While the traditional approach is to have a single numerical solver that employs a s uniform step sizing in all parts of models, the alternative, using particular solvers with particular step sizes in different segments of models with different dynamic behaviors is getting more attention. I would like to ask what methods and tools are frequently employed in research community, particularly for the cases that in which MATLAB/Simulink, modelica or Scilab/Xcos is used and realtime execution is a requirement.
if you look in the chemical engineering field you should find various examples. From memory there is a discussion of this in CW Gear's book on the numerical solution of initial value problems, and that is early to mid-1970s. Franks, Modelling and Simulation in Chemical Engineering, has a small discussion on this, and does, I think, Luyben, in his book on Process Modelling and Control. A common approach in chemical engineering is to go a step further and look to see if an equation is sufficiently fast compared to the main line that it can be treated as pseudo-equilibrium.
In all cases I don't think there is much focus on the choice of numerical algorithm - the emphasis is on recognising which equations can be partitioned, and then some are solved at one timestep, the others at the other, longer, timestep, but all using the same ODE solver. The interest is in choosing how to represent the state variables between the two systems - easiest with multistep methods, where interpolation takes place. Runge Kutta methods need more careful design to ensure that they are set up to permit interpolation, and the need for some local extrapolation.
From a practical implementation standpoint, you might consider the free Gnu Scientific Library: https://www.gnu.org/software/gsl/manual/html_node/Numerical-Integration.html . Look for "adaptive integration". Also, at a higher level, see: https://www.gnu.org/software/gsl/manual/html_node/Ordinary-Differential-Equations.html
I plan to incorporate this into our emerging MaxSim project based on Max/Msp. So far, we have employed Runge Kutta variants (RK1,2,4), but GSL looks promising.
For the course that I offered this semester, I had a close look at the Gnu Scientific Library. They cover a good range of numerical integration algorithms. I did not have the chance to try them for a real time application, but I cannot see any showstopper there. But in the scope of the question, as far as I understand, for the simulation model segments with different solvers and rates,GSL does not provide any particular feature set. But I would say, it shall be possible by manual coding.
Dear Himanshu, we also tried a similar approach using HLA (http://www.ep.liu.se/ecp/096/025/ecp14096025.pdf), but it is more favourable to my opinion, for loosely coupled system models. How do you think?
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