ODE schemes: a list of schemes to be implemented

Just a list of schemes that we would like to be implemented into the library:
- [x] explicit schemes:
- [x] Adams-Bashforth schemes see [7]:
- [x] 1 step, namely the forward explicit Euler scheme, 1st order accurate;
- [x] 2 steps, 2nd accurate;
- [x] 3 steps, 3rd accurate;
- [x] 4 steps, 4th accurate;
- [x] Euler scheme, 1st order accurate;
- [x] Leapfrog, 2nd order accurate:
- [x] unfiltered leapfrog, 2nd order accurate, mostly unstable, see [4];
- [x] Robert-Asselin filtered leapfrog, 1st order accurate, see [4, 5, 6];
- [x] Robert-Asselin-Williams filtered leapfrog, 3rd order accurate, see [5, 6];
- [ ] Runge-Kutta schemes:
- [ ] embedded (adaptive) schemes:
- [x] Heun-Euler, 2 stages, 2nd order accurate;
- [ ] Runge-Kutta-Fehlberg, 5 stages, 4th order accurate, see [8];
- [x] Cash-Karp, 6 stages, 4th order accurate, see[14];
- [x] Prince-Dormand, 7 stages, 4th order accurate, see[10];
- [x] Calvo et al., 9 stages, 6th order accurate, see[13];
- [x] Feagin, 17 stages, 10th order accurate, see[15];
- [ ] low-storage schemes, see [1, 2, 3]:
- [x] 1 stage, namely the forward explicit Euler scheme, 1st order accurate;
- [ ] 2 stages;
- [ ] 3 stages;
- [ ] 4 stages;
- [x] 5 stages, 4th order accurate, 2N registers, see [3];
- [x] 6 stages, 4th order accurate, 2N registers, see [11];
- [x] 7 stages, 4th order accurate, 2N registers, see [11];
- [x] 12 stages, 4th order accurate, 2N registers, see [12];
- [x] 13 stages, 4th order accurate, 2N registers, see [12];
- [x] 14 stages, 4th order accurate, 2N registers, see [12];
- [ ] TVD or SSP schemes, see [1]:
- [x] 1 stage, namely the forward explicit Euler scheme, 1st order accurate;
- [x] 2 stages, 2nd order accurate;
- [x] 3 stages, 3rd order accurate;
- [ ] 4 stages;
- [x] 5 stages, 4th order accurate;
- [ ] implicit schemes:
- [x] Adams-Moulton schemes:
- [x] 0 step, namely the backward implicit Euler scheme, 1st order accurate;
- [x] 1 step, 2nd accurate;
- [x] 2 steps, 3rd accurate;
- [x] 3 steps, 4th accurate;
- [ ] predictor-corrector schemes:
- [x] Adams-Bashforth-Moulton schemes:
- [x] 1 step, AB(1)-AM(0), 1st order accurate;
- [x] 2 steps, AB(2)-AM(1), 2nd accurate;
- [x] 3 steps, AB(3)-AM(2), 3rd accurate;
- [x] 4 steps, AB(4)-AM(3), 4th accurate;
- [ ] Runge-Kutta schemes;
#### Bibliography

[1] _High Order Strong Stability Preserving Time Discretizations_, Gottlieb, S., Ketcheson, D. I., Shu, C.W., Journal of Scientific Computing, vol. 38, N. 3, 2009, pp. 251-289.

[2] _Low-Storage Runge-Kutta Schemes_, J. H. Williamson, Journal of Computational Physics, vol. 35, 1980, pp. 48--56.

[3] _Fourth-Order 2N-Storage Runge-Kutta Schemes_, Mark H. Carpenter, Christopher A. Kennedy, NASA Technical Memorandum 109112, June 1994.

[4] Mesinger F. and A. Arakawa, 1976: Numerical methods used in atmospheric models. Global _Atmospheric Research Programme (GARP) Technical Report_. http://twister.ou.edu/CFD2003/Mesinger_ArakawaGARP.pdf

[5] Williams, P. D., 2009: A Proposed Modification to the Robert–Asselin Time Filter. _Mon. Wea. Rev._, 137, 2538–2546. doi: http://dx.doi.org/10.1175/2009MWR2724.1

[6] _The RAW filter: An improvement to the Robert–Asselin filter in semi-implicit integrations_, Williams, P.D., Monthly Weather Review, vol. 139(6), pages 1996--2007, June 2011.

[7] Linear multistep method, [wikipedia article](https://en.wikipedia.org/wiki/Linear_multistep_method)

[8] _Low-order classical Runge-Kutta formulas with step size control and their application to some heat transfer problems_, Erwin Fehlberg (1969), NASA Technical Report 315.

[9] _A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides_, J. R. Cash, A. H. Karp, ACM Transactions on Mathematical Software 16: 201-222, 1990. doi:10.1145/79505.79507

[10] _A family of embedded Runge-Kutta formulae_, Dormand, J. R.; Prince, P. J. (1980), , Journal of Computational and Applied Mathematics 6 (1): 19--26, doi:10.1016/0771-050X(80)90013-3

[11] _High-accuracy large-step explicit Runge–Kutta (HALE-RK) schemes for computational aeroacoustics_, Vasanth Allampalli and Ray Hixon and M. Nallasamy and Scott D. Sawyer, Journal of Computational Physics, vol. 228, 2009, pp. 3837--3850.

[12] _Efficient low-storage Runge–Kutta schemes with optimized stability regions_, Jens Niegemann and Richard Diehl and Kurt Busch, Journal of Computational Physics, vol. 231, 2012, pp. 364--372.

[13] _A New Embedded Pair of Runge-Kutta Formulas of orders 5 and 6_, M. Calvo, J.I. Montijano, L. Randez, Computers & Mathematics with Applications, Volume 20, Issue 1, 1990, Pages 15--24, ISSN 0898-1221, http://dx.doi.org/10.1016/0898-1221(90)90064-Q.

[14] _A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides_, J. R. Cash, A. H. Karp, ACM Transactions on Mathematical Software, vol. 16, pp. 201--222, 1990, doi:10.1145/79505.79507.

[15] _A tenth-order Runge-Kutta method with error estimate_, Feagin, T., Proceedings of the IAENG Conf. on Scientific Computing. 2007.