Discrete values from NDSolve?

Questions
,
1

NDSolve typically returns an interpolation function, which you can then build a table of values from.
However, if I only need a pre-determined set of discrete time steps, can I just directly obtain these from NDSolve without building the interpolating function?

Presumably if i do

state = NDSolve`ProcessEquations[ <stuff>, t ];
NDSolve`Iterate[state, 1 ] (* iterates from t=0 out to t=1 *)

then i already have a list of discrete values stored somewhere in “state”, and I don’t need to do

NDSolve`ProcessSolutions[state]

Any ideas on how to do this? I just want a list of values, say at t={0,0.2,.0,4,0.6,0.8,1}, for example.

2

Hi,

Try to use “EventLocator” method for NDSolve (or WhenEvent[] option).
Here is an example with “EventLocator” (for some reason values at 0. are not catched):

data = Flatten[
	Reap[
		NDSolve[
			{
			x''[s]+x[s]==0,
			x[0]==0,
			x'[0]==1
			},
			{},
			{s,0.,1.},
			Method->{
				"EventLocator",
				"Event"-> Thread[s-Table[x,{x,0.,1.,0.2}]],
				"EventAction" :> Sow[{s,x[s],x'[s]}],
				"Direction" -> 1
			}
		]
	]
]
Show[
	Plot[Sin[s],{s,0,1},Frame->True,Axes->False],
	ListPlot[Partition[data,3][[;;,1;;2]],PlotStyle->{Red,PointSize[Large]}]
]

Not sure if it is possible with NDSolve components, you can try to set EventLocator method for NDSolve`ProcessEquations with SetOptions[]

3

Here is a solution with NDSolve components:

state = First@NDSolve`ProcessEquations[{x''[s]+x[s]==0,x[0]==0,x'[0]==1},x,s];
data=Rest@Flatten@Reap@Do[
	NDSolve`Iterate[state,0.1];
	point=NDSolve`SolutionDataComponent[state@"SolutionData"["Forward"], "DependentVariables"];
	Sow[{t,point}];
	state=First@NDSolve`Reinitialize[state,Thread[{x[0],x'[0]}==point]];,
	{t,0.1,1.,0.1}
]
Show[Plot[Sin[s],{s,0,1},Frame->True,Axes->False],ListPlot[Partition[data,3][[;;,1;;2]],PlotStyle->{Red,PointSize[Large]}]]
4

Ok, thanks a lot. It turns out this is actually slower (about a factor of ten) than just Threading the interpolation function NDSolve gives you over a list of times. But I learned a lot from your example.