What's the theory behind ode14x solver?

I am trying to understand the theory behind the various solvers available in Simulink.

However, while for the majority of them several references and details are given, I'm not finding detailed and clear information about how ode14x works. The help states: "The ode14x solver uses a combination of Newton's method and extrapolation from the current value to compute the model state as an implicit function of the state and the state derivative at the next time step."

Where can I find precise details on the algorithm. How is the extrapolation used in the context and combination of an implicit method? Furthermore, what particular implicit method is used?

Thank you.

Answer 1

[Harsh Saxena]

Hi Antonio,

Two things that the documentation clearly mentions is the use of extrapolation and implicit method.

Extrapolation basically means using the known data in present to compute the future data. We can find the equation given in help

X(n+1) - X(n)- h dX(n+1) = 0 ( https://www.mathworks.com/help/simulink/gui/solver.html)

We can clearly see the use of extrapolation here as the general formula of extrapolation is: y(n+1) = y(n) + slope*(y(n+1) - y(n)).

Further, we can see that the function is implicit function because it is using a function of next state to calculate next state, x(n+1) = x(n) + f(x(n+1)). Which is the definition of implicit methods.

I am not sure the exact implicit method that is used by 'ode14x'. The documentation doesn’t helps that much in this case.

Hope this helps!

[Antonio Carlo Bertolino]

Hi Harsh,

yes, this is what the online documentation writes. The equation that you wrote is basically the Backward Newton method, which is implicit.

I was looking for some more details actually. For example, for the other solvers the paper "The MATLAB ODE suite" by Shampine et al. gives a lor of information (https://www.mathworks.com/help/pdf_doc/otherdocs/ode_suite.pdf).

I would like something like that also for the ode14x.

Thank you again for your answer!