How should I deal with messages being produced by my Manipulate?

Question

I have made a knockoff of John Polking's dfield for a web-embedded CDF (via Enterprise Mathematica). My code extends that from Belisarius who last year showed me how to use ClickPane to store and clear solution curves.

Although clicking on any point of the slope field produces an apparently correct solution curve, an error message NDSolve::ndsz is generated by NDSolve. Moreover, by choosing an initial value near the point with coordinates (2,-2) and following by moving the "b" slide from its default position, the slope field turns the color pink and I get the message "InterpolatingFunction::dmval: Input value {-1.99992} lies outside the range of data in the interpolating function. Extrapolation will be used."

sf := 
  VectorPlot[{1, diffEq /. {a -> A, b -> B}}, {t, -2, 2}, {x, -2, 2},
    VectorPoints -> 17,
    VectorScale -> {0.03, Automatic, None}, 
    VectorStyle -> {{Red, Arrowheads[0]}}, ImagePadding -> 1, 
    PerformanceGoal -> "Speed"];

f[t_, a_, b_, t0_, x0_] := 
  u[t] /. First[Quiet[NDSolve[{u'[t] == ODE, u[t0] == x0}, u, {t, -2, 2}, 
                  Method -> "StiffnessSwitching"]]];

Manipulate[
  ODE = diffEq /. {a -> A, b -> B} /. x -> u[t];
  ClickPane[
    Show[
      Plot[g, {t, -2, 2}, 
        PlotRange -> 2, Frame -> True, 
        ImageSize -> 400, AspectRatio -> 0.75], 
      sf, 
      Graphics[{PointSize[Large], Point[sp]}]],
    (AppendTo[g, f[t, A, B, #[[1]], #[[2]]]]; {t0, u0} = #; 
     AppendTo[sp, #]) &],
  Style["Enter f(t,x)"],
  {{diffEq, x^2 - a t + b, "dx/dt = "}},
  {{A, 1, "a"}, -4, 4, 0.01, Appearance -> "Labeled", ControlPlacement -> Bottom},
  {{B, 0, "b"}, -4, 4, 0.01, Appearance -> "Labeled", ControlPlacement -> Bottom},
  Button["clear", {g = {}; sp = {}}, ImageSize -> {40, 20}], 
  Initialization :> {(
    g = {}; sp = {}; {t0, x0} = {})},
  SaveDefinitions -> True]

FYI, the ODE that I have set as an input is the default used in dfield.

The two error messages that I get are probably related.

NDSolve::ndsz: At t == -1.71226, step size is effectively zero; singularity or stiff system suspected. >>

InterpolatingFunction::dmval: Input value {-1.99992} lies outside the range of data in the interpolating function. Extrapolation will be used. >>

I do not have sufficient Mathematica chops to fix my problem.

Answer 1

Re NDSolve::ndsz: Using WhenEvent, one can stop the integration before the solution reaches an infinite singularity, which is what is happening with the example differential equation. I removed the Quiet so one might test it. If the differential equation is changed, then it would be possible to get the message to appear. Use Quiet to suppress it, Check to deal with it.

Re InterpolatingFunction::dmval: Using the NDSolve option

"ExtrapolationHandler" -> {Indeterminate &, "WarningMessage" -> False}

prevents extrapolation and turns off the warning message.

Random changes: I incorporated needed definitions in the Initialization option instead of SaveDefinitions; see The dangers of SaveDefinitions --- should this really happen? I made the functions explicitly depend on the Manipulate variables. This is just a good habit when developing code, especially Dynamic code. Otherwise, one day it works, and the next a minor change breaks the dynamic updating. I localized g and sp. One might want to localize the functions, too. I probably did a few other idiosyncratic things as I went along. None of these had anything to do with the two messages the OP asked about.

