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There are many methods to solve ordinary differential equations, but what about systems of ODEs?

So far I've seen only two methods mentioned in this context: Euler's and Runge-Kutta 4th order. Both of them calculate new $y_{i,n+1}$ based only on set of $y_{*,n}$ from the previous iteration.

So are there other numerical methods to solve (possibly non-linear) systems of differential equations?

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  • $\begingroup$ Also, Spectral Methods. $\endgroup$ – NicNic8 Jun 5 '18 at 21:54
  • $\begingroup$ @RodrigodeAzevedo Non-linear with varying parameters. More specifically, $\frac{dM}{dt} = -m_{p}$, $\frac{dx}{dt} = \frac{m_{p}V_{p}}{M}$ $\endgroup$ – olegst Jun 5 '18 at 21:58
  • $\begingroup$ Well, from your question i guess you may only deal with initial value ODE at present, later for boundary value problem you'll need finite difference method,finite volume method and finite element method, lol. $\endgroup$ – J. Yu Jun 7 '18 at 6:11
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All Runge-Kutta methods, all multi-step methods can be easily extended to vector-valued problems, that is systems of ODE. Some of the order conditions for Runge-Kutta systems collapse for scalar equations, which means that the order for vector ODE may be smaller than for scalar ODE. Methods with that defect are usually not considered, you can find one famous instance, an order 5 method, in the original paper of Kutta.

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