# NDSolve and memory used

I am trying to solve a PDE with NDSolve, but it turns out that it uses a lot of RAM memory. In fact my PC stalled serveral times. I tried with two different computers and with Mathematica versions v11.0 and v11.1, with the same results. The memory used goes up till filling all the RAM. I think it is because I am trying to use a large resolution and over a large period of time.

As a solution I tried to reinitialize the code, with the hope of freeing some memory. My simplifyied code (the real one is more complex) is

    a = 10;
xl = yl = -a;
xr = yr = a;
tmax = 1;
ClearAll[vortex];
vortex[x_, y_] := (1/Sqrt[6 π ] ) Exp[-(x^2 + y^2)/(6)];
eqn = I*Derivative[0, 0, 1][ψ][x,y,t] == -0.5 Laplacian[ψ[x,y,t],{x,y}] +
(Abs[ψ[x, y, t]]^2 + (x^2 + y^2)/2)*ψ[x,y,t];
bcs ={ψ[xl,y,t]==vortex[xl, y], ψ[xr,y,t] == vortex[xr, y],
ψ[x,yl,t] ==vortex[x,yl], ψ[x,yr,t] == vortex[x,yr]};
ics = ψ[x,y,0] == vortex[x,y];
nxy = 200;
ndssdata =
First[NDSolveProcessEquations[{eqn,ics,bcs}, ψ, {x,xl,xr}, {y,yl,yr},
{t,0,tmax},
Method -> {"MethodOfLines",
"SpatialDiscretization" -> {"TensorProductGrid",
"MinPoints" -> nxy, "MaxPoints" -> 2 nxy}}]];


and

    NDSolveIterate[ndssdata, 0.1];
ndsol = NDSolveProcessSolutions[ndssdata, "Forward"]


Then I want to reinitialize the code using the solution at time t=0.1, I got:

   newstate = First[NDSolveReinitialize[ndssdata, {ics=ψ[x, y, 0]==
Evaluate[ψ[x,y,0.1]/.ndsol]}]]
(* {NDSolveStateData["<" 0. ">"]} *)


and now I want to iterate another 0.1 timestep:

   NDSolveIterate[newstate, 0.1]


The problem is that the original code use too much memory when I set a large resolution -nxy large or spatial resolution better than 0.01-. The procedure above seems to bound the total memory used if I reinitialize every small timestep. There is some other way to solve this issue? I mean to keep the memory bounded, or a more elegant way of reinitializing the code automatically. I also find different solutions depending on the method. The pseudoespectal method, I think is the method to use in my case, is the one which approach the correct solution. Thankx.

• Is the current b.c. an approximation for infinite domain? If so, there exists example where improper approximation for infinity causes trouble: mathematica.stackexchange.com/q/145478/1871 – xzczd Jun 22 '18 at 13:31
• @xzczd it is possible, I thought about that. Excellent method you suggested me, I will implement it and tell you. Thank you very much. – Gluoncito Jun 22 '18 at 13:47
• @xzczd I gave a try to the Scrinzi method, and modified your code but it didn't work with this equation. The "neweq" is really awfull, and not a surprise that Mathematica didn't integrate it -without any warnings-. However, I liked the methods and the subroutines involved. I wonder if I must post another question with the full PDE's (which is frustrating because I know I'm close to the solution and do not want other's peoples to waste time). – Gluoncito Jun 23 '18 at 23:28
• If the problem can't be reproduced with this simplified example, it's definitely OK to add a new example to your question. (Of course the new example should be as simple as possible. ) – xzczd Jun 24 '18 at 9:27