aa > 0 && r > 0 && Pi > theta > 0 &&
+   2 Pi >= phi >= 0 && t ~Element~ Reals &&
+   x ~Element~ Reals && y ~Element~ Reals && z ~Element~ Reals;
+(* Global Simplify time limit *)
+tc = 60
+
+(* Help functions *)
+pSimplify[list_] :=
+  ArrayReshape[
+    ParallelMap[
+      Simplify[#, TimeConstraint->tc]&, Flatten@list, Method->"FinestGrained"
+    ],
+    Dimensions@list
+];
+
+psi4analytical[g_, coordinates_, vecn_, vecmb_] :=
+  Module[
+    {dim = Length[coordinates], t1, t2,
+      t3, t4, ig, dg, gamma, dgamma, RiemR, psi4},
+    dg = D[g, {coordinates}];
+    ig = Inverse[g] // pSimplify;
+    t1 =
+      AbsoluteTiming[
+        gamma = pSimplify[
+          ParallelTable[
+            1/2 Plus @@ ((ig[[i, #]] (dg[[#, j, k]] + dg[[#, k, j]] -
+                    dg[[k, j, #]])) &) /@ Range[dim],
+            {i, 1, dim}, {j,1, dim}, {k, 1, dim}
+          ]
+        ];
+      ][[1]];
+    Print["Finish computing gamma, time cost:", t1, "s"];
+    t2 =
+      AbsoluteTiming[
+        dgamma = pSimplify[
+          D[gamma, {coordinates}]
+        ];
+      ][[1]];
+    Print["Finish computing dgamma, time cost:", t2, "s"];
+    t3 =
+      AbsoluteTiming[
+        RiemR = ParallelTable[
+            dgamma[[l, i, k, j]] - dgamma[[l, j, k, i]] +
+              Plus @@ ((gamma[[#, k, i]] gamma[[l, j, #]] -\
+                gamma[[#, k, j]] gamma[[l, i, #]]) &) /@ Range[dim],
+            {i, 1, dim}, {j, 1, dim}, {k, 1, dim}, {l, 1, dim}
+        ];
+      ][[1]];
+    Print["Finish computing RiemR, time cost:" , t3, "s"];
+    t4 =
+      AbsoluteTiming[
+        psi4 = ((((RiemR . g) . vecmb) . vecn) . vecmb) . vecn;
+      ][[1]];
+    Print["Finish computing psi4, time cost:" , t4, "s"];
+    psi4
+  ];
+
+(* set metric *)
+Switch[m,
+  0,
+    frr = 2 - 3 Sin[theta]^2;
+    frtheta = -3 Cos[theta] Sin[theta];
+    frphi = 0;
+    fthetatheta1 = 3 Sin[theta]^2;
+    fthetatheta2 = -1; fthetaphi = 0;
+    fphiphi1 = -fthetatheta1;
+    fphiphi2 = 3 Sin[theta]^2 - 1;,
+  1,
+    frr = Sin[2 theta] Cos[phi];
+    frtheta = Cos[2 theta] Cos[phi];
+    frphi = -Cos[theta] Sin[phi];
+    fthetatheta1 = -frr;
+    fthetatheta2 = 0;
+    fthetaphi = -Sin[theta] Sin[phi];
+    fphiphi1 = -fthetatheta1;
+    fphiphi2 = -frr;,
+  -1,
+    frr = Sin[2 theta] Sin[phi];
+    frtheta = Cos[2 theta] Sin[phi];
+    frphi = Cos[theta] Cos[phi];
+    fthetatheta1 = -frr;
+    fthetatheta2 = 0;
+    fthetaphi = Sin[theta] Cos[phi];
+    fphiphi1 = -fthetatheta1;
