Finding NDSolve method details












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









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    active

    oldest

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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]


    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



        share|improve this answer










        answered 1 hour ago









        zhkzhk

        10k11433




        10k11433






























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