Variation of heat equation with guessed initial condition

Hi I am trying to solve a variation of the heat equation with interaction terms and external source -- actually it's the Schrödinger-Newton equation if that's more familiar to you. \begin{align} i \dot \psi & = -\nabla^2 \psi + \psi \phi_N + |\psi|^2\psi, \cr \nabla^2 \phi_N & = |\psi|^2, \end{align}

The boundary condition is that $\psi, \phi$ both vanish at large radius and its first derivative vanishes at zero:

\begin{align} \frac{d \psi}{d r}(0,t) &= 0, \cr \psi(\infty,t) &= 0, \cr \frac{d \phi}{d r}(0,t) &= 0, \cr \phi(\infty,t) &= 0. \end{align}

The exact form of initial condition is not known and the requirement is the wave function being real at t=0, it vanishes at $r$ being large. Since NDSolve requires an explicit initial condition, I've been trying the following code with a guessed initial condition:

\begin{align} \psi(r,0) &= (1+r) \mathrm{e}^{-r}, \cr \phi(r,0) &=\phi_0(r), \end{align} where $\phi_0(r)$ is the solution of $\nabla^2 \phi = (1+r)^2 \mathrm{e}^{-2r}$, with $\frac{d \phi_0}{d r} (0) = 0$, and $\phi_0(\infty) = 0$.

So far I haven't got much success to run the code. The kernel complains about zero differential order(pdord), inconsistency (ibcinc), insufficient number of boundary conditions (bcart), unable to continue with complex values (mconly), and unable to find initial conditions that satisfy (icfail). Is there any way to solve it with NDSolve?

tmpψinitial[r_] := (1 + r)*E^(-r)
tmpϕinitial[rr_] := DSolveValue[{D[ϕ[r], r, r] == (1 + r)^2* E^(-2 r),(D[ϕ[r], r] /. r -> 0) == 0, ϕ[rEnd0] == 10^-4}, ϕ[r], r] /. r -> rr
rStart0 = 0.01;
rEnd0 = 5;
epsilon=10^-5;

eqn = {-I*D[ψ[r, t], t] - 1/2*(D[ψ[r, t], r, r]
+ 2/r*D[ψ[r, t], r]) + ψ[r, t]*ψ[r, t]* Conjugate[ψ[r, t]] + ψ[r, t]*ϕ[r, t] == 0
, 2/r*D[ϕ[r, t], r] + D[ϕ[r, t], r, r] == Conjugate[ψ[r, t]]*ψ[r, t]};

ic = {ψ[r, 0] == tmpψinitial[r], ϕ[r, 0] == tmpϕinitial[r]};
bc = {(D[ψ[r, t], r] /. r -> rStart0)== epsilon
, (D[ϕ[r, t], r]/. r -> rStart0) == epsilon
, ψ[rEnd0, t] == epsilon
, ϕ[rEnd0, t] == epsilon};

sol = NDSolveValue[Flatten@{eqn, ic, bc}, {ψ, ϕ}, {r, rStart0, rEnd0}, {t, 0, 10}, MaxSteps -> 10^6]
• (a) Is this 1D, or 2D/3D with radial symmetry? (b) your guessed initial condition does not have vanishing derivative at $r=0$, is that on purpose? (c) I suspect your question is not well defined. Is there a single possible initial condition? are you looking for any consistent initial condition or a specific one? – yohbs Jan 17 '18 at 20:57
• Hi, thanks for the comments. a) this is the S-N equation in 3D with spherical symmetry. So the laplacian is $2/r \frac{d}{dr} + \frac{d^2}{dr^2}$. b) actually the first derivative of $(1+r) e^{-r}$ at $r=0$ does vanish. c) Because this question is physically motivated, actually it is not possible to find the exact initial condition. In the literature, either $\sim (1+r) e^{-r}$, or simply Gaussian distribution is used. To put it another way, I'm looking for a solution for a type of consistent initial conditions (exponential function of power of $r$, function vanishes when r goes large.) – Boson Bear Jan 17 '18 at 21:16
• Sorry I mean the requirements for the initial condition are just 1) spherically symmetric, $i.e.$ it only depends on $r$, 2) the function goes to zero as $r$ goes to infinity. – Boson Bear Jan 17 '18 at 21:24

