Finding NDSolve method details
4
$begingroup$
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 Mar 24 at 4:14
xzczd
27.5k574256
asked Mar 24 at 1:09
sana alharbisana alharbi
456
$endgroup$
|
show 7 more comments
4
$begingroup$
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 Mar 24 at 4:14
xzczd
27.5k574256
asked Mar 24 at 1:09
sana alharbisana alharbi
456
$endgroup$
2
$begingroup$
Partial duplicate: mathematica.stackexchange.com/questions/145/…
$endgroup$
– Michael E2
Mar 24 at 1:17
1
$begingroup$
Another partial duplicate: mathematica.stackexchange.com/questions/102704/…
$endgroup$
– Michael E2
Mar 24 at 1:37
1
$begingroup$
You say you don't understand some technique or other, nor the output of yourTracecommand. 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
Mar 24 at 1:44
2
$begingroup$
"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
Mar 24 at 3:39
3
$begingroup$
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
Mar 24 at 3:59
|
show 7 more comments
4
4
4
0
$begingroup$
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 Mar 24 at 4:14
xzczd
27.5k574256
asked Mar 24 at 1:09
sana alharbisana alharbi
456
$endgroup$
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 Mar 24 at 4:14
xzczd
27.5k574256
asked Mar 24 at 1:09
sana alharbisana alharbi
456
edited Mar 24 at 4:14
xzczd
27.5k574256
asked Mar 24 at 1:09
sana alharbisana alharbi
456
edited Mar 24 at 4:14
xzczd
27.5k574256
edited Mar 24 at 4:14
xzczd
27.5k574256
edited Mar 24 at 4:14
xzczd
27.5k574256
27.5k574256
asked Mar 24 at 1:09
sana alharbisana alharbi
456
asked Mar 24 at 1:09
sana alharbisana alharbi
456
asked Mar 24 at 1:09
sana alharbisana alharbi
456
456
2
$begingroup$
Partial duplicate: mathematica.stackexchange.com/questions/145/…
$endgroup$
– Michael E2
Mar 24 at 1:17
1
$begingroup$
Another partial duplicate: mathematica.stackexchange.com/questions/102704/…
$endgroup$
– Michael E2
Mar 24 at 1:37
1
$begingroup$
You say you don't understand some technique or other, nor the output of yourTracecommand. 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
Mar 24 at 1:44
2
$begingroup$
"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
Mar 24 at 3:39
3
$begingroup$
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
Mar 24 at 3:59
|
show 7 more comments
2
$begingroup$
Partial duplicate: mathematica.stackexchange.com/questions/145/…
$endgroup$
– Michael E2
Mar 24 at 1:17
1
$begingroup$
Another partial duplicate: mathematica.stackexchange.com/questions/102704/…
$endgroup$
– Michael E2
Mar 24 at 1:37
1
$begingroup$
You say you don't understand some technique or other, nor the output of yourTracecommand. 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
Mar 24 at 1:44
2
$begingroup$
"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
Mar 24 at 3:39
3
$begingroup$
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
Mar 24 at 3:59
2
2
$begingroup$
Partial duplicate: mathematica.stackexchange.com/questions/145/…
$endgroup$
– Michael E2
Mar 24 at 1:17
$begingroup$
Partial duplicate: mathematica.stackexchange.com/questions/145/…
$endgroup$
– Michael E2
Mar 24 at 1:17
1
1
$begingroup$
Another partial duplicate: mathematica.stackexchange.com/questions/102704/…
$endgroup$
– Michael E2
Mar 24 at 1:37
$begingroup$
Another partial duplicate: mathematica.stackexchange.com/questions/102704/…
$endgroup$
– Michael E2
Mar 24 at 1:37
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
Mar 24 at 1:44
$begingroup$
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
Mar 24 at 1:44
2
2
$begingroup$
"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
Mar 24 at 3:39
$begingroup$
"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
Mar 24 at 3:39
3
3
$begingroup$
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
Mar 24 at 3:59
$begingroup$
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
Mar 24 at 3:59
|
show 7 more comments
1 Answer
1
active
oldest
votes
6
$begingroup$
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 Mar 24 at 5:09
zhkzhk
10.1k11533
$endgroup$
$begingroup$
Thank you so much @zhk but how can defend the axes for both figures? the first as x represented steps and y residual error. the second one x represents the t time and y residual error. sorry if my question is trivial but it is first time to see the code
$endgroup$
– sana alharbi
Mar 24 at 9:13
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
6
$begingroup$
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 Mar 24 at 5:09
zhkzhk
10.1k11533
$endgroup$
$begingroup$
Thank you so much @zhk but how can defend the axes for both figures? the first as x represented steps and y residual error. the second one x represents the t time and y residual error. sorry if my question is trivial but it is first time to see the code
$endgroup$
– sana alharbi
Mar 24 at 9:13
add a comment |
6
$begingroup$
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 Mar 24 at 5:09
zhkzhk
10.1k11533
$endgroup$
$begingroup$
Thank you so much @zhk but how can defend the axes for both figures? the first as x represented steps and y residual error. the second one x represents the t time and y residual error. sorry if my question is trivial but it is first time to see the code
$endgroup$
– sana alharbi
Mar 24 at 9:13
add a comment |
6
6
6
$begingroup$
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 Mar 24 at 5:09
zhkzhk
10.1k11533
$endgroup$
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 Mar 24 at 5:09
zhkzhk
10.1k11533
answered Mar 24 at 5:09
zhkzhk
10.1k11533
answered Mar 24 at 5:09
zhkzhk
10.1k11533
answered Mar 24 at 5:09
zhkzhk
10.1k11533
10.1k11533
$begingroup$
Thank you so much @zhk but how can defend the axes for both figures? the first as x represented steps and y residual error. the second one x represents the t time and y residual error. sorry if my question is trivial but it is first time to see the code
$endgroup$
– sana alharbi
Mar 24 at 9:13
add a comment |
$begingroup$
Thank you so much @zhk but how can defend the axes for both figures? the first as x represented steps and y residual error. the second one x represents the t time and y residual error. sorry if my question is trivial but it is first time to see the code
$endgroup$
– sana alharbi
Mar 24 at 9:13
$begingroup$
Thank you so much @zhk but how can defend the axes for both figures? the first as x represented steps and y residual error. the second one x represents the t time and y residual error. sorry if my question is trivial but it is first time to see the code
$endgroup$
– sana alharbi
Mar 24 at 9:13
$begingroup$
Thank you so much @zhk but how can defend the axes for both figures? the first as x represented steps and y residual error. the second one x represents the t time and y residual error. sorry if my question is trivial but it is first time to see the code
$endgroup$
– sana alharbi
Mar 24 at 9:13
add a comment |
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2
$begingroup$
Partial duplicate: mathematica.stackexchange.com/questions/145/…
$endgroup$
– Michael E2
Mar 24 at 1:17
1
$begingroup$
Another partial duplicate: mathematica.stackexchange.com/questions/102704/…
$endgroup$
– Michael E2
Mar 24 at 1:37
1
$begingroup$
You say you don't understand some technique or other, nor the output of your
Tracecommand. 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
Mar 24 at 1:44
2
$begingroup$
"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
Mar 24 at 3:39
3
$begingroup$
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
Mar 24 at 3:59