pathsim.solvers.euler module

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

Bases: ExplicitSolver

Explicit Forward Euler (FE) integration method.

This is the simplest explicit numerical integration method. It is first-order accurate (\(O(h)\)) and generally not suitable for stiff problems due to its limited stability region.

Method:

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

Characteristics

• Order: 1

• Stages: 1

• Explicit

• Fixed timestep only

• Not A-stable

• Low accuracy and stability, but computationally very cheap.

When to Use

• Educational purposes: Ideal for teaching basic numerical integration concepts

• Very smooth problems: When the function is extremely smooth and well-behaved

• Rapid prototyping: Quick initial testing before applying more sophisticated methods

• Resource-constrained scenarios: When computational cost must be minimized

Not recommended for production use, stiff problems, or when accuracy is important.

References

[1]

Euler, L. (1768). “Institutionum calculi integralis”. Impensis Academiae Imperialis Scientiarum, Vol. 1.

[2]

Butcher, J. C. (2016). “Numerical Methods for Ordinary Differential Equations”. John Wiley & Sons, 3rd Edition.

[3]

Hairer, E., Nørsett, S. P., & Wanner, G. (1993). “Solving Ordinary Differential Equations I: Nonstiff Problems”. Springer Series in Computational Mathematics, Vol. 8.

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 (BE) integration method.

This is the simplest implicit numerical integration method. It is first-order accurate (\(O(h)\)) and is A-stable and L-stable, making it suitable for very stiff problems where stability is paramount, although its low order limits accuracy for non-stiff problems or when high precision is required.

Method:

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

This implicit equation is solved iteratively using the internal optimizer.

Characteristics

• Order: 1

• Stages: 1 (Implicit)

• Implicit

• Fixed timestep only

• A-stable, L-stable

• Very stable, suitable for stiff problems, but low accuracy.

When to Use

• Highly stiff problems: Excellent stability for very stiff ODEs

• Robustness over accuracy: When stability is more critical than precision

• Long-time integration: For simulations over very long time periods where stability matters

• Initial testing of stiff systems: Simple method to verify problem setup

Trade-off: Sacrifices accuracy for exceptional stability. For higher accuracy on stiff problems, consider BDF or ESDIRK methods.

References

[1]

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

[2]

Hairer, E., & Wanner, G. (1996). “Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems”. Springer Series in Computational Mathematics, Vol. 14.

[3]

Butcher, J. C. (2016). “Numerical Methods for Ordinary Differential Equations”. John Wiley & Sons, 3rd Edition.

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