Yes, there is a new method which is called Piecewise Analytic Method (PAM). It does more than Runge-Kutta.

1. PAM gives a general analytic formula that can be used in differentiation and integration.

2. PAM can solve highly non-linear differential equation.

3. The accuracy and error can be controlled according to our needs very easily.

4. PAM can solve problems which other famous techniques can’t solve.

5. In some cases, PAM gives the exact solution.

6. ....

You can see :

Preprint Piecewise Analytic Method VS Runge-Kutta Method (Comparative Study)

Also, You can write your comments and follow the update of PAM in the discussion

https://www.researchgate.net/post/What_do_you_know_about_Piecewise_Analytic_Method_PAM-Published_January_2012_by_T_Abassy

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