+    fphiphi2 = -frr;,
+  2,
+    frr = Sin[theta]^2 Cos[2 phi];
+    frtheta = Sin[theta] Cos[theta] Cos[2 phi];
+    frphi = -Sin[theta] Sin[2 phi];
+    fthetatheta1 = (1 + Cos[theta]^2) Cos[2 phi];
+    fthetatheta2 = -Cos[2 phi];
+    fthetaphi = Cos[theta] Sin[2 phi];
+    fphiphi1 = -fthetatheta1;
+    fphiphi2 = Cos[theta]^2 Cos[2 phi];,
+  -2,
+    frr = Sin[theta]^2 Sin[2 phi];
+    frtheta = Sin[theta] Cos[theta] Sin[2 phi];
+    frphi = Sin[theta] Cos[2 phi];
+    fthetatheta1 = (1 + Cos[theta]^2) Sin[2 phi];
+    fthetatheta2 = -Sin[2 phi];
+    fthetaphi = -Cos[theta] Cos[2 phi];
+    fphiphi1 = -fthetatheta1; fphiphi2 = Cos[theta]^2 Sin[2 phi];,
+  _,
+    Print["Error: m must be one of {0,1,2,-1,-2}."];Abort[]
+]
+
+h = aa {
+     {0, 0, 0, 0},
+     {0, AA[t, r] frr, BB[t, r] frtheta r, BB[t, r] frphi r Sin[theta]},
+     {0, BB[t, r] frtheta r,
+        (CC[t, r] fthetatheta1 + AA[t, r] fthetatheta2) r^2,
+        (AA[t, r] - 2 CC[t, r]) fthetaphi r^2 Sin[theta]},
+     {0, BB[t, r] frphi r Sin[theta],
+        (AA[t, r] - 2 CC[t, r]) fthetaphi r^2 Sin[theta],
+        (CC[t, r] fphiphi1 + AA [t, r] fphiphi2) r^2 Sin[theta]^2}
+    } // Simplify;
+g = DiagonalMatrix[{-1, 1, r^2, r^2 Sin[theta]^2}] + h // Simplify;
+
+rule1 = Switch[type,
+  "in",
+    {
+      AA -> Function[{t, r},3 (F''[t + r]/r^3 - (3 F'[t + r])/r^4 + (3 F[t + r])/r^5)],
+      BB -> Function[{t, r}, -(-(F'''[t + r]/r^2) + (3 F''[t + r])/r^3 - (6 F'[t + r])/r^4 + (6 F[t + r])/r^5)],
+      CC -> Function[{t, r}, 1/4 (F''''[t + r]/r - (2 F'''[t + r])/r^2 + (9 F''[t + r])/r^3 - (21 F'[t + r])/r^4 + (21 F[t + r])/r^5)]
+    },
+  "out",
+    {
+      AA -> Function[{t, r}, 3 (F''[t - r]/r^3 + (3 F'[t - r])/r^4 + (3 F[t - r])/r^5)],
+      BB -> Function[{t, r}, -(F'''[t - r]/r^2 + (3 F''[t - r])/r^3 + (6 F'[t - r])/r^4 + (6 F[t - r])/r^5)],
+      CC -> Function[{t, r}, 1/4 (F''''[t - r]/r + (2 F'''[t - r])/r^2 + (9 F''[t - r])/r^3 + (21 F'[t - r])/r^4 + (21 F[t - r])/r^5)]
+    },
+  _,
+    Print["Error: check wave type, can only be in or out"];Abort[]
+]
+
+rule2 = F -> Function[L^5 (#/L)^n E^(-(#^2/L^2))]
+
+(* set tetrad *)
+vecn = 1/2 {1, -1, 0, 0};
+vecmb = 1/(Sqrt[2] r) {0, 0, 1, -(I/Sin[theta])};
+
+(* Calculate psi4 *)
+psi4 = psi4analytical[g, {t, r, theta, phi}, vecn, vecmb];
+psi4 = psi4 /. {aa->a};
+psi4 = psi4 /. rule1;
+psi4 = psi4 /. rule2;
+
+psi4 = psi4 /. {t->0};
+(* psi4 = Simplify[psi4, TimeConstraint->tc]; *)
+
+Put[psi4, filename]

