Correct syntax for working precision in functions version 12

Question

I just recently updated from version 11.2 -> 12...there apparently are quite a few changes since 11.2. $\unicode{952}\unicode{960}$

One being how version >11.3 deal with numbers that approach zero.

To simulate a physical impulse one can use such an equation:

bump[t_] := 7 E^(-10 (t - 3)^2) - 3/2 E^(-10 (t - 6)^2) + 3 E^(-10 (t - 9)^2)

Which looks like this plotted over time in 11.2 and 11.1 this didn't cause any issues for me, I use this function quite often for impulses in many systems.

Now however I am constantly getting an error that comes along with it:

After doing some duckduckgoing, I found in this post And with some more infos and what I thought would be a solution at this post

One can simply increase the precision using

Exp[-900`20]
(*3.6678745841776872*10^-348*)

However as I try in all combinations of adding the higher precision or using Working Precision -> 20 I get error messages, or barring error messages in the functions themselves, I get the old General error message in the plot,

When using NDSolve[] the system becomes stiff at a specific time and cannot be further calculated (which in prior versions was again, never an issue)

Example working code from 11.2 with error from v12 and NDSolve:

Plot[{θ[t] 10, ϕ'[t]} /. 
   NDSolve[{{{0.18884908961676594` Sin[θ[t]] + 
      0.0023324989011712697` θ''[
        t] == -0.000795` ArcTan[
        10.` θ'[t]] + 
      0.0920755` Cos[θ[t]] f[t] - 
      0.00016799999999999996` θ'[t] - 
      0.0004032497194086029` ϕ''[t], 
    0.0004032497194086029` ϕ''[t] == 
     0.0335` u[t] - 
      0.000039255383907286545` ϕ'[t] + 
      0.0004032497194086029` θ''[
        t]}, {θ[0] == 3.0915926535897933`, 
    θ'[0] == 0, ϕ[0] == 0, 
    ϕ'[0] == 0}} /. {u[t] -> 
    42.69342907334017` (-π + θ[t]) + 
     3.88853571287127` θ'[t] + 
     0.21937104569306368` ϕ'[t], 
   f[t] -> 3 E^(-10 (-9 + t)^2) - 3/2 E^(-10 (-6 + t)^2) + 
     7 E^(-10 (-3 + t)^2)}}, {θ[t], ϕ'[t]}, {t, 0, 
     14}] // Evaluate, {t, 0, 14}, PlotRange -> All, 
 ImageSize -> Large, PlotLegends -> {θ, ϕ'}]

f[t] being where I would normally add my external forces, as my impulse function.

An attempt to add higher working precision to my bumpy function with error:

bumpy[t_] := 7 Exp[-10 (t - 3)^2`20 ] - 3/2 Exp[-10 (t - 6)^2`20 ] + 3 Exp[-10 (t - 9)^2`20]

(*Syntax::sntxf: "(" cannot be followed by "-10 (t-3)^2 `20)".*)

What is the correct way/syntax to write a function with increased precision for further use?

Thanks for the help!

Answer 1

If working with machine precision produces machine underflow/overflow errors, than you need to work with extended precision numbers. This means that your input cannot contain any machine precision numbers. One possibility is to use Rationalize to convert your ode into exact numbers:

ode = Flatten @ Rationalize[{{{0.18884908961676594` Sin[θ[t]] + 
      0.0023324989011712697` θ''[
        t] == -0.000795` ArcTan[
        10.` θ'[t]] + 
      0.0920755` Cos[θ[t]] f[t] - 
      0.00016799999999999996` θ'[t] - 
      0.0004032497194086029` ϕ''[t], 
    0.0004032497194086029` ϕ''[t] == 
     0.0335` u[t] - 
      0.000039255383907286545` ϕ'[t] + 
      0.0004032497194086029` θ''[
        t]}, {θ[0] == 3.0915926535897933`, 
    θ'[0] == 0, ϕ[0] == 0, 
    ϕ'[0] == 0}} /. {u[t] -> 
    42.69342907334017` (-π + θ[t]) + 
     3.88853571287127` θ'[t] + 
     0.21937104569306368` ϕ'[t], 
   f[t] -> 3 E^(-10 (-9 + t)^2) - 3/2 E^(-10 (-6 + t)^2) + 
     7 E^(-10 (-3 + t)^2)}}, 0];
ode //TeXForm

$\left\{\frac{3559064 \theta ''(t)}{1525858811}+\frac{28353452 \sin (\theta (t))}{150138145}=-\frac{36921618901 \theta '(t)}{219771541077381}-\frac{159 \tan ^{-1}\left(10 \theta '(t)\right)}{200000}+\frac{184151 \left(3 e^{-10 (t-9)^2}-\frac{3}{2} e^{-10 (t-6)^2}+7 e^{-10 (t-3)^2}\right) \cos (\theta (t))}{2000000}-\frac{4587948 \phi ''(t)}{11377436311},\frac{4587948 \phi ''(t)}{11377436311}=\frac{4587948 \theta ''(t)}{11377436311}+\frac{67 \left(\frac{139127231 \theta '(t)}{35778823}+\frac{307971783 (\theta (t)-\pi )}{7213564}+\frac{79283177 \phi '(t)}{361411310}\right)}{2000}-\frac{183308 \phi '(t)}{4669626985},\theta (0)=\frac{118092245}{38197867},\theta '(0)=0,\phi (0)=0,\phi '(0)=0\right\}$

Now, there are no machine numbers polluting your equations. So, use NDSolveValue with a WorkingPrecision option:

sol = NDSolveValue[ode, {θ[t], ϕ'[t]}, {t, 0, 14}, WorkingPrecision->20];

N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating 82222031699886430677801536777088869542973260569651374914500 E^1260 π.

N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating 82222031699886430677801536777088869542973260569651374914500 E^1260 π.

N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating 82222031699886430677801536777088869542973260569651374914500 E^1260 π.

General::stop: Further output of N::meprec will be suppressed during this calculation.

Visualization:

Plot[sol, {t, 0, 14}]