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I am trying to solve the nonlinear PDE, recently. The formulas are given as

enter image description here

with the conditions

enter image description here

I write the numerical code to solve it. The code is written as

sys = {b*(u[r, t] - m[r, t]) + 
     D[u[r, t], {t}] == (-D[u[r, t], {r}])^((1 - n)/n)*
           ((kd^(1/n)*D[u[r, t], {r}, {r}])/
        n + (kd^(1/n)*D[u[r, t], {r}])/r), 
       l*D[m[r, t], {t}] == b*(u[r, t] - m[r, t]), 
   Derivative[1, 0][u][rw, t] == -(2^n/(kd*rw^n)), u[R, t] == e, 
       u[r, e] == e, m[r, e] == e}; 
sol = NDSolve[sys, {u, m}, {r, rw, R}, {t, e, T}, 
  MaxStepSize -> {0.01, 1}, 
     Method -> {"MethodOfLines", 
    "DifferentiateBoundaryConditions" -> {True, "ScaleFactor" -> T}}]

To verify the numerical results, the semianalytical solution is used. When n =1, the equations are linear. The code is

Vi[n_, i_] := 
 Vi[n, i] = (-1)^(i + n/2) Sum[ 
     k^(n/2) (2 k)! /( (n/2 - k)! k! (k - 1)! (i - k)! (2 k - 
            i)! ), { k, Floor[ (i + 1)/2 ], Min[ i, n/2] } ] // N; 
Stehfest[F_, s_, t_, n_: 16] :=
 If[n > 16, Message[Stehfest::optimalterms, n];
        If[ OddQ[n], Message[Stehfest::odd, n];
                "Enter an even number of terms",
                If[n > 32, Message[Stehfest::terms, n];
                    " Try a smaller value for n. Maximum allowable n is 32 ",
                    Log[2]/t Sum[ Vi[n, i]*F /. s -> i Log[2]/t , {i, 1, n} ] ]],
        If[ OddQ[n], Message[Stehfest::odd, n];
                "Enter an even number of terms", 
    If[n > 32, Message[Stehfest::terms, n];
                    " Try a smaller value for n. Maximum allowable n is 32.",
                    Log[2]/t Sum[ 
       Vi[n, i]*F /. s -> i Log[2]/t , {i, 1, n} ] ]]]  // N; 
s[r_, t_] := 
 Stehfest[(2^(2 + n) r^(1/2 - n/2)
      rw BesselK[(-1 + n)/(-3 + n), -((
       r^(3/2 - n/2) Sqrt[(2^(1 + n) n p (b + p l + b l))/(
        kd (b + p l))])/(-3 + n))])/(kd p (rw^2 Sqrt[(
        2^(1 + n) n p (b + p l + b l))/(kd (b + p l))]
         BesselK[
         2/(-3 + n), -((
          rw^(3/2 - n/2) Sqrt[(2^(1 + n) n p (b + p l + b l))/(
           kd (b + p l))])/(-3 + n))] + 
       rw^2 Sqrt[(2^(1 + n) n p (b + p l + b l))/(kd (b + p l))]
         BesselK[(
         2 (-2 + n))/(-3 + n), -((
          rw^(3/2 - n/2) Sqrt[(2^(1 + n) n p (b + p l + b l))/(
           kd (b + p l))])/(-3 + n))] + 
       2 (-1 + n) rw^((1 + n)/2)
         BesselK[(-1 + n)/(-3 + n), -((
          rw^(3/2 - n/2) Sqrt[(2^(1 + n) n p (b + p l + b l))/(
           kd (b + p l))])/(-3 + n))])), p, t, 16]

I compared the results and found that the curves predicted by numerical solution and semianalytical solution are close to each other. It represents the MaxStepSize is valid.

The parameters is given as

kd = 10; 
n = 1; 
T = 1000; 
rw = 0.025; 
R = rw + Sqrt[(Pi*T)/1.4]; 
b = 0.5; 
l = 50; 
e = 5./10^10; 

With the result

S0 = {#, s[5/10, #]} & /@ Exp@N@Range[-2, Round@Log@T, 1/4];
ap1 = ListLogLogPlot[S0]; np1 = 
 LogLogPlot[u[5/10, t] /. sol, {t, 0.1, T}]; Show[ap1, np1, 
 PlotRange -> All]

enter image description here

However, as n is larger than 1 (e.g., n = 1.1, 1.5, and 2), the procedure of the solving the nonlinear equation is stuck. Is there anything wrong in my numerical code? How can I fix it? Thank you.

