tests for system ode solvers

drudox · July 16, 2018, 08:37

Hi everybody !
I'm looking for a place (into the web of course) in which I can find a differential problem (system of ode or ode) in order to testing an ode solver that I wrote just for me ...

the best thing is to find a series of a differential problem with or without the analytical solution

please let me know !

sbaffini · July 16, 2018, 09:36

You could try with the method of manufactured solution as I don't think you will be able to find non trivial solutions for anything beyond 3 equations

drudox · July 16, 2018, 09:41

yes but for the system, I don't have any idea ... I would like to find some academic text (pdf) in which there is some exercise ... but i didn't found yet

sbaffini · July 16, 2018, 09:45

Let say that your dependent variables are yi and the independent ones are xi, for i from 1 to n.

Pick up a typical system and a solution yi(xi) you think is representative. Put the solution in the system and use the resulting unbalanced terms as source terms in the system.

That's it

drudox · July 16, 2018, 10:00

I'm sorry .. but I really didn't get the point

could you explain me with an example?

sbaffini · July 16, 2018, 10:33

Let's say your equation is:

dy/dx + y = s

and you want your solution to be, say, y=sin(x-1). Then, inserting your wanted solution in your equation, you get:

s=sin(x-1)+cos(x-1)

If you use this s and the initial condition y(0)=sin(-1), then your solution to the given problem will be y=sin(x-1).

For a system of equations you can just parameterize the solution you want on the ewuation index i.

drudox · July 16, 2018, 10:54

ok now I got the point !! thank you very much !

FMDenaro · July 16, 2018, 12:14

Many classical exercises are present in basic textbooks of numerical analysis, I suggest starting from simple test cases like the integration of linear ode, such as

dx/dt =u(x,y)
dy/dt= v(x,y)

for example using the velocity components of a circular motion. Then, you can add the time variability of the components.

July 16, 2018, 08:37	tests for system ode solvers	#1
drudox Member Marco Ghiani Join Date: Jan 2011 Posts: 35 Rep Power: 15	Hi everybody ! I'm looking for a place (into the web of course) in which I can find a differential problem (system of ode or ode) in order to testing an ode solver that I wrote just for me ... the best thing is to find a series of a differential problem with or without the analytical solution please let me know !

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July 16, 2018, 09:36		#2
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,151 Blog Entries: 29 Rep Power: 39	You could try with the method of manufactured solution as I don't think you will be able to find non trivial solutions for anything beyond 3 equations

July 16, 2018, 09:41		#3
drudox Member Marco Ghiani Join Date: Jan 2011 Posts: 35 Rep Power: 15	yes but for the system, I don't have any idea ... I would like to find some academic text (pdf) in which there is some exercise ... but i didn't found yet

July 16, 2018, 09:45		#4
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,151 Blog Entries: 29 Rep Power: 39	Let say that your dependent variables are yi and the independent ones are xi, for i from 1 to n. Pick up a typical system and a solution yi(xi) you think is representative. Put the solution in the system and use the resulting unbalanced terms as source terms in the system. That's it

July 16, 2018, 10:00		#5
drudox Member Marco Ghiani Join Date: Jan 2011 Posts: 35 Rep Power: 15	I'm sorry .. but I really didn't get the point could you explain me with an example?

July 16, 2018, 10:33		#6
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,151 Blog Entries: 29 Rep Power: 39	Let's say your equation is: dy/dx + y = s and you want your solution to be, say, y=sin(x-1). Then, inserting your wanted solution in your equation, you get: s=sin(x-1)+cos(x-1) If you use this s and the initial condition y(0)=sin(-1), then your solution to the given problem will be y=sin(x-1). For a system of equations you can just parameterize the solution you want on the ewuation index i.

July 16, 2018, 10:54		#7
drudox Member Marco Ghiani Join Date: Jan 2011 Posts: 35 Rep Power: 15	ok now I got the point !! thank you very much !

July 16, 2018, 12:14		#8
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,768 Rep Power: 71	Many classical exercises are present in basic textbooks of numerical analysis, I suggest starting from simple test cases like the integration of linear ode, such as dx/dt =u(x,y) dy/dt= v(x,y) for example using the velocity components of a circular motion. Then, you can add the time variability of the components.