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.
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 codeDDEABM. Matlab's methodode113is related.
Three explicit Runge Kutta methods of order 5 are implemented:
BS5: efficient fifth order method by Bogacki and Shampine [1]. Three interpolants are included: the original accurate fifth order interpolant, a lower cost fifth order one, and a 'free' fourth order one.CK5: fifth order method with the coefficients from [2], for general use.Ts5: relatively new solver (2011) by Tsitouras, optimized with fewer simplifying assumptions [3].
Three higher order explicit Runge Kutta methods by Prince [4] are implemented:
Pr7: a seventh order discrete method with fifth order error estimate, derived from a sixth order continuous method.Pr8: an eighth order discrete method with sixth order error estimate, derived from a seventh order continuous method.Pr9: a ninth order discrete method with seventh order error estimate, derived from an eighth order continuous method.
One method for a specific type of problem is available:
CKdisc: variable order solver by Cash and Karp, tailored to solve non-smooth problems efficiently [2].
The numbers in the names refer to the discrete methods, while the orders in [4] refer to the continuous methods. These methods are relatively efficient when dense output is needed, because the interpolants are free. (Other high-order methods typically need several additional function evaluations for dense output.)
The initial step size, when not supplied by you, is estimated using the method of Watts [7]. This method analyzes your problem with a few (3 to 4) evaluations and carefully estimates a safe stepsize to start the integration with.
Most of extensisq Runge Kutta methods have stiffness detection. If many steps fail, or if the integration needs a lot of steps, the power iteration method of Shampine [8] is used to test your problem for stiffness. You will get a warning if your problem is diagnosed as stiff. The kind of roots (real, complex or nearly imaginary) is also reported, such that you can select a stiff solver that better suits your problem.
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 BS5
def exponential_decay(t, y): return -0.5 * y
sol = solve_ivp(exponential_decay, [0, 10], [2, 4, 8], method=BS5)
print(sol.t)
print(sol.y)
Notice that the class BS5 is passed to solve_ivp, not the string "BS5". The other methods (CK5, CKdisc, Ts5, Pr7, Pr8, Pr9 and 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
[1] P. Bogacki, L.F. Shampine, "An efficient Runge-Kutta (4,5) pair", Computers & Mathematics with Applications, Vol. 32, No. 6, 1996, pp. 15-28. https://doi.org/10.1016/0898-1221(96)00141-1
[2] 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. https://doi.org/10.1145/79505.79507
[3] Ch. Tsitouras, "Runge-Kutta pairs of order 5(4) satisfying only the first column simplifying assumption", Computers & Mathematics with Applications, Vol. 62, No. 2, 2011, pp. 770 - 775. https://doi.org/10.1016/j.camwa.2011.06.002
[4] 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
[5] L.F. Shampine and M.K. Gordon, "Computer solution of ordinary differential equations: The initial value problem", San Francisco, W.H. Freeman, 1975.
[6] H.A. Watts and L.F. Shampine, "Smoother Interpolants for Adams Codes", SIAM Journal on Scientific and Statistical Computing, Vol. 7, No. 1, 1986, pp. 334-345. https://doi.org/10.1137/0907022
[7] H.A. Watts, "Starting step size for an ODE solver", Journal of Computational and Applied Mathematics, Vol. 9, No. 2, 1983, pp. 177-191. https://doi.org/10.1016/0377-0427(83)90040-7
[8] L.F. Shampine, "Diagnosing Stiffness for Runge–Kutta Methods", SIAM Journal on Scientific and Statistical Computing, Vol. 12, No. 2, 1991, pp. 260-272, https://doi.org/10.1137/0912015