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  • $\begingroup$ The code contains the undefined quantity \[Epsilon], and api is an empty plot. $\endgroup$ – bbgodfrey May 6 '18 at 14:40
  • $\begingroup$ Sorry, [Epsilon] should be replaced by e. the semianalytical solution should start the kernel thus api would display the result. $\endgroup$ – LingLong May 7 '18 at 3:53
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The computation has two difficulties. First, there is an inconsistency between initial and boundary conditions at r == t == 0, and second, (-D[u[r, t], r])^((1 - n)/n) sometimes is very large or yields complex numbers for n > 1. With sys as in the question, and T == 10 (to save computational time and focus on small t) in

kd = 10; n = 1; T = 10; rw = 0.025; R = rw + Sqrt[(Pi*T)/1.4]; b = 0.5; l = 50; 
    e = 5./10^10;

yields an apparently well behaved solution, although there are some slight irregularities visible at very small t.

sol = NDSolve[sys, {u, m}, {r, rw, R}, {t, e, T}, MaxStepSize -> {0.01, 1}, 
    Method -> {"MethodOfLines",  "DifferentiateBoundaryConditions" -> 
    {True, "ScaleFactor" -> T}}]
Plot3D[u[r, t] /. sol, {r, rw, R}, {t, e, T}, AxesLabel -> {r, t, u}, 
    LabelStyle -> Directive[Black, Bold, Medium], ImageSize -> Large, PlotRange -> All]

enter image description here

despite the warning messages,

NDSolve::ibcinc: Warning: boundary and initial conditions are inconsistent.

NDSolve::eerr: Warning: scaled local spatial error estimate of 45.650 at t = 10. in the direction of independent variable r is much greater than the prescribed error tolerance...

However, when n is increased to 1.5 (but all other parameters the same), problems become obvious at early times.

Plot3D[Im[u[r, t] /. sol], {r, rw, R}, {t, e, .4}, AxesLabel -> {r, t, u}, 
    LabelStyle -> Directive[Black, Bold, Medium], ImageSize -> Large, 
    PlotRange -> All]

enter image description here

Even though the imaginary part of u is small compared to the real part, NDSolve takes many minutes and produces about 4 MB of data to resolve the complicated structure shown and compute as far as T == 10. I attempted to repeat this calculation for n == 2 (and e == 5 10^-3), but NDSolve required a few hours and 10000 time steps just to get to t == 0.01. Almost 5000000 mesh points were involved. Even 3D plots of this data are very slow. Here are two 1D plots.

Plot[Evaluate@ReIm[u[5/10, t] /. sol], {t, e, .01}, AxesLabel -> {t, u}, 
    LabelStyle -> Directive[Black, Bold, Medium], ImageSize -> Large]

enter image description here

Plot[Evaluate@ReIm[u[r, .01] /. sol], {r, rw, R}, AxesLabel -> {r, u},
    LabelStyle -> Directive[Black, Bold, Medium], ImageSize -> Large, 
    PlotRange -> All]

enter image description here

I would offer two suggestions. First, resolve the inconsistency at r == t == 0. The precise solution depends on the details of the physical situation to be modeled. Second, something needs to be done about (-D[u[r, t], r])^((1 - n)/n). Replacing (-D[u[r, t], r]) by Abs[D[u[r, t], r]] would eliminate complex numbers in the solution but would not eliminate potential singularities as D[u[r, t], r] approaches zero. Perhaps, it would be possible to obtain a power series solution of the equations at small t to get past what appears the be the region of greatest problems. Alternatively, if physically reasonable, modify the small r boundary condition to cause the solution to approach its asymptotic value more slowly.

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  • $\begingroup$ Thank you. It is interesting to know that NDSolve takes many time to produce the data of imaginary part of u. Replacing (-D[u[r, t], r]) by Abs[D[u[r, t], r]] will change the numerical result? I will try this one. $\endgroup$ – LingLong May 9 '18 at 1:57
  • $\begingroup$ @JOJO There are irregularities in the real part of u too for larger n. By the way, do you need to use "DifferentiateBoundaryConditions" -> {True, "ScaleFactor" -> T}? $\endgroup$ – bbgodfrey May 9 '18 at 2:24
  • $\begingroup$ I'm not pretty sure. But, the use of "DifferentiateBoundaryConditions" -> {True, "ScaleFactor" -> T} seems making the result better by comparing the semianalytical result. $\endgroup$ – LingLong May 9 '18 at 11:02
  • $\begingroup$ I found that replacing (-D[u[r, t], r]) by Abs[D[u[r, t], r]] would not work. The error message will display. $\endgroup$ – LingLong May 10 '18 at 4:02
  • $\begingroup$ @JOJO Replacing Abs by RealAbs eliminates the error that you probably encountered but does nothing to eliminate potential singularities as D[u[r, t], r] approaches zero, as I had expected. I tried a few other things as well but without success. You should try resolving NDSolve::ibcinc: Warning: boundary and initial conditions are inconsistent. $\endgroup$ – bbgodfrey May 13 '18 at 13:05

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