DIRK2

class pathsim.solvers.dirk2.DIRK2(*solver_args, **solver_kwargs)

Bases: DiagonallyImplicitRungeKutta

Two-stage, 2nd order Diagonally Implicit Runge-Kutta (DIRK) method.

This specific method is SSP-optimal (largest radius of absolute monotonicity for a 2-stage, 2nd order DIRK), symplectic, and A-stable. It’s a robust choice for moderately stiff problems where second-order accuracy is sufficient.

Characteristics

• Order: 2

• Stages: 2 (Implicit)

• Implicit (DIRK)

• Fixed timestep only

• A-stable, SSP-optimal, Symplectic

When to Use

• Moderately stiff problems: Good entry-level implicit method for stiff ODEs

• SSP requirements with stiffness: Combines strong stability preservation with A-stability

• Symplectic integration: Preserves geometric structure in Hamiltonian systems

• Low-order implicit needs: When 2nd order implicit accuracy is sufficient

Trade-off: Lower order than ESDIRK or BDF methods but has SSP and symplectic properties. For higher accuracy on stiff problems, consider ESDIRK43 or BDF methods.

References

[1]

Ferracina, L., & Spijker, M. N. (2008). “Strong stability of singly-diagonally- implicit Runge-Kutta methods”. Applied Numerical Mathematics, 58(11), 1675-1686.

[2]

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