physics – How to solve this complicated differential equation


I am working on electromagnetic perturbations of regular black holes by solving the Regge-Wheeler equations. I have to solve a dynamic equation with the initial conditions that $dfrac{dA_3}{dt}Big|_{(0,t)}=0$ and $A_3(0,t)=S(t)$ where $S(t)$ is the solution to the static problem. Here is my code:

First some definitions

g = 1;
M = 1;
m(r_) := M r^3/(r^3 + 2 M g^2);
J(r_) := 10^-14/r^2;
mt(t_) := m((t + 1)/(1 - t))
Jt(t_) := J((t + 1)/(1 - t))

The solution of the static problem

sol = ParametricNDSolveValue({-(((t + 1)/(1 - t) - 2 mt(t))^2/((
         t + 1)/(1 - t))^2) 1/2 (t - 1)^3 D(As(t), {t, 2}) + 
     1/2 (t - 1)^2 D(As(t), 
       t) (-7 g^4 (t + 1)/(1 - t) + 2^(1/4) g ((t + 1)/(1 - t))^4 + 
        3 mt(t) (4 g^4 - 2^(1/4) g ((t + 1)/(1 - t))^3) + (t + 1)/(
         1 - t) (2 g^4 + 2^(1/4) g ((t + 1)/(1 - t))^3) ((*1/
          2(t-1)^2D(mt(t),t)*)mt'(t))) (
      2^(3/4) ((t + 1)/(1 - t) - 2 mt(t)))/(
      2^(3/4) g^4 ((t + 1)/(1 - t))^3 + g ((t + 1)/(1 - t))^6) + (
      9 l (l + 1) (g/((t + 1)/(1 - t)))^3 ((t + 1)/(1 - t) - 
         2 mt(t)))/((2^(1/4) + 2 g^3/((t + 1)/(1 - t))^3) ((t + 1)/(
         1 - t))^3) As(t) == 4 Pi Jt(t), As(0.4) == 0, 
   As(0.99999) == 0}, As, {t, 0.4, 0.99999}, l, 
  WorkingPrecision -> 30)

The dynamic problem

din = ParametricNDSolveValue({D(
      A3(T, t), {T, 
       2}) - ((t + 1)/(1 - t) - 2 mt(t))^2/((t + 1)/(1 - t))^2 1/
      2 (t - 1)^3 D(A3(T, t), {t, 2}) + 
     1/2 (t - 1)^2 D(A3(T, t), 
       t) (-7 g^2 (t + 1)/(1 - t) + 2^(1/4) g ((t + 1)/(1 - t))^4 + 
        3 mt(t) (4 g^2 - 2^(1/4) g ((t + 1)/(1 - t))^3) + (t + 1)/(
         1 - t) (2 g^4 + 2^(1/4) g ((t + 1)/(1 - t))^3) mt'(t)) (
      2^(3/4) ((t + 1)/(1 - t) - 2 mt(t)))/(
      2^(3/4) g^4 ((t + 1)/(1 - t))^3 + g ((t + 1)/(1 - t))^6) + (
      9 l (l + 1) (g/((t + 1)/(1 - t)))^3 ((t + 1)/(1 - t) - 
         2 mt(t)))/((2^(1/4) + 2 g^3/((t + 1)/(1 - t))^3) ((t + 1)/(
         1 - t))^3) A3(T, t) == 4 Pi Jt(t), 
   Derivative(1, 0)(A3)(0, t) == 0, A3(0, t) == sol(l)(t), 
   A3(T, 0.4) == 0, A3(T, 0.99999) == 0}, 
  A3, {T, 0, 10}, {t, 0.4, 0.99999}, l, 
  Method -> {"MethodOfLines", 
    "SpatialDiscretization" -> {"TensorProductGrid", 
      "MaxPoints" -> 50, "MinPoints" -> 50}})

I could get the static problem to work, but the dynamic keeps giving me problems: with the above code I get the warning:

Warning: scaled local spatial error estimate of 54549.16088776184 at T = 10. in the direction of independent variable t is much greater
than the prescribed error tolerance. Grid spacing with 50 points may
be too large to achieve the desired accuracy or precision. A
singularity may have formed or a smaller grid spacing can be
specified using the MaxStepSize or MinPoints method options

I have tried to variate the Minpoints and MaxPoints options and I have also used MaxStepSize, but I get different results every time: where a the solution grows with a set of parameters, it decreases with others and the warning never goes away.

How can I solve this problem?