Manipulate[
 ClickPane[
  Show[
   Plot[g, {t, -2, 2}, PlotRange -> 2, Frame -> True, 
    ImageSize -> 400, AspectRatio -> 0.75],
   sf@dx[diffEq, A, B],
   Graphics[{PointSize[Large], Point[sp]}]],
  (AppendTo[g, f[dx[diffEq, A, B], #]];
    AppendTo[sp, #]) &],

 Style["Enter f(t,x)"],
 {{diffEq, x^2 - a t + b, "dx/dt = "}},
 {{A, 1, "a"}, -4, 4, 0.01, Appearance -> "Labeled", ControlPlacement -> Bottom},
 {{B, 0, "b"}, -4, 4, 0.01, Appearance -> "Labeled", ControlPlacement -> Bottom},
 {{g, {}}, ControlType -> None}, {{sp, {}}, ControlType -> None},
 Button["clear", g = {}; sp = {}, ImageSize -> {Automatic, 20}],

 Initialization :> (    (* alternative to SaveDefinitions -> True; *)
   Clear[f, sf, dx]; 
   g = {};
   sp = {};
   dx[de_, a0_, b0_] := de /. {a -> a0, b -> b0};
   sf[dx_] := VectorPlot[{1, dx},
     {t, -2, 2}, {x, -2, 2},
     VectorPoints -> 17, VectorScale -> {0.03, Automatic, None}, 
     VectorStyle -> {{Red, Arrowheads[0]}}, ImagePadding -> 1, 
     PerformanceGoal -> "Speed"];
   f[dx_, {t0_, x0_}] := u[t] /. 
      First@ NDSolve[            (* consider `Quiet`, `Check` to handle NDSolve::ndsz *)
       {u'[t] == dx /. {x -> u[t]}, u[t0] == x0,
        WhenEvent[Abs[u[t]] > 3, "StopIntegration"]},
       u, {t, -2, 2},
       Method -> "StiffnessSwitching", 
       "ExtrapolationHandler" -> {Indeterminate &, "WarningMessage" -> False}];
   )]

---

A somewhat more robust way might be to use the functionality for this sort of thing built into VectorPlot. One significant difference is that it will not draw solutions when they get too close to one another, and no solution if it starts too close to another solution. It allows one to simplify the code considerably since VectorPlot handles all the NDSolve stuff under the hood.

Another difference is that the curves automatically change when the a or b sliders are changed, since VectorPlot recomputes the stream lines.

Manipulate[
 ClickPane[
  Show[
   VectorPlot[{1, dx[diffEq, A, B]}, {t, -2, 2}, {x, -2, 2}, 
    VectorPoints -> 17, VectorScale -> {0.03, Automatic, None}, 
    VectorStyle -> {{Red, Arrowheads[0]}}, ImagePadding -> 1, 
    PerformanceGoal -> "Speed", PlotRange -> 2, Frame -> True, 
    ImageSize -> 400, AspectRatio -> 0.75,
    StreamPoints -> {Thread[
       {sp,
        Table[
         Directive[ColorData[97, i], AbsoluteThickness[1.6]], {i, 
          Length@sp}]}]},
    StreamScale -> None],
   Graphics[{PointSize[Large], Point[sp]}]],
  AppendTo[sp, #] &],

 Style["Enter f(t,x)"],
 {{diffEq, x^2 - a t + b, "dx/dt = "}},
 {{A, 1, "a"}, -4, 4, 0.01, Appearance -> "Labeled", 
  ControlPlacement -> Bottom},
 {{B, 0, "b"}, -4, 4, 0.01, Appearance -> "Labeled", 
  ControlPlacement -> Bottom},
 Button["clear", sp = {}, ImageSize -> {Automatic, 20}],

 Initialization :> (
   Clear[dx];
   sp = {};
   dx[de_, a0_, b0_] := de /. {a -> a0, b -> b0};
   )]

Answer 2

I think your Quiet is not in the right place, and you may need another one.

For f:

f[t_, a_, b_, t0_, x0_] := 
  Quiet[u[t] /. 
    First[NDSolve[{u'[t] == ODE, u[t0] == x0}, u, {t, -2, 2}, 
      Method -> "StiffnessSwitching"]]];

gets rid of the NDSolve warning. Then add another Quiet around the Plot:

Quiet[Plot[g, {t, -2, 2}, PlotRange -> 2, Frame -> True, 
  ImageSize -> 400, AspectRatio -> 0.75]]

This removes the InterpolatingFunction warning.