calculate_modes.wl

@@ -0,0 +1,61 @@
+(*
+  Usage: Please set the fowllowing variables from outside as
+  command line arguments, like this:
+    wolframscript -file calculate_modes.wl psi4_file r rfilename ifilename
+  for example:
+    wolframscript -file calculate_modes.wl psi4.txt 2.5 rmodes.txt imodes.txt
+*)
+
+(* Parsing arguments *)
+argv = Rest@$ScriptCommandLine;
+(* Print[argv] *)
+Check[
+  psi4file = argv[[1]];
+  (* {l, m, r} = Interpreter["Number"]/@argv[[2;;4]]; *)
+	r = Interpreter["Number"]@argv[[2]];
+	rfilename = argv[[3]];
+	ifilename = argv[[4]];
+,
+  Print["Error: Check your command line arguments!"];Abort[]
+];
+
+(* Set Assumptions *)
+$Assumptions = Pi > theta > 0 && 2 Pi >= phi >= 0 &&\
+                Element[l, Reals] && Element[m, Reals] && -l<=m<=l;
+
+Y[s_, l_, m_, theta_, phi_] :=
+    (-1)^m*
+      (Sqrt[(((l + m)!*(l - m)!)*(2*l + 1))/((((l + s)!*(l - s)!)*4)*Pi)]*
+        Sin[theta/2]^(2*l))*
+        Sum[
+          (((Binomial[l - s, r]*Binomial[l + s, r + s - m])*(-1)^(l - r - s))*
+          E^((I*m)*phi))*Cot[theta/2]^(2*r + s - m),
+          {r, 0, l - s}
+        ]
+
+(* Y[l_, m_, theta_, phi_] := ((-1)^(l + m) E^(I m phi)
+  Binomial[-2 + l, -2 - m] Cot[theta/2]^(-2 -m)
+  Sqrt[((1 + 2 l) (l - m)! (l + m)!)/((-2 + l)! (2 + l)!)]
+  Hypergeometric2F1[-2 - l, -l - m, -1 - m,
+          -Cot[theta/2]^2] Sin[theta/2]^(2 l))/(2 Sqrt[Pi]) *)
+
+psi4[theta_, phi_] = ToExpression[Import[psi4file, "Text"] ]/.{t->0};
+
+mode[l_, m_] := Module[{f},
+  (* f = Simplify[Conjugate[Y[-2, l,m,theta,phi] ]*psi4[r,theta,phi]*Sin[theta] ]; *)
+  f = Conjugate[Y[-2, l,m,theta,phi] ]*psi4[theta,phi]*Sin[theta];
+  Quiet[NIntegrate[f,{theta, 0, Pi}, {phi, 0, 2 Pi}] ]
+]
+
+(* mode[0,0,2.2]//Print *)
+
+(* result = ParallelTable[mode[l,m],{l,1,8},{m,-l,l}]//Flatten; *)
+lmlist = Flatten[Table[{l, m}, {l, 1, 8}, {m, -l, l}], 1]
+result = ParallelMap[mode@@#&, lmlist]
+rmodes = Re/@result;
+imodes = Im/@result;
+Export[rfilename, rmodes, "Table"]
+Export[ifilename, imodes, "Table"]
+
+(* result = mode[l,m]
+Print[Re@result, " ", Im@result] *)

modes_wolfram/imode_a1e-06-m0-N1-l0.5-r0.4.txt

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modes_wolfram/imode_a1e-06-m0-N1-l0.5-r1.3.txt

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modes_wolfram/imode_a1e-06-m0-N1-l0.5-r1.8.txt

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modes_wolfram/imode_a1e-06-m0-N1-l1-r0.4.txt

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modes_wolfram/rmode_a1e-06-m1-N1-l0.5-r0.4.txt

@@ -0,0 +1,80 @@
+0.
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modes_wolfram/rmode_a1e-06-m1-N1-l0.5-r1.3.txt

@@ -0,0 +1,80 @@
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modes_wolfram/rmode_a1e-06-m1-N1-l0.5-r1.8.txt

@@ -0,0 +1,80 @@
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psi4_value.wl

@@ -0,0 +1,22 @@
+(*
+  Usage: Please set the fowllowing variables from outside as
+  command line arguments, like this:
+    wolframscript -file psi4_value.wl psi4_file x y z
+  for example:
+    wolframscript -file psi4_value.wl psi4.txt 1.2 2.1 3.4
+*)
+
+(* Parsing arguments *)
+argv = Rest@$ScriptCommandLine;
+(* Print[argv] *)
+Check[
+  psi4file = argv[[1]];
+	{x, y, z} = ToExpression[argv[[2]]];
+,
+  Print["Error: Check your command line arguments!"];Abort[]
+];
+
+psi4[r_, theta_, phi_] = ToExpression[Import[psi4file, "Text"] ]/.{t->0};
+
+result = psi4@@ToSphericalCoordinates[{x,y,z}]
+Print[Re@result, " ", Im@result]

