Euler

class pathsim.solvers.euler.EUF(*solver_args, **solver_kwargs)

Bases: ExplicitSolver

Explicit forward Euler method. First-order, single-stage.

\[x_{n+1} = x_n + h \, f(x_n, t_n)\]

Characteristics

• Order: 1

• Stages: 1

• Explicit, fixed timestep

• Not A-stable

Note

The cheapest solver per step but also the least accurate. Its small stability region requires very small timesteps for moderately dynamic block diagrams, which usually makes higher-order methods more efficient overall. Prefer RK4 for fixed-step or RKDP54 for adaptive integration of non-stiff systems. Only practical when computational cost per step must be absolute minimum and accuracy is secondary.

References

[1]

Hairer, E., Nørsett, S. P., & Wanner, G. (1993). “Solving Ordinary Differential Equations I: Nonstiff Problems”. Springer Series in Computational Mathematics, Vol. 8. :doi:`10.1007/978-3-540-78862-1`

[2]

Butcher, J. C. (2016). “Numerical Methods for Ordinary Differential Equations”. John Wiley & Sons, 3rd Edition. :doi:`10.1002/9781119121534`

step(f, dt)

performs the explicit forward timestep for (t+dt) based on the state and input at (t)

Parameters:

• f (array_like) – evaluation of function

• dt (float) – integration timestep

Returns:

• success (bool) – timestep was successful

• err (float) – truncation error estimate

• scale (float) – timestep rescale from error controller

class pathsim.solvers.euler.EUB(*solver_args, **solver_kwargs)

Bases: ImplicitSolver

Implicit backward Euler method. First-order, A-stable and L-stable.

\[x_{n+1} = x_n + h \, f(x_{n+1}, t_{n+1})\]

The implicit equation is solved iteratively by the internal optimizer.

Characteristics

• Order: 1

• Stages: 1 (implicit)

• Fixed timestep

• A-stable, L-stable

Note

Maximum stability at the cost of accuracy. L-stability fully damps parasitic high-frequency modes, making this a safe fallback for very stiff block diagrams (e.g. high-gain PID loops or fast electrical dynamics coupled to slow mechanical plant). Because each step requires solving a nonlinear equation, the cost per step is higher than explicit methods. For better accuracy on stiff systems, use BDF2 (fixed-step) or ESDIRK43 (adaptive).

References

[1]

Curtiss, C. F., & Hirschfelder, J. O. (1952). “Integration of stiff equations”. Proceedings of the National Academy of Sciences, 38(3), 235-243. :doi:`10.1073/pnas.38.3.235`

[2]

Hairer, E., & Wanner, G. (1996). “Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems”. Springer Series in Computational Mathematics, Vol. 14. :doi:`10.1007/978-3-642-05221-7`

solve(f, J, dt)

Solves the implicit update equation using the internal optimizer.

Parameters:

• f (array_like) – evaluation of function

• J (array_like) – evaluation of jacobian of function

• dt (float) – integration timestep

Returns:

err – residual error of the fixed point update equation

Return type:

float