Inconsistent of initial and boundary condition in 1D PDE

Question

I am now dealing with the 1D PDE with periodic boundary condition given by the following:

$$ \partial_{t}u(t,x) = \partial_{x}u(t,x). \quad \text{with}\quad u(0,x)=\frac{1}{\sqrt{2\pi}}e^{-(x-\pi/4)^2/2}\quad u(t,-\pi)=u(t,\pi) $$

ufun = NDSolveValue[{\!\(
\*SubscriptBox[\(\[PartialD]\), \(t\)]\(u[t, \[Theta]]\)\) + \!\(
\*SubscriptBox[\(\[PartialD]\), \(\[Theta]\)]\(u[t, \[Theta]]\)\) == 
 0, u[0, \[Theta]] == Exp[-(\[Theta] - \[Pi]/4)^2], 
u[t, -\[Pi]] == u[t, \[Pi]]}, 
u, {t, 0, 0.1, 20}, {\[Theta], -\[Pi], \[Pi]}, 
Method -> {"MethodOfLines", 
 "SpatialDiscretization" -> {"TensorProductGrid", 
   "MinPoints" -> 200}}, PrecisionGoal -> 1];

plots = Table[Plot[ufun[t, \[Theta]], {\[Theta], -\[Pi], \[Pi]},PlotRange -> {0, 1}, ColorFunction -> "LakeColors"], {t, 0, 
20, .1}];
ListAnimate[plots]

Although there is no error, the solution is not expected.

I have tried the solution here 2D Heat equation: inconsistent boundary and initial conditions.

 Method -> {"MethodOfLines", 
           "SpatialDiscretization" -> {"TensorProductGrid", "MinPoints" -> 20}}

It seems doesn't work. Any suggestion?

How about using "SpatialDiscretization" -> {"TensorProductGrid", "MinPoints" -> 200, "MaxPoints" -> 200}? That fixed it for me. Also, is PrecisionGoal -> 1 necessary? — Chris K, Aug 15 '18 at 17:20
@ChrisK It works! But I still didn't get it how does the these mesh affect the stability and why mathematica does not do these automatically? — Bob Lin, Aug 15 '18 at 17:47
I can't say for sure, but when I run your code it gives a warning Using maximum number of grid points 10000. Such a fine mesh size may mean that time steps need to be quite small to satisfy the Courant condition. I suppose numerically solving PDEs is a hard problem for Mathematica to make the right decisions on, so sometimes the user has to make the appropriate choice of numerical parameters. — Chris K, Aug 15 '18 at 17:58
BTW, welcome to Mathematica.SE! I suggest the following:

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Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign! — Chris K, Aug 15 '18 at 17:59
Thanks for answering! I am wondering if there is a way to adaptively adjust the mesh to save memory and time (in 2D problem). — Bob Lin, Aug 15 '18 at 18:04
Sorry, I don't know about that. You might open another question on it if the 2D problem actually turns out to be problematic. — Chris K, Aug 15 '18 at 18:15
The time step of PDE solver is adaptive, but the spatial step is not (at least now). You can refer to tutorial/NDSolveMethodOfLines for more information — xzczd, Aug 16 '18 at 06:53
@xzczd Thanks! Btw, I am wondering if the unbounded transport equation can be solved by NDSolve perfectly or not? Do you think specifying a periodic boundary condition at some far away point like x=100 & -100 can mimic an unbounded problem? — Bob Lin, Aug 16 '18 at 08:12
To solve unbounded problem numerically, as you've noticed, artificial b.c. is often needed, but I never found material about artificial b.c. for transport equation. (This is a related post, don't miss the links there. ) The reason might be that, the analytic solution for transport equation is known and very simple. So I think it's better to solve transport equation analytically. — xzczd, Aug 16 '18 at 08:43
@xzczd Yes, you are right. Dsolve always treats this problem well. However, when I run into 2D problem (link), it seems to be inevitable to deal with this problem numerically like convection-dominated problem. — Bob Lin, Aug 16 '18 at 08:52

score 1 · Accepted Answer · answered Aug 17 '18 at 14:25

As I commented above, without setting "MaxPoints", NDSolve uses 10000 grid points, which makes it hard to solve, maybe due to the Courant condition. Adding "MaxPoints"->200 fixes it. I also removed PrecisionGoal->1 and changed the t range specification in NDSolve.

ufun = NDSolveValue[{
  D[u[t, θ] , t] + D[u[t, θ], θ] == 0, 
  u[0, θ] == Exp[-(θ - π/4)^2], 
  u[t, -π] == u[t, π]}, 
  u, {t, 0, 20}, {θ, -π, π}, 
  Method -> {"MethodOfLines", "SpatialDiscretization" -> {"TensorProductGrid", 
    "MinPoints" -> 200, "MaxPoints" -> 200}}];
Plot[ufun[20, θ], {θ, -π, π}, ColorFunction -> "LakeColors"]

Mathematica graphics

Inconsistent of initial and boundary condition in 1D PDE

1 Answers1