test_Inv_modes.py

@@ -0,0 +1,304 @@
+import os
+import subprocess
+import csv
+import itertools
+import numpy as np
+
+from datetime import datetime
+
+# from wolframclient.evaluation import WolframLanguageSession
+# from wolframclient.language import wl, wlexpr
+
+nmesh_exe = "/home/wyj/Code/NR/nmesh/exe/nmesh"
+
+def timestr():
+  now = datetime.now()
+  return now.strftime('%y-%m-%d %H:%M:%S')
+
+class nmesh_tester:
+  def __init__(self, a=1e-8, m=0, N=1, l=1, type="out", ntheta=16, nphi=32,
+               interp="yes", n=20, luni=0, rlist="", rmodes=[], imodes=[],
+               rpsi4s=[], ipsi4s=[], mode_file="mode.csv", psi4_file="psi4.csv",
+               l2_file="l2.csv",
+               outlist="0.05 0.08 0.1 -0.4 0.42 0.33 1.4 0.2 0.35"):
+    self.a = a
+    self.m = m
+    self.N = N
+    self.l = l
+    self.type = type
+    self.ntheta = ntheta
+    self.nphi = nphi
+    self.interp = interp
+    self.n = n
+    self.luni = luni
+    self.rlist = rlist
+    self.rarray = list(map(float, rlist.split()))
+    self.outlist = outlist
+    self.outarray = list(map(float, outlist.split()))
+    self.rmodes = rmodes
+    self.imodes = imodes
+    self.rpsi4s = rpsi4s
+    self.ipsi4s = ipsi4s
+    self.mode_file = mode_file
+    self.psi4_file = psi4_file
+    self.l2_file = l2_file
+    if max(max(self.rarray),max(self.outarray)) < 2.0:
+      self.mesh_type = "CubedSpheres"
+    else:
+      self.mesh_type = "Shell"
+    print(f"[{timestr()}]Start a now tester.")
+  def run_test(self):
+    parfile = self.write_parfile()
+    basename = parfile[:-4]
+    print(f"[{timestr()}]Written to parfile {parfile}")
+    print(f"[{timestr()}]Starting running nmesh")
+    if os.path.exists(basename):
+      print(f"[{timestr()}]\tHave already runed {parfile}, skipping.")
+    else:
+      print(f"[{timestr()}]running: {nmesh_exe} {parfile}")
+      self.run_nmesh(parfile)
+      print(f"[{timestr()}]Finished running nmesh")
+    self.collect_result(basename)
+    print(f"[{timestr()}]Finished result analysis.")
+  def write_parfile(self):
+    filename = f"parfiles/Inv-a{self.a}-m{self.m}-N{self.N}-l{self.l}-{self.type}-"+\
+         f"nt{self.ntheta}-np{self.nphi}-{self.interp}-"+\
+               f"n{self.n}-{self.luni}-{self.rlist}.par"
+    preparfile =\
+        '''###################################################################
+# modules we use
+
+physics = Invariants ADM Teukolsky
+
+###################################################################
+# parameters for ADM
+
+ADM_Set1stDerivsInADMconstraints = yes
+
+'''
+    if self.mesh_type == "Shell":
+      mesh_parfile =\
+        f'''###################################################################
+# initial mesh
+
+amr_mesh_type = Shell
+amr_Shell_rin = {np.floor(min(self.rarray+self.outarray))}
+amr_Shell_rout = {np.ceil(max(self.rarray+self.outarray))}
+amr_Shell_r1 = 0
+amr_n0 = {self.n}
+amr_n1 = {self.n}
+amr_n2 = {self.n}
+amr_luni = {self.luni}
+
+'''
+    else:
+      mesh_parfile =\
+        f'''###################################################################
+# initial mesh
+
+amr_mesh_type = 7 CubedSpheres
+amr_n0 = {self.n}
+amr_n1 = {self.n}
+amr_n2 = {self.n}
+amr_luni = {self.luni}
+
+'''
+    teukolsky_parfile =\
+         f'''###################################################################
+# parameters for Teukolsky
+
+Teukolsky_waves_amplitude = {self.a}
+Teukolsky_wave_width = {self.l}
+Teukolsky_wave_mm = {self.m}
+Teukolsky_wave_N = {self.N}
+Teukolsky_wave_type = {self.type}
+
+'''
+    invariants_parfile =\
+         f'''###################################################################
+# parameters for Invariants
+
+Invariants_ntheta = {self.ntheta}
+Invariants_nphi = {self.nphi}
+Invariants_modes_lmin = 1
+Invariants_modes_lmax = 8
+Invariants_modes_test = no
+Invariants_compute_modes = yes
+Invariants_compute_modes_just_interp = {self.interp}
+Invariants_modes_r = {self.rlist}
+Invariants_psi4_test_0doutput = yes
+Invariants_psi4_test_0dout_coordinates = {self.outlist}
+
+'''
+    output_parfile =\
