ESDIRK85

class pathsim.solvers.esdirk85.ESDIRK85(*solver_args, **solver_kwargs)

Bases: DiagonallyImplicitRungeKutta

Sixteen-stage, 8th order L-stable Embedded Singly Diagonally Implicit Runge-Kutta method.

Features an embedded 5th order method for adaptive step size control. The first stage is explicit. Designed for very stiff problems requiring very high accuracy. Computationally expensive due to 16 stages, but can take very large timesteps with tight tolerances. This is the ESDIRK(16,8)[2]SAL-[(16,5)] method.

Characteristics

• Order: 8

• Embedded Order: 5

• Stages: 16 (1 Explicit, 15 Implicit)

• Implicit (ESDIRK)

• Adaptive timestep

• L-stable, Stiffly accurate

When to Use

• Extremely high accuracy on stiff problems: When very tight tolerances are essential

• Expensive right-hand sides: When large timesteps justify the 16 stages

• Benchmark computations: Reference solutions for stiff problems

• Specialized applications: Only when 8th order accuracy is truly needed

Warning: Very expensive (16 implicit stages). Only use when extremely high accuracy is essential and large timesteps are possible. For most applications, ESDIRK54 is more practical.

References

[1]

Alamri, Y., & Ketcheson, D. I. (2019). “Very high-order A-stable stiffly accurate diagonally implicit Runge-Kutta methods with error estimators”. arXiv preprint arXiv:1905.11370.

[2]

Kennedy, C. A., & Carpenter, M. H. (2019). “Diagonally implicit Runge-Kutta methods for stiff ODEs”. Applied Numerical Mathematics, 146, 221-244.

[3]

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