desolver 4.4.1

Differential Equation System Solver

DESolver

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

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

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

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

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

5. Adaptive Heun-Euler Method

Fixed Step Methods

1. Symplectic BABs9o7H Method (Mads & Nielsen, 2015, BAB’s9o7H)

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

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

4. Runge-Kutta 4 - The classic RK4 integrator

5. Midpoint Method

6. Heun’s Method

7. Euler’s Method

8. Euler-Trapezoidal Method

Implicit Methods [NEW]

Adaptive Methods

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

2. Radau IIA 5(2) (Kroulíková, 2017)

Fixed Step Methods

1. Backward Euler

2. Implicit Midpoint

3. Crank-Nicolson

4. Lobatto IIIA 2

5. Lobatto IIIB 2

6. Lobatto IIIC 2

7. Radau IA 3

8. Radau IIA 3

9. Lobatto IIIA 4

10. Lobatto IIIB 4

11. Gauss-Legendre 4

12. Radau IA 5

13. Radau IIA 6

Minimal Working Example

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

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