+         '''###################################################################
+# output
+
+logfile_creation = yes
+0douttime = 0.1
+0doutput = Invariants_rpsi4 Invariants_ipsi4
+
+1douttime = 0.1
+1doutput = Invariants_rpsi4 Invariants_ipsi4
+'''
+    with open(filename, "w") as f:
+      f.write(preparfile)
+      f.write(mesh_parfile)
+      f.write(teukolsky_parfile)
+      f.write(invariants_parfile)
+      f.write(output_parfile)
+    return filename
+  def run_nmesh(self, parfile):
+    subprocess.run([nmesh_exe, parfile])
+  def collect_result(self, basename):
+    print(f"[{timestr()}]Starting collect and analysis result...")
+    if not os.path.exists(self.mode_file):
+      with open(self.mode_file, "w") as f:
+        csv_writer = csv.writer(f)
+        csv_writer.writerow(["a", "m", "N", "L", "type", "nt", "np", "interp",
+                             "mesh", "n", "luni", "r", "l", "m", "re_mode",
+                                "im_mode", "re_mode_err", "im_mode_err",
+                                  "re_mode_rel_error", "im_mode_rel_err"])
+    if not os.path.exists(self.psi4_file):
+      with open(self.psi4_file, "w") as f:
+        csv_writer = csv.writer(f)
+        csv_writer.writerow(["a", "m", "N", "L", "type", "mesh", "n", "luni",
+                             "x", "y", "z", "rpsi4", "ipsi4", "rpsi4_err",
+                             "ipsi4_err", "rpsi4_rel_err", "ipsi4_rel_err"])
+    if not os.path.exists(self.l2_file):
+      with open(self.l2_file, "w") as f:
+        csv_writer = csv.writer(f)
+        csv_writer.writerow(["a", "m", "N", "L", "type", "nt", "np", "interp",
+                             "mesh", "n", "luni", "r", "l2-norm",
+                             "l2-distance", "l2-rel_err"])
+
+    print(f"[{timestr()}]\tAnalysing modes...")
+    with open(self.mode_file, "a") as mode_fp:
+      csv_writer = csv.writer(mode_fp)
+      ind = 0
+      for r in self.rarray:
+        l2norm = np.dot(self.rmodes, self.rmodes)+\
+          np.dot(self.imodes, self.imodes)
+        l2distance = 0
+        for l in range(1,9):
+          for m in range(-l, l+1):
+            output_line = [self.a, self.m, self.N, self.l, self.type,\
+                           self.ntheta, self.nphi, self.interp, self.mesh_type,\
+                                          self.n, self.luni, r, l, m]
+            mode_file = f'{basename}/modes/psi4_r{r}_l{l}_m{m}.t'
+            with open(mode_file, 'r') as f:
+                lines = f.readlines()
+                output_line.extend(map(float,lines[-1].split()[1:]))
+            rmode = self.rmodes[ind]
+            imode = self.imodes[ind]
+            rmode_error = output_line[-2] - rmode
+            imode_error = output_line[-1] - imode
+            rmode_rel_error = 'nan' if rmode==0 else rmode_error/rmode
+            imode_rel_error = 'nan' if imode==0 else imode_error/imode
+            l2distance += rmode_error**2 + imode_error**2
+            output_line.extend([rmode_error, imode_error,
+                                rmode_rel_error, imode_rel_error])
+            ind += 1
+            csv_writer.writerow(output_line)
+        with open(self.l2_file, "a") as l2_fp:
+          l2_writer = csv.writer(l2_fp)
+          l2_writer.writerow([self.a, self.m, self.N, self.l, self.type,
+                              self.ntheta, self.nphi, self.interp,
+                              self.mesh_type, self.n, self.luni, r, l2norm,
+                              l2distance, l2distance/l2norm])
+
+    print(f"[{timestr()}]\tAnalysing 0doutput...")
+    with open(self.psi4_file, "a") as psi4_fp:
+      csv_writer = csv.writer(psi4_fp)
+      n_points = len(self.outarray)//3
+      for i in range(n_points):
+        output_line = [self.a, self.m, self.N, self.l, self.type,\
+                        self.mesh_type, self.n, self.luni]
+        psi4_files = [f"{basename}/Invariants_rpsi4_pt{i}.t",
+                      f"{basename}/Invariants_ipsi4_pt{i}.t"]
+        with open(psi4_files[0]) as f:
+          line = f.readline()
