9. The higher order Runge-Kutta methods are: a.self-starting g. easy to find coe

Question

9. The higher order Runge-Kutta methods are: a.self-starting g. easy to find coefficients b. difficult to use d. unstable numerically when compared to Adams methods 10. A second-order ordinary differential equation can always be: a, solved analytically converted to a first order system of ordinary differential equations d. linearized and regularized b. c. made to be oscillatory in nature 11. The high-order Adams methods are a.self-starting c. difficult to find coefficients b. easy to change stepsize d. often susceptible to numerical instability

Explanation / Answer

9.

Note also that high-order extrapolation and deferred correction methods are Runge-Kutta methods.

If your accuracy is limited by rounding errors then you should use a higher-order method. This is because higher-order methods require fewer steps (and fewer function evaluations, even though there are more evaluations per step), so they commit fewer rounding errors. You can easily verify this yourself with simple experiments; it is a good homework problem for a first course in numerical analysis.

Tenth-order methods are extremely useful in double-precision arithmetic. On the contrary, if all we had was Euler's method, then rounding error would be a major issue and we would need very high-precision floating point numbers for many problems where high-order solvers do just fine.

There are Runge-Kutta methods of every order that are not only AA-stable, but also BB-stable (a stability property useful for some nonlinear problems). To learn about these methods, see for instance the text of Hairer & Wanner.

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