Extend scipy.integrate with various methods for solve_ivp
This package extends scipy.integrate with various methods (OdeSolver classes) for the solve_ivp function.
Currently, several explicit methods (for non-stiff problems) are provided.
Three explicit Runge Kutta methods of order 5 and two variants are implemented:
BS45: efficient solver with an accurate high order interpolant by Bogacki and Shampine . The variant
BS45_ihas a free, lower order interpolant.
CK45: variable order solver by Cash and Karp, tailored to solve non-smooth problems efficiently . The variant
CK45_ois a fixed (fifth) order method with the same coefficients.
Ts45: relatively new solver (2011) by Tsitouras, optimized with fewer simplifying assumptions .
Three higher order explicit Runge Kutta methods by Prince  are implemented:
Pri6: a seventh order discrete method with fifth order error estimate, derived from a sixth order continuous method.
Pri7: an eighth order discrete method with sixth order error estimate, derived from a seventh order continuous method.
Pri8: a ninth order discrete method with seventh order error estimate, derived from an eighth order continuous method.
The numbers in the names refer to the continuous methods. These higher order methods, unlike conventional discrete methods, do not require additional function evaluations for dense output.
One multistep method is implemented:
SWAG: the variable order Adams-Bashforth-Moulton predictor-corrector method of Shampine, Gordon and Watts [5-7]. This is a translation of the Fortran code
DDEABM. Matlab's method
The initial step size estimator  is also used for all other exensisq methods.
You can install extensisq from PyPI:
pip install extensisq
Or, if you'd rather use conda:
conda install -c conda-forge extensisq
Borrowed from the the scipy documentation:
from scipy.integrate import solve_ivp from extensisq import BS45_i def exponential_decay(t, y): return -0.5 * y sol = solve_ivp(exponential_decay, [0, 10], [2, 4, 8], method=BS45_i) print(sol.t) print(sol.y)
Notice that the class
BS45_i is passed to
solve_ivp, not the string
"BS45_i". The other methods (
SWAG) can be used in a similar way.
More examples are available as notebooks:
- Duffing's equation, Bogacki Shampine method
- Non-smooth problem, Cash Karp method
- Lotka Volterra equation, all fifth order methods
- Riccati equation, higher order Prince methods
- Van der Pol's equation, Shampine Gordon Watts method
 P. Bogacki, L.F. Shampine, "An efficient Runge-Kutta (4,5) pair", Computers & Mathematics with Applications, Vol. 32, No. 6, 1996, pp. 15-28, ISSN 0898-1221. https://doi.org/10.1016/0898-1221(96)00141-1
 J. R. Cash, A. H. Karp, "A Variable Order Runge-Kutta Method for Initial Value Problems with Rapidly Varying Right-Hand Sides", ACM Trans. Math. Softw., Vol. 16, No. 3, 1990, pp. 201-222, ISSN 0098-3500. https://doi.org/10.1145/79505.79507
 Ch. Tsitouras, "Runge-Kutta pairs of order 5(4) satisfying only the first column simplifying assumption", Computers & Mathematics with Applications, Vol. 62, No. 2, pp. 770 - 775, 2011. https://doi.org/10.1016/j.camwa.2011.06.002
 P.J. Prince, "Parallel Derivation of Efficient Continuous/Discrete Explicit Runge-Kutta Methods", Guisborough TS14 6NP U.K., September 6 2018. http://www.peteprince.co.uk/parallel.pdf
 L.F. Shampine and M.K. Gordon, "Computer solution of ordinary differential equations: The initial value problem", San Francisco, W.H. Freeman.
 H.A. Watts and L.F. Shampine, "Smoother Interpolants for Adams Codes", SIAM Journal on Scientific and Statistical Computing, 1986, Vol. 7, No. 1, pp. 334-345. ISSN 0196-5204. https://doi.org/10.1137/0907022
 H.A. Watts, "Starting step size for an ODE solver", Journal of Computational and Applied Mathematics, Vol. 9, No. 2, 1983, pp. 177-191, ISSN 0377-0427. https://doi.org/10.1016/0377-0427(83)90040-7
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