ibcinc warning isn't a big deal here, because the i.c. and b.c. are almost consistent. The hard part is pdord warning. (bcart, mconly and icfail are probably by-products of it. )

pdord pops up because the equation system doesn't explicitly contain derivative of $$\phi$$ with respect to $$t$$, and several previous questions in this site suggest that, NDSolve just can't directly handle this type of problem well at the moment (probably due to the not-strong-enough DAE solver). So we need to transform the equation a bit to help NDSolve to use an ODE solver to solve it.

We first separate real and imaginary part of the equation because Conjugate turns out to be hard to handle in later step:

rStart0 = 1/100;
rEnd0 = 5;
epsilon = 0(*1/10^5*);

tmpψinitial[r_] := (1 + r) E^-r
tmpϕinitial[r_] =
DSolve[{D[ϕ[r], r, r] == (1 + r)^2 E^(-2 r), (D[ϕ[r], r] /. r -> 0) ==
epsilon, ϕ[rEnd0] == epsilon}, ϕ[r], r][[1, 1, -1]];

Unevaluated[
eqn = {-I*D[ψ[r, t], t] -
1/2*(D[ψ[r, t], r, r] + 2/r*D[ψ[r, t], r]) + ψ[r, t]*ψ[r, t]*
Conjugate[ψ[r, t]] + ψ[r, t]*ϕ[r, t] == 0,
2/r*D[ϕ[r, t], r] + D[ϕ[r, t], r, r] ==
Conjugate[ψ[r, t]]*ψ[r, t]};

ic = {ψ[r, 0] == tmpψinitial[r], ϕ[r, 0] == tmpϕinitial[r]};
bc = {(D[ψ[r, t], r] /. r -> rStart0) ==
epsilon, (D[ϕ[r, t], r] /. r -> rStart0) == epsilon, ψ[rEnd0, t] ==
epsilon, ϕ[rEnd0, t] ==
epsilon};] /. {ψ -> (psiR[#, #2] + I psiI[#, #2] &), ϕ -> (phiR[#, #2] +
I phiI[#, #2] &)};

systemnew = {repart, impart} = ComplexExpand@Map[#, {eqn, ic, bc}, {3}] & /@ {Re, Im};

Next we add derivative with respect to t to the 2nd equation:

addD = MapAt[D[#, t] &, #, {1, 2}] &;
{eqnfinal, icfinal, bcfinal} = Transpose[addD /@ systemnew];

Still, NDSolve isn't able to handle eqnfinal (I believe it's related to this problem), so we need to discretize the system to an ODE system all by ourselves. I'll use pdetoode for the task:

points = 25; domain = {rStart0, rEnd0}; grid = Array[# &, points, domain];
difforder = 4;
(* Definition of pdetoode isn't included in this post,
please find it in the link above. *)
ptoofunc = pdetoode[{psiR, psiI, phiR, phiI}[r, t], t, grid, difforder];

del = #[[2 ;; -2]] &;

ode = Map[del, ptoofunc[eqnfinal], {2}];

odeic = ptoofunc@icfinal;

odebc = With[{sf = 1}, Map[sf # + D[#, t] &, ptoofunc@bcfinal, {3}]];

tend = 10;

The last step is to solve the discretized system. Directly solving {ode, odeic, odebc} with NDSolve is OK, but when points becomes large (increasing points is perhaps the most efficient way for increasing precision in this case), the pre-processing inside NDSolve will become very slow (it internally use Solve to transform the system, for more information you may check this post), so, again, we need to transform the system by ourselves outside of NDSolve, with the help of CoeffcientArrays and LinearSolve, given the system is a linear system for derivatives with respect to t:

