Forcing at least $n$ spatial steps in solving a 1D spatio-temporal PDE problem with NDSolve

Question

I want to solve the telegraph equation with a spatial discretization forced at 200 points. I tried:

reg = Line[{{0}, {1}}];
shape = Cos[16 π (x - 0.5)] D[0.125 Erf[(x - 0.5)/0.125], x];
op = D[u[t, x], {x, 2}] - D[u[t, x], {t, 2}];
ics = {u[0, x] == shape, Derivative[1, 0][u][0, x] == 0};
sol = NDSolveValue[{op == 0, ics}, u, {t, 0, 2}, {x} ∈ reg, MaxStepSize -> 0.005];
Plot[shape, {x, 0, 1}, PlotRange -> All,Frame -> True]
Plot[sol[0, x], {x, 0, 1}, PlotRange -> {-1, 1.5}, Frame -> True]

The plots show that NDSolve has chosen 20 spatial points. My MaxStepSize option is used for the temporal variable t and not for the spatial variable x.

1. What is the NDSolve option or syntax for forcing 200 spatial points?

2. Where is this information in the NDSolve documentation?

3. Is there a NDSolve method that would have by default discretize more cleverly?

Edited question given the first answers that were posted:

5. How to force the initial spatial meshing at 200 points, keeping the region specification for x and consequently the finiteElement method ?

Answer 1

shape = Cos[16 \[Pi] (x - 0.5)] D[0.125 Erf[(x - 0.5)/0.125], x];
op = D[u[t, x], {x, 2}] - D[u[t, x], {t, 2}];
ics = {u[0, x] == shape, Derivative[1, 0][u][0, x] == 0};
bc = {u[t, 0] == shape /. x -> 0, u[t, 1] == shape /. x -> 1};
sol = NDSolveValue[{op == 0, ics, bc}, u, {t, 0, 2}, {x, 0, 1}, 
   Method -> {"MethodOfLines", 
     "SpatialDiscretization" -> {"TensorProductGrid", 
       "MinPoints" -> 200, "MaxPoints" -> 213, 
       "DifferenceOrder" -> "Pseudospectral"}}];
{Plot[shape, {x, 0, 1}, PlotRange -> All, Frame -> True],
 Plot[sol[0, x], {x, 0, 1}, PlotRange -> {-1, 1.5}, Frame -> True]}

Plot3D[sol[t, x], {t, 0, 2}, {x, 0, 1}, Mesh -> None, 
 ColorFunction -> Hue]

Answer 2

Just to point out several issues.

The plots show that NDSolve has chosen 20 spatial points.

Well, I don't know why you think the plots show the number of grid points is 20, but you're wrong. The number of mesh coordinates (yeah the FiniteElement Method has been chosen for spatial discretization because of the {x} ∈ reg, check this post for more information ) can be checked by

coord = Flatten@sol["Coordinates"][[2]]["Coordinates"] // Sort
(* {0., 0.025, 0.05, 0.075, 0.1, 0.125, 0.15, 0.175, 0.2, 0.225, 0.25, 0.275, 0.3, 
0.325, 0.35, 0.375, 0.4, 0.425, 0.45, 0.475, 0.5, 0.525, 0.55, 0.575, 0.6, 0.625, 0.65, 
0.675, 0.7, 0.725, 0.75, 0.775, 0.8, 0.825, 0.85, 0.875, 0.9, 0.925, 0.95, 0.975, 1.} *)

coord // Length
(* 41 *)

The next important issue is boundary condition (b.c.). I'm not sure why you don't add b.c. to the system, but you're wrong again if you think NDSolveValue will be forced to solve an initial value problem (IVP) i.e. a problem defined in $x \in (-\infty,\infty)$. The b.c. at $x=0$ and $x= 1$ is Neumann 0 condition i.e. NeumannValue[0,…] in this case, for more information check the Details section of document of NeumannValue. If you need to approximate infinity, consider starting from this post.

Finally I'd like to add, it's possible to use MaxStepSize to control grid size in $x$ direction if TensorProductGrid method is used for spatial discretization. We just need to use a List as the option value, here is an example. Nevertheless, I don't recommend using this method to control spatial grid, because there's no MinStepSize for NDSolve. (The MinStepSize option mentioned in e.g. NDSolveValue::mxsst warning is an option for TensorProductGrid method. )

Answer 3

The answer I was looking for is the following: Add a Method option in NDSolve that is compatible with the method that NDSolve chooses by itself given your PDEs syntax. Then add your discretization specification in the sub-options. Here, the solution is:

NDSolveValue[{op == 0, ics}, u, {t, 0, 2}, {x} \[Element] reg, 
Method -> {"FiniteElement", "MeshOptions" -> {"MaxCellMeasure" -> 0.005}}]

With and without the Method option the final result is either correct (regeneration of the shape at time t=2, after two reflections on the region boundaries), or completely False:

Now, there is a correlative question: In the solution above I specify the stepsize manually to get sufficient meshing and a correct solution. A much better solution would be to force NDSolve to check the validity of its solution and refine the spatial stepsize by itself... I create a separate post for this question.