Do-loop with ParametricNDSolveValue not giving expected results

Question

I have written code that using a Do-loop. In the loop I am changing the value of x, v, l and R, and looking at the computed value of tot, which should equal Z. It does not matter what is the values of x, v, l or R are, tot should equal Z. However, the loop gives me a different value of tot. Can anyone help, please?

l1 = 0.81
Z = 500; 
x0 = 10; 
v0 = 0.02; 
ϵ = $MachineEpsilon;
l0 = 0.0714`20.;

ps = 
  ParametricNDSolveValue[
    {y''[r] + 2 y'[r]/r == -4 π l k Exp[-y[r]], 
     y[ϵ] == y0, y'[ϵ] == 0, 
     WhenEvent[r == 1, y'[r] -> y'[r] + Z l]}, 
    {y, y'}, {r, ϵ, R}, {k}, 
    Method -> "StiffnessSwitching", 
    WorkingPrecision -> 30];

Do[  
  x = i x0;
  v = i^3 v0; 
  R = Rationalize[v^(-1/3), 0];
  l = Rationalize[l1/(i x0), 0];
  nn = FindRoot[Last[ps[y0]][R], {y0, -10, 0}, Evaluated -> False][[1, 2]];
  tot = 4 π nn NIntegrate[r^2  Exp[-First[ps[nn]][r]], {r, 0, R}];
  Print[NumberForm[i, 5], "  ", NumberForm[tot, 10]];,
  {i, 2.92, 3.1, 0.01}]

Answer 1

I corrected the code so that in this problem all values should be calculated with a given accuracy. There is an overflow in intermediate calculations, but it apparently does not affect the final result.

l1 = 0.81;
Z = 500;
x0 = 10;
v0 = 0.02;
\[Epsilon] = $MachineEpsilon;

l0 = 0.0714`20.;

ps = ParametricNDSolveValue[{y''[r] + 
      2 y'[r]/r == -4 \[Pi] l k Exp[-y[r]], y[\[Epsilon]] == y0, 
    y'[\[Epsilon]] == 0, WhenEvent[r == 1, y'[r] -> y'[r] + Z l]}, {y,
     y'}, {r, \[Epsilon], R}, {k, l}, 
   Method -> {"StiffnessSwitching"}, AccuracyGoal -> 5, 
   PrecisionGoal -> 4, WorkingPrecision -> 15];

Do[x = i x0;
  v = i^3 v0;
  R = Rationalize[v^(-1/3), 0];
  l = Rationalize[l1/(i x0), 0];
  nn = FindRoot[Last[ps[y0, l]][R], {y0, -1}, Evaluated -> False][[1, 
    2]];
  Tot = 4 \[Pi] nn NIntegrate[
     r^2 Exp[-First[ps[nn, l]][r]], {r, \[Epsilon], R}, 
     PrecisionGoal -> 4];
  Print[NumberForm[i*1., 5], "  ", NumberForm[Tot, 5]];, {i, 292/100, 
   31/10, 1/100}] // Quiet

2.92 500.05

2.93 500.05

2.94 500.05

2.95 500.05

2.96 500.05

2.97 500.06

2.98 500.06

2.99 500.06

3. 500.06

3.01 500.06

3.02 500.06

3.03 500.06

3.04 500.06

3.05 500.06

3.06 500.06

3.07 500.06

3.08 500.06

3.09 500.05

3.1 500.05