I have two equations:

$$ \frac{2}{3}\thinspace\left(\frac{\varepsilon}{\Delta}\right)^{\frac{3}{2}}=\sum_{i=1,2} \sigma_i\lambda_{i}\left(1+\lambda_{i}^{2}\right)^{\frac{1}{4}}E\left(\frac{\pi}{2},k_{i}\right)+\sum_{i=1,2} \sigma_i \frac{(1+\lambda_i^2)^{1/4} }{ 2 ( \lambda_{i}+\sqrt{1+\lambda_{i}^{2} })} F(\frac{\pi}{2}, k_i ) $$


$$ \sqrt{\Delta}\sum_{i=1,2}\biggl[-\left(1+\lambda_{i}^{2}\right)^{\frac{1}{4}}E\left(\frac{\pi}{2},k_{i}\right)+\frac{F\left(\frac{\pi}{2},k_{i}\right)}{2\left(1+\lambda_{i}^{2}\right)^{\frac{1}{4}}}\left( \lambda_{i} + \frac{1}{\sqrt{1+\lambda_{i}^{2}}+\lambda_{i}} \right)\biggr]=\sum_{i=1,2}\biggl[-\left(1+\widetilde{\lambda}_{i}^{2}\right)^{\frac{1}{4}}E\left(\frac{\pi}{2},\widetilde{k}_{i}\right)+\frac{F\left(\frac{\pi}{2},\widetilde{k}_{i}\right)}{2\left(1+\widetilde{\lambda}_{i}^{2}\right)^{\frac{1}{4}}}\left(\widetilde{\lambda}_{i}+\frac{1}{\sqrt{1+\widetilde{\lambda}_{i}^{2}}+\widetilde{\lambda}_{i}} \right)\biggr] $$

where $\sigma_1=1$, $\sigma_2=-1$, $k_{i}^{2}=\frac{\sqrt{1+\lambda_{i}^{2}}+\lambda_{i}}{2\sqrt{\left(1+\lambda_{i}^{2}\right)}}$ for $i=1,2$ with $\lambda_{1}\equiv \frac{\mu-\Delta_{0}}{\Delta}$, $\lambda_{2}\equiv \frac{-\Delta_{0}-\mu}{\Delta} $, $\widetilde{k}_{i}^{2}=\frac{\sqrt{1+\widetilde{\lambda}_{i}^{2}}+\widetilde{\lambda}_{i}}{2\sqrt{\left(1+\widetilde{\lambda}_{i}^{2}\right)}}$ for $i=1,2$ with $\widetilde{\lambda}_{1}\equiv \mu$, $\widetilde{\lambda}_{2}\equiv -\mu $. $F(\frac{\pi}{2},k)$ and $E(\frac{\pi}{2},k)$ are the complete elliptic integral of the first and the second kind respectively which in Mathematica are EllipticK[] and EllipticE[]. In the code below $\mu$, $\Delta_0$, $\Delta$, $\varepsilon$ are represented by u, o, d, e.

By first converting those two equations in the form of differential equations and then using NDSolveValue to obtain plots of $\Delta$ and $\mu$ as a function of $\Delta_0$ at $\varepsilon=0.001$, I encounter the error:

lambda1[u_, o_, d_] := (u - o)/d;
lambda2[u_, o_, d_] := (-u - o)/d;
k1sq[u_, o_, d_] := (Sqrt[1 + ((lambda1[u, o, d])^2)] + lambda1[u, o, d])/(2*(Sqrt[1 + ((lambda1[u, o, d])^2)]));
k2sq[u_, o_, d_] := (Sqrt[1 + ((lambda2[u, o, d])^2)] + lambda2[u, o, d])/(2*(Sqrt[1 + ((lambda2[u, o, d])^2)]));
a1[u_, o_, d_] := (1 + (lambda1[u, o, d])^2)^(1/4);
a2[u_, o_, d_] := (1 + (lambda2[u, o, d])^2)^(1/4);
b1[u_, o_, d_] := Sqrt[1 + (lambda1[u, o, d])^2] + lambda1[u, o, d];
b2[u_, o_, d_] := Sqrt[1 + (lambda2[u, o, d])^2] + lambda2[u, o, d];

equat1[u_, o_, d_, e_] = 
(2/3)*(((e)/d)^(3/2)) == ((lambda1[u, o, d]*a1[u, o, d]*EllipticE[k1sq[u, o, d]]) 
-((lambda1[u, o, d]*EllipticK[k1sq[u, o, d]])/(2*a1[u, o, d]*b1[u, o, d]))+
((EllipticK[k1sq[u, o, d]])/(2*a1[u, o, d])) - 
(lambda2[u, o, d]*a2[u, o, d]* EllipticE[k2sq[u, o, d]]) 
+ ((lambda2[u, o, d]*EllipticK[k2sq[u, o, d]])/(2*a2[u, o, d]*b2[u, o, d]))-
((EllipticK[k2sq[u, o, d]])/(2*a2[u, o, d])));

