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Finding NDSolve method details

How to find out which method Mathematica selected?inspecting step size and order of $tt NDSolve$What does MaxStepFraction do?How does Mathematica resolve symbolic systems of inequalities?NDSolve and strange “nonlinear coefficients problem”The idea behind Stiffness switching method with NDsolveProblems when solving a nonlinear PDE system with NDSolveSingularity treatment in a simple problemPDEs : automatic method choice : TensorProductGrid or FiniteElement?NDSolve struggling with tricky boundary conditionsNDSolve and memory usedDetails of NDSolve calling LSODA

2

$begingroup$

I have eqs about the NDSolve, I know this code given the solving automatically.

How can I find out what method is used behind the scenes? How can I gauge the reliability level, find how many iterations have been used, the order of method. How can I estimate the error?

I found hints on this site, but I still do not fully understand.

It is impossible to say NDSolve has automatically solution for publishing paper?

I used this code related to my system:

r = 0.431201; β = 2.99 *10^-6; σ = 0.7; δ = 0.57;

{m = 0.3, η = 0.1, μ = 0.1, ρ = 0.3};





S = {N1'[t] == r N1[t] (1 - β N1[t]) - η  N1[t] I1[t],

     I1'[t] == σ + (ρ  N1[t]  I1[t])/( m + N1[t]) - δ I1[t] - μ  N1[t] I1[t]};



c = {N1[0] == 1, I1[0] == 1.22};



Select[Flatten[

  Trace[

    NDSolve[{S, c}, {N1, I1}, {t, 0, 30}], 

    TraceInternal -> True]], 

  !FreeQ[#, Method | NDSolve`MethodData] &]

but I don't understand the output.

differential-equations implementation-details

share|improve this question

edited 2 hours ago

xzczd

27.4k573254

asked 5 hours ago

sana alharbisana alharbi

356

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Partial duplicate: mathematica.stackexchange.com/questions/145/…
  $endgroup$
  – Michael E2
  5 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Another partial duplicate: mathematica.stackexchange.com/questions/102704/…
  $endgroup$
  – Michael E2
  5 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  You say you don't understand some technique or other, nor the output of your Trace[] command. But the first is a very general statement about things already explained and the second is about a command that no one else can reproduce
  $endgroup$
  – Michael E2
  5 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  "It is impossible to say NDSolve has automatically solution for publishing paper. " Simply saying "I've used NDSolve function of software Mathematica" is enough in many cases, AFAIK.
  $endgroup$
  – xzczd
  3 hours ago
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Well, if the reviewer insists on such stuff, given that your system isn't that difficult, a possible workaround at this point is to choose a primary method like classical RK4 to solve the problem. The way to choose classical RK4 in NDSolve can be found in tutorial/NDSolveExplicitRungeKutta#1456351317, then you just need to set Method -> {"ExplicitRungeKutta", "DifferenceOrder" -> 4, "Coefficients" -> ClassicalRungeKuttaCoefficients}, StartingStepSize -> 1/20000, MaxSteps -> Infinity in NDSolve. The solving process is slower but gives the same result as given by default.
  $endgroup$
  – xzczd
  2 hours ago
  

  
  
  
  

 | 
show 5 more comments

2

$begingroup$

I have eqs about the NDSolve, I know this code given the solving automatically.

How can I find out what method is used behind the scenes? How can I gauge the reliability level, find how many iterations have been used, the order of method. How can I estimate the error?

I found hints on this site, but I still do not fully understand.

It is impossible to say NDSolve has automatically solution for publishing paper?