Answer 3

The following turns off the warning messages, but is not really a good solution, since it doesn't deal with the root problem of the trouble NDSolve is having with your differential equation.

sf[eq_, A_, B_] := 
  VectorPlot[{1, eq /. {a -> A, b -> B}}, {t, -2, 2}, {x, -2, 2}, 
    VectorPoints -> 17, VectorScale -> {0.03, Automatic, None}, 
    VectorStyle -> {{Red, Arrowheads[0]}}, ImagePadding -> 1,
    PerformanceGoal -> "Speed"]

f[t_, a_, b_, t0_, x0_] := 
   u[t] /. 
     First[Quiet @ NDSolve[{u'[t] == ODE, u[t0] == x0}, u, {t, -2, 2}, 
                     Method -> "StiffnessSwitching"]]

Manipulate[
  ODE = diffEq /. {a -> A, b -> B} /. x -> u[t];
  ClickPane[
    Show[
      Quiet @ Plot[g, {t, -2, 2},
        PlotRange -> 2, Frame -> True, ImageSize -> 400, AspectRatio -> 0.75],
      sf[diffEq, A, B],
      Graphics[{PointSize[Large], Point[sp]}]], 
    (AppendTo[g, f[t, A, B, #[[1]], #[[2]]]]; {t0, u0} = #;
     AppendTo[sp, #]) &],
  Style["Enter f(t,x)", 12],
  {{diffEq, x^2 - a t + b, Style["dx/dt = ", 12]}}, 
  {{A, 1, "a"}, -4, 4, 0.01, Appearance -> "Labeled", ControlPlacement -> Bottom}, 
  {{B, 0, "b"}, -4, 4, 0.01, Appearance -> "Labeled", ControlPlacement -> Bottom}, 
  Button[Style["Clear", 10], {g = {}; sp = {}}, ImageSize -> {50, 25}],
  Initialization :> (g = {}; sp = {}; {t0, x0} = {}),
  SaveDefinitions -> True]

Answer 4

I've not answered any questions with this example, but I added tmin, tmax, xmin, xmax, and adjusted when to halt the integration. I don't know yet how to handle the Check command in MichaelE2's comment, so that still remains in the code.

Manipulate[ClickPane[Show[Plot[g, {t, tmin, tmax},
    PlotRange -> {{tmin, tmax}, {xmin, xmax}},
    Frame -> True, ImageSize -> 400, AspectRatio -> 0.75], 
   sf@dx[diffEq, A, B], 
   Graphics[{PointSize[Large], Point[sp]}]], (AppendTo[g, 
     f[dx[diffEq, A, B], #]];
    AppendTo[sp, #]) &], 
 Style["Enter f(t,x)"], {{diffEq, x^2 - a t + b, "dx/dt = "}},
 Row[{
   Control[{{tmin, -2, "tmin = "}}],
   Spacer[10],
   Control[{{tmax, 10, "tmax = "}}]
   }],
 Row[{
   Control[{{xmin, -4, "xmin = "}}],
   Spacer[10],
   Control[{{xmax, 4, "xmax = "}}]
   }],
 {{A, 1, "a"}, -4, 4, 0.01, Appearance -> "Labeled", 
  ControlPlacement -> Bottom}, {{B, 0, "b"}, -4, 4, 0.01, 
  Appearance -> "Labeled", ControlPlacement -> Bottom}, {{g, {}}, 
  ControlType -> None}, {{sp, {}}, ControlType -> None}, 
 Button["clear", g = {}; sp = {}, ImageSize -> {Automatic, 20}], 
 Initialization :> ((*alternative to SaveDefinitions\[Rule]True;*)
   Clear[f, sf, dx];
   g = {};
   sp = {};
   dx[de_, a0_, b0_] := de /. {a -> a0, b -> b0};
   sf[dx_] := 
    VectorPlot[{1, dx}, {t, tmin, tmax}, {x, xmin, xmax}, 
     VectorPoints -> 17, VectorScale -> {0.03, Automatic, None}, 
     VectorStyle -> {{Red, Arrowheads[0]}}, ImagePadding -> 1, 
     PerformanceGoal -> "Speed"];
   f[dx_, {t0_, x0_}] := 
    u[t] /. First@
      NDSolve[(*consider `Quiet`,`Check` to handle NDSolve::ndsz*){u'[
           t] == dx /. {x -> u[t]}, u[t0] == x0, 
        WhenEvent[Abs[u[t]] > 2.5 Max[{Abs[xmin], Abs[xmax]}], 
         "StopIntegration"]}, u, {t, tmin, tmax}, 
       Method -> "StiffnessSwitching", 
       "ExtrapolationHandler" -> {Indeterminate &, 
         "WarningMessage" -> False}];)] 

I've also adjusted tmin, tmax, xmin, xmax to match the default image in John Polking's dfield program.

I like the use of keeping the parameters a and b at the bottom, as you can use them in other functions. For example, try (a-x)(b-x).