I have a project to animate heat equation in 3D. I have the following problem

$$u_t=u_{xx},\qquad 0\leq x\leq 1 \land t>0$$ $$u(0,t)=u(1,t)=0$$ $$u(x,0)=\begin{cases} 2x, & 0\leq x<\frac{1}{2}\\ 2-2x, & \frac{1}{2}\leq x<1 \end{cases}$$

And then animate the comparison between exact solution and my manual solution within the time interval $0 \leq t \leq 2$. I try to solve using Mathematica with the command NDSolve, here's my code

sol = NDSolve[{D[u[x, t], t] == D[u[x, t], {x, 2}],
   u[0, t] == 0, u[1, t] == 0,
   u[x, 0] == Piecewise[{{2 x, 0 <= x <= 1/2}, {2 - 2 x, 1/2 <= x <= 1}}]}, 
   u, {t, 0, 2}, {x, 0, 1}]
Manipulate[Plot3D[u[x, t] /. sol, {x, 0, 1}, {t, 0, 2}, PlotRange -> All, PlotPoints -> 250, Mesh -> None], {t, 0, 2}]

But my Wolfram Mathematica seems heavily working, it even hardly change the plot3D image, when I change the control value in Manipulate. Is there a way to fix this? Or is there any alternative better command?

  • $\begingroup$ I'm sorry i have a trouble editing the equation it keeps getting me the "Your post appears to contain code that is not properly formatted" please help me edit it if you can $\endgroup$
    – Aji Wibowo
    May 30, 2022 at 3:24
  • $\begingroup$ You're using {t, 0, 2} for both Plot3D's range specification and Manipulate's range specification, and I'm pretty sure that's not what you're trying to animate here? Edit: Conceptually, once Plot3D uses its t range, there should be no more t values to Manipulate. $\endgroup$
    – eyorble
    May 30, 2022 at 3:28
  • $\begingroup$ @eyorble I change it to $u$, but still can't export it to .avi, well I can and it sometimes not responding but the result video is just the plot at t=0 not moving anywhere $\endgroup$
    – Aji Wibowo
    May 30, 2022 at 3:41
  • 1
    $\begingroup$ PlotPoints -> 250 is not your friend, nor the friend of speed. Manipulate (Animate just makes a specialized Manipulate) is designed for computations that can be updated quickly. If you need an initial 250 plot point grid to make a reasonable plot, you probably cannot do it quickly. But as @eyorble seems to hint, maybe you want Plot[u[x, t] /. sol, {x, 0, 1}] instead with t controlled by Manipulate? $\endgroup$
    – Michael E2
    May 30, 2022 at 4:04
  • $\begingroup$ The problem comes from the fact that the initial condition is not compatible with your ODE. The second derivative D[u[x, 0], {x, 2}] at t==1/2 is infinite. Try to give an initial condition that gives a finite second derivative. $\endgroup$ May 30, 2022 at 14:41

1 Answer 1


it even hardly change the plot3D image, when I change the control value in Manipulate

I made some changes to make it run much faster now.

enter image description here

ClearAll[u, x, t];
pde = D[u[x, t], t] == .1*D[u[x, t], {x, 2}];
bc = {u[0, t] == 0, u[1, t] == 0};
ic = u[x, 0] == Piecewise[{{2 x, 0 <= x <= 1/2}, {2 - 2 x, 1/2 <= x <= 1}}];
sol = NDSolveValue[{pde, ic, bc}, u, {t, 0, 2}, {x, 0, 1}];
 Plot3D[sol[x, t], {x, 0, 1}, {t, 0, t0}, PerformanceGoal -> "Speed", 
  PlotRange -> {Automatic, {0, 2}, {-.1, 1}}, 
  AxesLabel -> {"x", "time", "u(x,t)"}, ImageSize -> 400],
 {{t0, 0.01, "time"}, .01, 2, .01, Appearance -> "Labeled"},
 TrackedSymbols :> {t0}

Removed PlotPoints -> 250 and added PerformanceGoal -> "Speed" and other minor cleanup.

  • $\begingroup$ Thank you so much, but i think what i need is just changing to that 'PerformanceGoal->"Speed"', only 1 thing still stumble me, how do I export the file so its smooth on the video ? Seems like the 'Export' doesnt render all the value in time $\endgroup$
    – Aji Wibowo
    May 30, 2022 at 9:12
  • $\begingroup$ You might try making a list of frames, then exporting that. (Also, ListAnimate might work better, because your frames are pre-rendered). $\endgroup$
    – Chris K
    May 30, 2022 at 15:50
  • $\begingroup$ You could precompute the graphic frames and use ListAnimate. ListAnimate will also let you specify the frames per second. reference.wolfram.com/language/ref/… $\endgroup$ Jun 2, 2022 at 1:36

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