I used this code related to my system:

r = 0.431201; β = 2.99 *10^-6; σ = 0.7; δ = 0.57;

{m = 0.3, η = 0.1, μ = 0.1, ρ = 0.3};





S = {N1'[t] == r N1[t] (1 - β N1[t]) - η  N1[t] I1[t],

     I1'[t] == σ + (ρ  N1[t]  I1[t])/( m + N1[t]) - δ I1[t] - μ  N1[t] I1[t]};



c = {N1[0] == 1, I1[0] == 1.22};



Select[Flatten[

  Trace[

    NDSolve[{S, c}, {N1, I1}, {t, 0, 30}], 

    TraceInternal -> True]], 

  !FreeQ[#, Method | NDSolve`MethodData] &]

but I don't understand the output.

differential-equations implementation-details

share|improve this question

edited 2 hours ago

xzczd

27.4k573254

asked 5 hours ago

sana alharbisana alharbi

356

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Partial duplicate: mathematica.stackexchange.com/questions/145/…
  $endgroup$
  – Michael E2
  5 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Another partial duplicate: mathematica.stackexchange.com/questions/102704/…
  $endgroup$
  – Michael E2
  5 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  You say you don't understand some technique or other, nor the output of your Trace[] command. But the first is a very general statement about things already explained and the second is about a command that no one else can reproduce
  $endgroup$
  – Michael E2
  5 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  "It is impossible to say NDSolve has automatically solution for publishing paper. " Simply saying "I've used NDSolve function of software Mathematica" is enough in many cases, AFAIK.
  $endgroup$
  – xzczd
  3 hours ago
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Well, if the reviewer insists on such stuff, given that your system isn't that difficult, a possible workaround at this point is to choose a primary method like classical RK4 to solve the problem. The way to choose classical RK4 in NDSolve can be found in tutorial/NDSolveExplicitRungeKutta#1456351317, then you just need to set Method -> {"ExplicitRungeKutta", "DifferenceOrder" -> 4, "Coefficients" -> ClassicalRungeKuttaCoefficients}, StartingStepSize -> 1/20000, MaxSteps -> Infinity in NDSolve. The solving process is slower but gives the same result as given by default.
  $endgroup$
  – xzczd
  2 hours ago
  

  
  
  
  

 | 
show 5 more comments

$begingroup$

I have eqs about the NDSolve, I know this code given the solving automatically.

How can I find out what method is used behind the scenes? How can I gauge the reliability level, find how many iterations have been used, the order of method. How can I estimate the error?

I found hints on this site, but I still do not fully understand.

It is impossible to say NDSolve has automatically solution for publishing paper?

I used this code related to my system:

r = 0.431201; β = 2.99 *10^-6; σ = 0.7; δ = 0.57;

{m = 0.3, η = 0.1, μ = 0.1, ρ = 0.3};





S = {N1'[t] == r N1[t] (1 - β N1[t]) - η  N1[t] I1[t],

     I1'[t] == σ + (ρ  N1[t]  I1[t])/( m + N1[t]) - δ I1[t] - μ  N1[t] I1[t]};



c = {N1[0] == 1, I1[0] == 1.22};



Select[Flatten[

  Trace[

    NDSolve[{S, c}, {N1, I1}, {t, 0, 30}], 

    TraceInternal -> True]], 

  !FreeQ[#, Method | NDSolve`MethodData] &]

but I don't understand the output.

differential-equations implementation-details

share|improve this question

edited 2 hours ago

xzczd

27.4k573254

asked 5 hours ago

sana alharbisana alharbi

356

$endgroup$

r = 0.431201; β = 2.99 *10^-6; σ = 0.7; δ = 0.57;

{m = 0.3, η = 0.1, μ = 0.1, ρ = 0.3};





S = {N1'[t] == r N1[t] (1 - β N1[t]) - η  N1[t] I1[t],

     I1'[t] == σ + (ρ  N1[t]  I1[t])/( m + N1[t]) - δ I1[t] - μ  N1[t] I1[t]};



c = {N1[0] == 1, I1[0] == 1.22};



Select[Flatten[

  Trace[

    NDSolve[{S, c}, {N1, I1}, {t, 0, 30}], 

    TraceInternal -> True]], 

  !FreeQ[#, Method | NDSolve`MethodData] &]

share|improve this question

edited 2 hours ago

xzczd

27.4k573254

asked 5 hours ago

sana alharbisana alharbi

356

share|improve this question

edited 2 hours ago

xzczd

27.4k573254

asked 5 hours ago

sana alharbisana alharbi

356

edited 2 hours ago

xzczd

27.4k573254

edited 2 hours ago

xzczd

27.4k573254

asked 5 hours ago

sana alharbisana alharbi

356

asked 5 hours ago

sana alharbisana alharbi

356

1 Answer
1

active

oldest

votes

3

$begingroup$

Comment

In response to your question, you already got very valuable comments. I will just try to comment on

How can I estimate the error?

