How to fix 'NIntegrate: Overflow/Indeterminate/infinity for all sampling points in the region (...)"

Question

I have a function Nφ0 which depends on another two functions G1 and G2 that are being integrated numerically in a row as:

G1[r1_, r_, L_, φ0_] := 2 Sqrt[r^2 + r1^2 - 2 r r1 Cos[φ0]] - 2 Sqrt[r^2 + r1^2 - 2 r r1 Cos[φ0] + L^2] - L Log[1 + (2 L (L - Sqrt[r^2 + r1^2 - 2 r r1 Cos[φ0] + L^2]))/(r^2 + r1^2 - 2 r r1 Cos[φ0])]

and

G2[r1_, r_, L_] := 2 r r1 (-Sqrt[L^2 + (r - r1)^2] + Sqrt[(r - r1)^2] + L ArcTanh[L/Sqrt[L^2 + (r - r1)^2]])

so

Nφ0[a0_, b0_, φ0_, L_] := 1/(π (φ0 (b0^2 - a0^2) L)) NIntegrate[G2[r1, r, L] - G1[r1, r, L, φ0], {r, a0, b0}, {r1, a0, r,b0}, MinRecursion -> 9, MaxRecursion -> 40, Method -> {"GlobalAdaptive", "MaxErrorIncreases" -> 100000}]

My intention is to make a table for a certain range of values of Nφ0. I tried the following code (which has worked before for another functions) a3 = 8000;b3 = 15000

Nφ0vsLatφ0Dat = ParallelTable[{L, Nφ0[a3, b3, (10 π)/180, L 10^3], 
Nφ0[a3, b3, (30 π)/180, L 10^3], 
Nφ0[a3, b3, (60 π)/180, L 10^3], 
Nφ0[a3, b3, (90 π)/180, L 10^3], 
Nφ0[a3, b3, (120 π)/180, L 10^3], 
Nφ0[a3, b3, (150 π)/180, L 10^3], 
Nφ0[a3, b3, π, L 10^3], 
Nφ0[a3, b3, (210 π)/180, L 10^3], 
Nφ0[a3, b3, (240 π)/180, L 10^3]}, {L, 1, 100, 2}];

But I get the following error message and I don't know how to fix it

NIntegrate: The integrand (....) has evaluated to Overflow, Indeterminate, or Infinity for all \sampling points in the region with boundaries \{{0.96875`,1.`},{1.`,0.\99999991132826218264676919307505685921566573881591466488316655159`65.\954589770191}}.

Answer 1

Use LocalAdaptive instead of GlobalAdaptive. This allowes to zoom in to the points where you have problems.

The following works for me:

Nφ0[a0_, b0_, φ0_, L_] := 
 1/(π (φ0 (b0^2 - a0^2) L)) NIntegrate[
   G2[r1, r, L] - G1[r1, r, L, φ0], {r, a0, b0}, {r1, a0, r,
     b0}, MinRecursion -> 9, MaxRecursion -> 40, 
   Method -> {"LocalAdaptive"}];

Nφ0vsLatφ0Dat = 
  ParallelTable[{L, Nφ0[a3, b3, (10 π)/180, L 10^3], 
    Nφ0[a3, b3, (30 π)/180, L 10^3], 
    Nφ0[a3, b3, (60 π)/180, L 10^3], 
    Nφ0[a3, b3, (90 π)/180, L 10^3], 
    Nφ0[a3, b3, (120 π)/180, L 10^3], 
    Nφ0[a3, b3, (150 π)/180, L 10^3], 
    Nφ0[a3, b3, π, L 10^3], 
    Nφ0[a3, b3, (210 π)/180, L 10^3], 
    Nφ0[a3, b3, (240 π)/180, L 10^3]}, {L, 1, 100, 2}];