Which is the efficient implicit method for non linear coupled ode's?
Hi, any system of nonlinear odes can be solved numerically by means of Runge-Kutta methods for example. In MATLAB, you can use the popular ode45 routine. Just to check could you please give us the general form of your system?
You can use pplane or ode solve for matlab
see
http://math.rice.edu/~dfield/
https://www.youtube.com/watch?v=8XILeDBwXys
or xppaut
http://www.math.pitt.edu/~bard/bardware/tut/start.html
Dear Nisha, Mathematica built-in internal function NDSolve uses arbitrary‐order implicit Runge–Kutta methods; at link below there are examples of coupled nonlinear odes systems, e.g. Lorenz's system for meteo mathematical model.
http://reference.wolfram.com/language/ref/NDSolve.html
In my point of view, the Newton-type methods can be used for any nonlinear equations. And it provides a robust and an efficient implicit suite for all problems. And the SNES library of PETSc provides a powerful suite for such problem. This is the home page of PETSc: http://www.mcs.anl.gov/petsc/
Nisha, as long it is an initial value problem, most numerical methods for solving ODEs can be used. I say most to exclude stiff equations. Fixed order, variable step methods like Runge-Kutta or variable order, variable step methods like Shampine-Gordon and so forth should all work well. The good old IMSL library has some really robust routines for these. Matlab does a reasonable job as well.
You can read the help file of MATLAB, "ODE". There are functions to solve stiff/non-stiff problems.
Actually my doubt is regarding STIFF SYSTEMS. What is the speciality that ode15s possess which ode45 doesn't have so as to use the former one to deal with stiff systems? Is it merely the difference between explicit scheme and implicit scheme which makes them different?
Suppose I have an example for the so called 'STIFF SYSTEMS. If I use both solvers ode45 and ode15s to solve it, definitely the system being stiff ode45 either takes long time to give a solution or may not give also. Other than the computation time, can I track some other parameter whose performance given by both the solvers be compared ,which tells me exactly what this stiff property means?
ode45 is RK 4th order method, but ode15s is Gear's method. They are different. ode15s, ode23tb, can be used to solve stiff problem but ode45, sometimes, cannot.
Dear Nisha
You can also use the Mathcad software (version 14, 15, etc.), which has routines specifically devoted to differential systems of stiff systems. For the analysis of finite difference schemes and the stability function formation, see: https://en.wikipedia.org/wiki/Stiff_equation.
To solve the system on long intervals (the independent variable), I recommend you an alternative solution with adaptable step, which you also find it. After solving the problem on an adaptive step routine, you can return to the constantly step routines. In any case, for the systems of differential equations susceptible to stiff system behavior (but also to others), we recommend alternative solutions for the safety of the solution (convergence, stability, convergence to the true solution, if this solution exist). Generally, it is not good to have confidence in one solution and in one only numerical solution.
With great esteem,
Cardei Petru
Most numerical packages include integrators that are designed to solve coupled systems of stiff ODEs. They usually employ implicit techniques, which means that they find solutions using Newton or other iterative methods. The Internet provides many resources and a search on "implicit ODE integrator" should bring up a lot of useful information. Usually, stiff integrators are much more efficient and robust at solving stiff systems than non-stiff integrators, with often order of magnitude reduction in execution times. For some stiff systems, non-stiff integrators may become unstable or fail all together.
My website provides a lot of free Matlab and R code for download which provides many application examples.
www.pdecomp.net
There are also detailed explanations of the mathematical techniques used and code analysis for a selection of integrators, including various BDF and NDF stiff integrators, in my book "Numerical Analysis Using R: Solutions to ODEs and PDEs", which is also detailed on my website.
I hope the above is useful.
Regards,
Graham W Griffiths
Dear Nisha, Numerical Method, Existence and Uniqueness for
Thermoelasticity System with Moving Boundary , starting on page 8. There are two numeric methods one couple and one uncoupled.
- DOI
- 10.1590/S0101-82052005000300007
Just a small comment. As you begin to understand your system more deeply, you will find ways to simplify it. E.g. you could decouple the fast from the slow processes. You could perhaps derive an analytical formula to solve the slow ones and at the same time use input from the fast ones. Many physicists and engineers do such things all the time for their specific problems. Good luck.
You can adopt MATLAB - ode 45 (R K Method of fourth order) for non-linear coupled equations. Also, ode15s and ode23tb are good options ,in case, ode45 does not work.
'predictor–corrector methods' is another option for the same.
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