For this I am going to plot residual error at steps and time, which will show the reliability and accuracy of NDSolve,

r = 0.431201; [Beta] = 2.99*10^-6; [Sigma] = 0.7; [Delta] = 0.57;

m = 0.3; [Eta] = 0.1; [Mu] = 0.1; [Rho] = 0.3;   



ode = {N1'[t] == r N1[t] (1 - [Beta] N1[t]) - [Eta] N1[t] I1[t], 

   I1'[t] == [Sigma] + ([Rho] N1[t] I1[t])/(m + N1[t]) - [Delta] I1[t] - [Mu] N1[t] I1[t]};



bcs = {N1[0] == 1, I1[0] == 1.22};



residuals = ode /. Equal -> Subtract;



{s} = NDSolve[{ode, bcs}, {N1, I1}, {t, 20}, InterpolationOrder -> All];



N1["Coordinates"] /. s;



residuals /. t -> N1["Coordinates"] /. s;



ListPlot[Abs[Flatten /@ (residuals /. t -> N1["Coordinates"] /. s)], Frame -> True]

With[{data = {Table[{t, Abs@residuals[[1]]} /. s, {t, N1["Coordinates"] /. s // Flatten}]}}, 

 ListLogPlot[data, Frame -> True, PlotRange -> All]]

Note: I adopted the above from this website but unable to find the link.

share|improve this answer

answered 1 hour ago

zhkzhk

10k11433

$endgroup$

add a comment | 

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3

$begingroup$

Comment

In response to your question, you already got very valuable comments. I will just try to comment on

How can I estimate the error?

For this I am going to plot residual error at steps and time, which will show the reliability and accuracy of NDSolve,

r = 0.431201; [Beta] = 2.99*10^-6; [Sigma] = 0.7; [Delta] = 0.57;

m = 0.3; [Eta] = 0.1; [Mu] = 0.1; [Rho] = 0.3;   



ode = {N1'[t] == r N1[t] (1 - [Beta] N1[t]) - [Eta] N1[t] I1[t], 

   I1'[t] == [Sigma] + ([Rho] N1[t] I1[t])/(m + N1[t]) - [Delta] I1[t] - [Mu] N1[t] I1[t]};



bcs = {N1[0] == 1, I1[0] == 1.22};



residuals = ode /. Equal -> Subtract;



{s} = NDSolve[{ode, bcs}, {N1, I1}, {t, 20}, InterpolationOrder -> All];



N1["Coordinates"] /. s;



residuals /. t -> N1["Coordinates"] /. s;



ListPlot[Abs[Flatten /@ (residuals /. t -> N1["Coordinates"] /. s)], Frame -> True]

With[{data = {Table[{t, Abs@residuals[[1]]} /. s, {t, N1["Coordinates"] /. s // Flatten}]}}, 

 ListLogPlot[data, Frame -> True, PlotRange -> All]]

Note: I adopted the above from this website but unable to find the link.

share|improve this answer

answered 1 hour ago

zhkzhk

10k11433

$endgroup$

add a comment | 

3

$begingroup$

Comment

In response to your question, you already got very valuable comments. I will just try to comment on

How can I estimate the error?