(* The following can be used but is slow: *)
(*
sollst = NDSolveValue[{ode, odeic, odebc},
Outer[#[#2] &, {psiR, psiI, phiR, phiI}, grid], {t, 0, tend},
Method -> {"EquationSimplification" -> "Solve"} ];
*)
(* More advanced but faster approach: *)
lhs = D[Flatten@Outer[#[#2][t] &, {psiR, psiI, phiR, phiI}, grid], t];
{b, m} = CoefficientArrays[Flatten[{ode, odebc}], lhs]; // AbsoluteTiming
rhs = LinearSolve[m, -b]; // AbsoluteTiming

sollst = NDSolveValue[{odeic, lhs == rhs // Thread},
Outer[#[#2] &, {psiR, psiI, phiR, phiI}, grid], {t, 0, tend},
Method -> {"EquationSimplification" -> "Solve"},
MaxSteps -> Infinity]; // AbsoluteTiming

sol = rebuild[#, grid, 2] & /@ sollst

label = {Re@ψ, Im@ψ, Re@ϕ, Im@ϕ};

Partition[Table[
Plot3D[sol[[i]][r, t], #, {t, 0, tend}, PlotRange -> All, PlotLabel -> label[[i]]] &@
Flatten@{r, domain}, {i, 4}], 2] // GraphicsGrid BTW, the following is a even more advanced but even faster approach to solve for sol:

varlst = Flatten@Outer[#[#2][t] &, {psiR, psiI, phiR, phiI}, grid];
varmid = CompileGetElement[lst, #] & /@ Range@Length@varlst;
crhs = Compile[{{lst, _Real, 1}}, #,
RuntimeOptions -> "EvaluateSymbolically" -> False] &[
rhs /. Thread[varlst -> varmid]];
iclst = LinearSolve[#2, -#] & @@ CoefficientArrays[Flatten@odeic, varlst /. t -> 0];
solvector =
NDSolveValue[{vector == iclst, vector'[t] == crhs@vector[t]}, vector, {t, 0, tend},
MaxSteps -> Infinity]; // AbsoluteTiming

sol = ListInterpolation[#, {grid, solvector["Coordinates"][]}] & /@
Partition[Transpose@DeveloperToPackedArray@solvector["ValuesOnGrid"], points];
• Thank you much for the detailed explanation. This is very helpful! Actually I did notice that simply adding a time derivative term could solve the system although it altered the equation. Am I right to think that you added a total derivative on both sides of the second equation just to tell Mathematica to choose the right PDE solver? – Boson Bear Jan 18 '18 at 16:00
• @BosonBear To be precise, it's for making Mathematica use an ODE solver rather than a DAE solver for solving the discretized system. If we don't differential the 2nd equation, after discretization the 2nd equation will become a set of algebraic equations that doesn't involve derivative of t, so NDSolve will choose a DAE solver for solving it, which doesn't work well on this problem. (You'll see icfail then. ) BTW the seemingly strange definition of odebc is also for adding derivative of t to the boundary condition. – xzczd Jan 18 '18 at 16:34
• I see. So when I plot it out at a slice of r with respect to t, the plot is a bit zig-zag due to numerical noise. I wonder what the most efficient way is in order to increase the precision in the t direction? – Boson Bear Jan 19 '18 at 13:57
• @BosonBear Increasing the spatial grid points (in this case points) is almost the only way, as far as I know, but then the previous approach turns out to be slow. I've edited the post the include a more efficient approach, check the edit. BTW, a quick test shows, difforder=2 seems to be accurate enough for the system and make the calculation faster. – xzczd Jan 19 '18 at 14:50
• Thanks a lot. I'm working on a project where solving this efficiently is part of it. Since you've helped me a lot in the past a few days, I wonder if there's any way to acknowledge it properly in my paper if the work comes to fruition. Cheers! – Boson Bear Jan 21 '18 at 15:28