Finding NDSolve method detailsHow 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

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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











$endgroup$







  • 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















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











$endgroup$







  • 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













2












2








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











$endgroup$




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




differential-equations implementation-details






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited 2 hours ago









xzczd

27.4k573254




27.4k573254










asked 5 hours ago









sana alharbisana alharbi

356




356







  • 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












  • 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







2




2




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




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




1




1




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




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




1




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




$begingroup$
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




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




$begingroup$
"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




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




$begingroup$
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










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]


enter image description here



With[data = Table[t, Abs@residuals[[1]] /. s, t, N1["Coordinates"] /. s // Flatten], 
 ListLogPlot[data, Frame -> True, PlotRange -> All]]


enter image description here



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






share|improve this answer









$endgroup$












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    1 Answer
    1






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    oldest

    votes









    active

    oldest

    votes






    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]


    enter image description here



    With[data = Table[t, Abs@residuals[[1]] /. s, t, N1["Coordinates"] /. s // Flatten], 
     ListLogPlot[data, Frame -> True, PlotRange -> All]]


    enter image description here



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






    share|improve this answer









    $endgroup$

















      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]


      enter image description here



      With[data = Table[t, Abs@residuals[[1]] /. s, t, N1["Coordinates"] /. s // Flatten], 
       ListLogPlot[data, Frame -> True, PlotRange -> All]]


      enter image description here



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






      share|improve this answer









      $endgroup$















        3












        3








        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]


        enter image description here



        With[data = Table[t, Abs@residuals[[1]] /. s, t, N1["Coordinates"] /. s // Flatten], 
         ListLogPlot[data, Frame -> True, PlotRange -> All]]


        enter image description here



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






        share|improve this answer









        $endgroup$



        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]


        enter image description here



        With[data = Table[t, Abs@residuals[[1]] /. s, t, N1["Coordinates"] /. s // Flatten], 
         ListLogPlot[data, Frame -> True, PlotRange -> All]]


        enter image description here



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







        share|improve this answer












        share|improve this answer



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