For this I am going to plot residual error at steps and time, which will show the reliability and accuracy of NDSolve,

r = 0.431201; [Beta] = 2.99*10^-6; [Sigma] = 0.7; [Delta] = 0.57;

m = 0.3; [Eta] = 0.1; [Mu] = 0.1; [Rho] = 0.3;   



ode = {N1'[t] == r N1[t] (1 - [Beta] N1[t]) - [Eta] N1[t] I1[t], 

   I1'[t] == [Sigma] + ([Rho] N1[t] I1[t])/(m + N1[t]) - [Delta] I1[t] - [Mu] N1[t] I1[t]};



bcs = {N1[0] == 1, I1[0] == 1.22};



residuals = ode /. Equal -> Subtract;



{s} = NDSolve[{ode, bcs}, {N1, I1}, {t, 20}, InterpolationOrder -> All];



N1["Coordinates"] /. s;



residuals /. t -> N1["Coordinates"] /. s;



ListPlot[Abs[Flatten /@ (residuals /. t -> N1["Coordinates"] /. s)], Frame -> True]

With[{data = {Table[{t, Abs@residuals[[1]]} /. s, {t, N1["Coordinates"] /. s // Flatten}]}}, 

 ListLogPlot[data, Frame -> True, PlotRange -> All]]

Note: I adopted the above from this website but unable to find the link.

share|improve this answer

answered 1 hour ago

zhkzhk

10k11433

$endgroup$

add a comment | 

$begingroup$

Comment

In response to your question, you already got very valuable comments. I will just try to comment on

How can I estimate the error?

For this I am going to plot residual error at steps and time, which will show the reliability and accuracy of NDSolve,

r = 0.431201; [Beta] = 2.99*10^-6; [Sigma] = 0.7; [Delta] = 0.57;

m = 0.3; [Eta] = 0.1; [Mu] = 0.1; [Rho] = 0.3;   



ode = {N1'[t] == r N1[t] (1 - [Beta] N1[t]) - [Eta] N1[t] I1[t], 

   I1'[t] == [Sigma] + ([Rho] N1[t] I1[t])/(m + N1[t]) - [Delta] I1[t] - [Mu] N1[t] I1[t]};



bcs = {N1[0] == 1, I1[0] == 1.22};



residuals = ode /. Equal -> Subtract;



{s} = NDSolve[{ode, bcs}, {N1, I1}, {t, 20}, InterpolationOrder -> All];



N1["Coordinates"] /. s;



residuals /. t -> N1["Coordinates"] /. s;



ListPlot[Abs[Flatten /@ (residuals /. t -> N1["Coordinates"] /. s)], Frame -> True]

With[{data = {Table[{t, Abs@residuals[[1]]} /. s, {t, N1["Coordinates"] /. s // Flatten}]}}, 

 ListLogPlot[data, Frame -> True, PlotRange -> All]]

Note: I adopted the above from this website but unable to find the link.

share|improve this answer

answered 1 hour ago

zhkzhk

10k11433

$endgroup$

r = 0.431201; [Beta] = 2.99*10^-6; [Sigma] = 0.7; [Delta] = 0.57;

m = 0.3; [Eta] = 0.1; [Mu] = 0.1; [Rho] = 0.3;   



ode = {N1'[t] == r N1[t] (1 - [Beta] N1[t]) - [Eta] N1[t] I1[t], 

   I1'[t] == [Sigma] + ([Rho] N1[t] I1[t])/(m + N1[t]) - [Delta] I1[t] - [Mu] N1[t] I1[t]};



bcs = {N1[0] == 1, I1[0] == 1.22};



residuals = ode /. Equal -> Subtract;



{s} = NDSolve[{ode, bcs}, {N1, I1}, {t, 20}, InterpolationOrder -> All];



N1["Coordinates"] /. s;



residuals /. t -> N1["Coordinates"] /. s;



ListPlot[Abs[Flatten /@ (residuals /. t -> N1["Coordinates"] /. s)], Frame -> True]

With[{data = {Table[{t, Abs@residuals[[1]]} /. s, {t, N1["Coordinates"] /. s // Flatten}]}}, 

 ListLogPlot[data, Frame -> True, PlotRange -> All]]

share|improve this answer

answered 1 hour ago

zhkzhk

10k11433

answered 1 hour ago

zhkzhk

10k11433

answered 1 hour ago

zhkzhk

10k11433