Skip to main content

differential equations - Piecewise causes ndnum warning in NDSolve


Bug introduced in 8.0.4 or earlier.





I try to solve the heat transfer equation with boundary condition that depends on time:


r0 = 0.75 10^-3;(*Beam spot size, m*)
ω = π ν;
ν = 1; (*pulse repetition rate, Hz*)

c = 1710;(*Heat capacity, W/(m·K)*)
ρ = 879; (*Density, kg/m^3*)
λ = 0.111;(*Heat conductivity, W/(m·K)*)
T0 = 300;(*initial temperature,K*)

T1 = 1000 - T0; (*Hot state temperature, K*)
Rm = 3 10^-3;(*Sample Radius, m*)
zm = 2 10^-3 ;(*Sample thickness, m*)

eq = D[T[R, z, t], t] == λ/(
c ρ) (1/(R + 10^-20) D[R D[T[R, z, t], R], R] +
D[T[R, z, t], {z, 2}]);

init1 = T[R, z, 0] == T0;
bc1 = D[T[R, z, t], {R, 1}] == 0 /. R -> 0;

bc2 = D[T[R, z, t], {R, 1}] == 0 /. R -> Rm;
bc3 = T[R, z, t] ==
Piecewise[{{T0 + T1 (1 - Abs@Sin[ω t])^500,
0 <= R <= r0}, {T0, True}}] /. z -> 0;
bc4 = T[R, z, t] == T0 /. z -> zm;

sol = NDSolveValue[{eq, init1, bc1, bc2, bc3, bc4},
T[R, z, t], {R, 0, Rm}, {z, 0, zm}, {t, 0, 10},
AccuracyGoal -> 30, MaxStepFraction -> 0.05]


The piecewise function defines the temperature at the left side of the z-interval as a short square pulses coming with frequency ω to central part of the disk.


However, solver returns the error-messages:



NDSolveValue::ibcinc: Warning: boundary and initial conditions are inconsistent. NDSolveValue::ndnum: Encountered non-numerical value for a derivative at t == 0.



What's wrong? I've tried with FEM package but it produce even more error-messages :)




Comments

Popular posts from this blog

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...

plotting - Magnifying Glass on a Plot

Although there is a trick in TEX magnifying glass but I want to know is there any function to magnifying glass on a plot with Mathematica ? For example for a function as Sin[x] and at x=Pi/6 Below, this is just a picture desired from the cited site. the image got huge unfortunately I don't know how can I change the size of an image here! Answer Insetting a magnified part of the original Plot A) by adding a new Plot of the specified range xPos = Pi/6; range = 0.2; f = Sin; xyMinMax = {{xPos - range, xPos + range}, {f[xPos] - range*GoldenRatio^-1, f[xPos] + range*GoldenRatio^-1}}; Plot[f[x], {x, 0, 5}, Epilog -> {Transparent, EdgeForm[Thick], Rectangle[Sequence @@ Transpose[xyMinMax]], Inset[Plot[f[x], {x, xPos - range, xPos + range}, Frame -> True, Axes -> False, PlotRange -> xyMinMax, ImageSize -> 270], {4., 0.5}]}, ImageSize -> 700] B) by adding a new Plot within a Circle mf = RegionMember[Disk[{xPos, f[xPos]}, {range, range/GoldenRatio}]] Show...