equat2[u_, o_, d_] = 
Sqrt[d]*(((lambda1[u, o, d])*
      EllipticK[k1sq[u, o, d]]/(2*a1[u, o, d])) - (a1[u, o, d]*
       k1sq[u, o, d]]) + ((EllipticK[k1sq[u, o, d]])/(2*a1[u, o, d]*
        b1[u, o, d])) + ((lambda2[u, o, d])*
      EllipticK[k2sq[u, o, d]]/(2*a2[u, o, d])) - (a2[u, o, d]*
       k2sq[u, o, d]]) + ((EllipticK[k2sq[u, o, d]])/(2*a2[u, o, d]*
        b2[u, o, d]))) == ((lambda1[u, 0, 1])*
     EllipticK[k1sq[u, 0, 1]]/(2*a1[u, 0, 1])) - (a1[u, 0, 1]*
      k1sq[u, 0, 1]]) + ((EllipticK[k1sq[u, 0, 1]])/(2*a1[u, 0, 1]*
       b1[u, 0, 1])) + ((lambda2[u, 0, 1])*
     EllipticK[k2sq[u, 0, 1]]/(2*a2[u, 0, 1])) - (a2[u, 0, 1]*
      k2sq[u, 0, 1]]) + ((EllipticK[k2sq[u, 0, 1]])/(2*a2[u, 0, 1]*
       b2[u, 0, 1]));

{dk, uk} = {d, u} /. FindRoot[{equat2[u, o, d], equat1[u, o, d, .001]} /. o -> 0, {{d, .5}, {u, .1}}];

{dsolk, usolk} = 
NDSolveValue[{D[equat2[u, o, d] /. {d -> d[o], u -> u[o]}, o], 
D[equat1[u, o, d, .001] /. {d -> d[o], u -> u[o]}, o], 
u[0] == uk, d[0] == dk}, {d, u}, {o, 0, 1}, 
Method -> {"EquationSimplification" -> "Residual"}];

Plot[{dsolk[t], usolk[t]}, {t, 0, 1}, PlotRange -> {-.1, 1.1}, Frame -> True]

NDSolveValue::mxst: Maximum number of 100000 steps reached at the point o == 0.6097033678408679`.

enter image description here

I expect the blue curve ($\Delta$) to continue levelling off and the yellow curve ($\mu$) to continue increasing. Increasing MaxSteps->Infinity probably helps but the run-time took too long to evaluate and I gave up. Eventually, I will be obtaining these plots for different values of $\varepsilon$ and so increasing the run-time by increasing MaxSteps does not seem to be the way to go.

Using FindRoot directly gives somwhat the plot I want and the run-time is relatively shorter, but fails around the sharp turn at $\Delta_0=0.29$ and throws the error:

   FindRoot[{equat2[u, o, d], 
     equat1[u, o, d, .001]}, {{d, .5}, {u, .1}}], {o, 0, 1, .001}];
ListPlot[Transpose@%, DataRange -> {0, 1}, Joined -> True, 
 AxesLabel -> {o, "d, u"}, 
 LabelStyle -> Directive[Bold, Black, Medium], ImageSize -> Large]

FindRoot::lstol: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the merit function. You may need more than MachinePrecision digits of working precision to meet these tolerances.

enter image description here


I think the problem is a 0/0 term somewhere in the expression for equat2[] that arises at the time the integration begins to fail, from around 0.60 to 0.62. This causes numeric noise in computing u'[o], which leads to NDSolve[] thinking there's an error. A common factor can be eliminated for o -> 0 via Simplify. I let Simplify run for a several minutes on general o to no avail. So I resorted to Limit, which is slow. To resolve the issue, I needed higher WorkingPrecision, so I had to avoid Method -> {"EquationSimplification" -> "Residual"}, which can only be used with the machine-precision IDA method. I also replaced the parameter e value 0.001 with 1/1000 to avoid machine precision. I timed all the time-consuming computations.

{dk, uk} = {d, u} /. 
   FindRoot[{equat2[u, o, d], equat1[u, o, d, 1/1000]} /. 
     o -> 0, {{d, .5}, {u, .1}}, WorkingPrecision -> 100];

newODE = First@Solve[
      {D[equat2[u, o, d] /. {d -> d[o], u -> u[o]}, o], 
       D[equat1[u, o, d, 1/1000] /. {d -> d[o], u -> u[o]}, o]},
      {d'[o], u'[o]}
      ] /. Rule -> Equal; // AbsoluteTiming
(*  {7.57532, Null}  *)

uprime[o1_?NumericQ, d1_, u1_] := Block[{o},
     newODE[[2, 2]] /. {d[o] -> d1, u[o] -> u1, o -> o1},
     Limit[newODE[[2, 2]] /. {d[o] -> d1, u[o] -> u1}, o -> o1],
     {Power::infy, Infinity::indet}],
    {Power::infy, Infinity::indet}]

newODE2 = {First@newODE, u'[o] == uprime[o, d[o], u[o]]};

nlim = 0;
{dsolk, usolk} = 
   NDSolveValue[{newODE2, u[0] == uk, d[0] == dk}, {d, u}, {o, 0, 1}, 
    PrecisionGoal -> 8, AccuracyGoal -> 8, 
    WorkingPrecision -> 50]; // AbsoluteTiming
  {99.1584, Null}
  3  <-- number of times Limit[] was called

ListLinePlot@{dsolk, usolk}

enter image description here

In sum, equat2[] is numerically ill-conditioned in places. Perhaps simplification can help, or manual analysis, although the equations are too unwieldy for a quick look-see. However, I don't have any more time to look into it or the FindRoot problem, which is likely to be related, at this point.

  • $\begingroup$ Note also that the total number of steps according to dsolk@"Grid" // Length is only 264. $\endgroup$ – Michael E2 Jul 1 at 0:34

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