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https://github.com/microno95/desolver

A Python library for solving Initial Value Problems using various numerical integration methods.
https://github.com/microno95/desolver

initial-value-problem numerical-integrators numerical-methods numpy ode ordinary-differential-equations python pytorch

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A Python library for solving Initial Value Problems using various numerical integration methods.

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README 

        

DESolver

========



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This is a python package for solving Initial Value Problems using various numerical integrators.

Many integration routines are included ranging from fixed step to symplectic to adaptive integrators.



Documentation

=============



Documentation is now available at `desolver docs `_! This will be updated with new examples as they are written, currently the examples show the use of ``pyaudi``.



Latest Release

==============



**4.2.0** - Improved performance of implicit methods, added embedded implicit methods following Kroulíková (2017) for fully implicit adaptive integration.



**4.1.0** - Initial release of implicit integration schemes that use a basic newton-raphson algorithm to solve for the intermediate states.



**3.0.0** - PyAudi support has been finalised. It is now possible to do numerical integrations using ``gdual`` variables such as ``gdual_double``\ , ``gdual_vdouble`` and ``gdual_real128`` (only on select platforms, refer to `pyaudi docs `_ for more information). Install desolver with pyaudi support using ``pip install desolver[pyaudi]``. Documentation has also been added and is available at `desolver docs `_.



**2.5.0** - Event detection has been added to the module. It is now possible to do numerical integration with terminal and non-terminal events.



**2.2.0** - PyTorch backend is now implemented. It is now possible to numerically integrate a system of equations that use pytorch tensors and then compute gradients from these.



Use of PyTorch backend requires installation of PyTorch from `here `_.



To Install:

===========



Just type



``pip install desolver``



Implemented Integration Methods

-------------------------------



Explicit Methods

~~~~~~~~~~~~~~~~



Adaptive Methods

^^^^^^^^^^^^^^^^



#. Runge-Kutta 14(12) (Feagin, 2009)

#. Runge-Kutta 10(8) (Feagin, 2009)

#. Runge-Kutta 8(7) (Dormand & Prince, 1980)

#. Runge-Kutta 4(5) with Cash-Karp Coefficients

#. Adaptive Heun-Euler Method



Fixed Step Methods

^^^^^^^^^^^^^^^^^^



#. Symplectic BABs9o7H Method  (Mads & Nielsen, 2015, BAB's9o7H)

#. Symplectic ABAs5o6HA Method (Mads & Nielsen, 2015, ABAs5o6H)

#. Runge-Kutta 5 - The 5th order integrator from RK45 with Cash-Karp Coefficients.

#. Runge-Kutta 4 - The classic RK4 integrator

#. Midpoint Method

#. Heun's Method

#. Euler's Method

#. Euler-Trapezoidal Method



Implicit Methods [\ **NEW**\ ]

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~



Adaptive Methods

^^^^^^^^^^^^^^^^



#. Lobatto IIIC 4(2) (Kroulíková, 2017)

#. Radau IIA 5(2) (Kroulíková, 2017)



Fixed Step Methods

^^^^^^^^^^^^^^^^^^



#. Backward Euler

#. Implicit Midpoint

#. Crank-Nicolson

#. Lobatto IIIA 2

#. Lobatto IIIB 2

#. Lobatto IIIC 2

#. Radau IA 3

#. Radau IIA 3

#. Lobatto IIIA 4

#. Lobatto IIIB 4

#. Gauss-Legendre 4

#. Radau IA 5

#. Radau IIA  6



Minimal Working Example

=======================



This example shows the integration of a harmonic oscillator using DESolver.



.. code-block:: python



   import desolver as de

   import desolver.backend as D



   def rhs(t, state, k, m, **kwargs):

       return D.array([[0.0, 1.0], [-k/m,  0.0]])@state



   y_init = D.array([1., 0.])



   a = de.OdeSystem(rhs, y0=y_init, dense_output=True, t=(0, 2*D.pi), dt=0.01, rtol=1e-9, atol=1e-9, constants=dict(k=1.0, m=1.0))



   print(a)



   a.integrate()



   print(a)



   print("If the integration was successful and correct, a[0].y and a[-1].y should be near identical.")

   print("a[0].y  = {}".format(a[0].y))

   print("a[-1].y = {}".format(a[-1].y))



   print("Maximum difference from initial state after one oscillation cycle: {}".format(D.max(D.abs(a[0].y-a[-1].y))))



References

==========



Feagin, T. (2009). High-Order Explicit Runge-Kutta Methods. Retrieved from `https://sce.uhcl.edu/rungekutta/ `_



Dormand, J. R. and Prince, P. J. (1980) A family of embedded Runge-Kutta formulae. *Journal of Computational and Applied Mathematics*, 6(1), 19-26. `https://doi.org/10.1016/0771-050X(80)90013-3 `_



Mads, K. and Nielsen, E. (2015). *Efficient fourth order symplectic integrators for near-harmonic separable Hamiltonian systems*. Retrieved from `https://arxiv.org/abs/1501.04345 `_



Kroulíková, T. (2017). RUNGE-KUTTA METHODS (Master's thesis, BRNO UNIVERSITY OF TECHNOLOGY, Brno, Czechia). Retrieved from `https://www.vutbr.cz/www_base/zav_prace_soubor_verejne.php?file_id=174714 `_