+          line_list = line.split()
+          Nrpsi4 = float(line_list[1])
+          output_line.extend(line_list[-3:])
+          output_line.append(Nrpsi4)
+        with open(psi4_files[1]) as f:
+          line = f.readline()
+          Nipsi4 = float(line.split()[1])
+        output_line.append(Nipsi4)
+        rpsi4 = self.rpsi4s[i]
+        ipsi4 = self.ipsi4s[i]
+        rpsi4_error = Nrpsi4 - rpsi4
+        ipsi4_error = Nipsi4 - ipsi4
+        rpsi4_rel_error = 'nan' if rpsi4==0 else rpsi4_error/rpsi4
+        ipsi4_rel_error = 'nan' if ipsi4==0 else ipsi4_error/ipsi4
+        output_line.extend([rpsi4_error, ipsi4_error,
+                            rpsi4_rel_error, ipsi4_rel_error])
+        csv_writer.writerow(output_line)
+
+rlist = [0.4, 1.3, 1.8]
+pointlist = [[0.05, 0.08, 0.1], [-0.4, 0.42, 0.33], [1.4, 0.2, 0.35]]#, [0.01, 0.02, 0.48]]
+
+rlist_str = ' '.join(str(r) for r in rlist)
+outlist = ' '.join(str(coord) for point in pointlist for coord in point)
+
+mode_file = "result_modes.csv"
+psi4_file = "result_psi4.csv"
+l2_file = "result_l2.csv"
+
+for a in [1e-6, 1e-8, 1e-10, 1e-12]:
+  for m in [0, 1, -1, 2, -2]:
+    for N in [1, 2, 3, 4]:
+      for l in [0.5, 1, 2]:
+        for type in ["in", "out"]:
+          psi4_analytical = f"psi4_analytical/psi4-m{m}-a{a}-L{l}-N{N}.txt"
+          if not os.path.exists(psi4_analytical):
+            print(f"[{timestr()}]Staring calculate psi4 analytically...")
+            subprocess.run(["wolframscript", "-file", "Teuk_psi4.wl", f"{m}",
+                            f"{a}", f"{l}", f"{N}", f"{type}",
+                            f"{psi4_analytical}"])
+            print(f"[{timestr()}]Finished calculate psi4 and saved in {psi4_analytical}.")
+          rmodes = []
+          imodes = []
+          for r in rlist:
+            rmodes_file = f"modes_wolfram/rmode_a{a}-m{m}-N{N}-l{l}-r{r}.txt"
+            imodes_file = f"modes_wolfram/imode_a{a}-m{m}-N{N}-l{l}-r{r}.txt"
+            if (not os.path.exists(rmodes_file)) or \
+              (not os.path.exists(imodes_file)):
+              print(f"[{timestr()}]Starting calculate modes at r={r}...")
+              subprocess.run(["wolframscript", "-file", "calculate_modes.wl",
+                              f"{psi4_analytical}", f"{r}",
+                              f"{rmodes_file}", f"{imodes_file}"])
+              print(f"[{timestr()}]Finished calculate modes and saved in \
+                    {rmodes_file} and {imodes_file}.")
+            with open(rmodes_file) as f:
+              rmodes.extend([float(line.strip()) for line in f.readlines()])
+            with open(imodes_file) as f:
+              imodes.extend([float(line.strip()) for line in f.readlines()])
+          rpsi4s = []
+          ipsi4s = []
+          for point in pointlist:
+            print(f"[{timestr()}]Starting calculate psi4 at {point}")
+            result = subprocess.run(["wolframscript", "-file",
+                                      "psi4_value.wl", f"{psi4_analytical}",
+                                      "{"+
+                                      f"{point[0]}, {point[1]}, {point[2]}"+
+                                        "}"],
+                                      capture_output=True, text=True)
+            try:
+              output = result.stdout.strip().split()
+              rpsi4s.append(float(output[0].replace("*^", "e")))
+              ipsi4s.append(float(output[1].replace("*^", "e")))
+            except:
+              print(result, output)
+              raise SystemExit("wrong in wolframscript.")
+          for ntheta in [16, 32, 64]:
+            nphi = 2*ntheta
+            for interp in ["yes"]:
+              for n in [10, 20, 40]:
+                for luni in [0, 1, 2]:
+                  tester = nmesh_tester(a, m, N, l, type, ntheta, nphi,
+                                        interp, n, luni, rlist_str, rmodes,
+                                        imodes, rpsi4s, ipsi4s, mode_file,
+                                        psi4_file, l2_file, outlist)
+                  tester.run_test()