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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
$begingroup$
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
edited 2 hours ago
xzczd
27.4k573254
asked 5 hours ago
sana alharbisana alharbi
356
$endgroup$
|
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
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 yourTrace[]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 usedNDSolvefunction 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 inNDSolvecan be found intutorial/NDSolveExplicitRungeKutta#1456351317, then you just need to setMethod -> "ExplicitRungeKutta", "DifferenceOrder" -> 4, "Coefficients" -> ClassicalRungeKuttaCoefficients, StartingStepSize -> 1/20000, MaxSteps -> InfinityinNDSolve. 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
edited 2 hours ago
xzczd
27.4k573254
asked 5 hours ago
sana alharbisana alharbi
356
$endgroup$
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
edited 2 hours ago
xzczd
27.4k573254
asked 5 hours ago
sana alharbisana alharbi
356
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
edited 2 hours ago
xzczd
27.4k573254
27.4k573254
asked 5 hours ago
sana alharbisana alharbi
356
asked 5 hours ago
sana alharbisana alharbi
356
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 yourTrace[]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 usedNDSolvefunction 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 inNDSolvecan be found intutorial/NDSolveExplicitRungeKutta#1456351317, then you just need to setMethod -> "ExplicitRungeKutta", "DifferenceOrder" -> 4, "Coefficients" -> ClassicalRungeKuttaCoefficients, StartingStepSize -> 1/20000, MaxSteps -> InfinityinNDSolve. The solving process is slower but gives the same result as given by default.
$endgroup$
– xzczd
2 hours ago
|
show 5 more comments
2
$begingroup$
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 yourTrace[]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 usedNDSolvefunction 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 inNDSolvecan be found intutorial/NDSolveExplicitRungeKutta#1456351317, then you just need to setMethod -> "ExplicitRungeKutta", "DifferenceOrder" -> 4, "Coefficients" -> ClassicalRungeKuttaCoefficients, StartingStepSize -> 1/20000, MaxSteps -> InfinityinNDSolve. 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
|
show 5 more comments
1 Answer
1
active
oldest
votes
$begingroup$
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.
answered 1 hour ago
zhkzhk
10k11433
$endgroup$
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
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active
oldest
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active
oldest
votes
$begingroup$
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.
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.
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.
answered 1 hour ago
zhkzhk
10k11433
$endgroup$
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]
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.
answered 1 hour ago
zhkzhk
10k11433
answered 1 hour ago
zhkzhk
10k11433
answered 1 hour ago
zhkzhk
10k11433
answered 1 hour ago
zhkzhk
10k11433
10k11433
add a comment |
add a comment |
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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
NDSolvefunction 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
NDSolvecan be found intutorial/NDSolveExplicitRungeKutta#1456351317, then you just need to setMethod -> "ExplicitRungeKutta", "DifferenceOrder" -> 4, "Coefficients" -> ClassicalRungeKuttaCoefficients, StartingStepSize -> 1/20000, MaxSteps -> InfinityinNDSolve. The solving process is slower but gives the same result as given by default.$endgroup$
– xzczd
2 hours ago