Theseus.jl

ImplicitEuler()

The backward (implicit) Euler method: a first-order, single-stage, A-stable, and L-stable nonlinear implicit Runge-Kutta method.

The stage equation is

\[u^{n+1} = u^n + \Delta t \, f(u^{n+1},\, t^{n+1}).\]

Each time step requires solving one nonlinear system via Newton-Krylov.

ImplicitMidpoint()

The implicit midpoint method: a second-order, single-stage, A-stable nonlinear implicit Runge-Kutta method.

The stage equation evaluates $f$ at the midpoint $(u^n + u^{n+1})/2$:

\[u^{n+1} = u^n + \Delta t \, f\!\left(\frac{u^n + u^{n+1}}{2},\; t + \frac{\Delta t}{2}\right).\]

Each time step requires solving one nonlinear system via Newton-Krylov.

ImplicitTrapezoid()

The implicit trapezoidal rule (Crank–Nicolson): a second-order, single-stage, A-stable (but not L-stable) nonlinear implicit method.

The update averages the RHS at both endpoints:

\[u^{n+1} = u^n + \frac{\Delta t}{2}\left[f(u^n, t) + f(u^{n+1}, t + \Delta t)\right].\]

Each time step requires solving one nonlinear system via Newton-Krylov.

TRBDF2

TR-BDF2 based solver after [1]. Using the formula given in [2] eq (1). See [3] for how it relates to implicit RK methods

LobattoIIIA2()

A second-order, two-stage, A-stable DIRK method from the general class of Lobatto IIIA methods.

References

See Table (213) on p. 69 for the Butcher tableau.

• Christopher A. Kennedy and Mark H. Carpenter (2016) Diagonally Implicit Runge–Kutta Methods for Ordinary Differential Equations: A Review. NASA Technical Memorandum NASA/TM-2016-219173, Langley Research Center, Hampton, VA, United States.

Crouzeix32()

A third-order, two-stage, A- and L-stable diagonally implicit Runge-Kutta (DIRK) method developed by Nørsett (1974) and Crouzeix (1975). The nodes and weights are the ones of the two-point Gauss–Legendre quadrature. Thus, this method has order four when applied to quadrature problems.

References

See Table 7.2 on p. 207 for the Butcher tableau.

• Ernst Hairer, Syvert P. Nørsett, and Gerhard Wanner (1993) Solving Ordinary Differential Equations I: Nonstiff Problems. Springer Series in Computational Mathematics, 2nd edition. DOI: 10.1007/978-3-540-78862-1

DIRK43()

A fourth-order, three-stage, A-stable DIRK method.

References

• D. Fränken and Karlheinz Ochs (2003) Passive Runge–Kutta Methods—Properties, Parametric Representation, and Order Conditions. BIT Numerical Mathematics 43(2):339–361. DOI: 10.1023/A:1026039820006

SP111()

The symplectic Euler method, a first-order, one-stage type I IMEX method combining an explicit and an implicit Euler method.

References

• Sebastiano Boscarino and Giovanni Russo (2024) Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review. DOI: 10.1007/s40324-024-00351-x

H222()

A second-order, two-stage type I IMEX method. The explicit part is strong stability preserving (SSP), the implicit part is A-stable but not L-stable.

References

• Sebastiano Boscarino, Lorenzo Pareschi, and Giovanni Russo (2025) Implicit-explicit methods for evolutionary partial differential equations. DOI: 10.1137/1.9781611978209

SSP2222()

A second-order, two-stage type I IMEX method developed by Pareschi and Russo (2005). The explicit part is strong stability preserving (SSP), the implicit part is L-stable.

References

• Lorenzo Pareschi and Giovanni Russo (2005) Implicit–Explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation. Journal of Computational Physics 203(2):469–491. DOI: 10.1007/s10915-004-4636-4
• Sebastiano Boscarino and Giovanni Russo (2024) Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review. DOI: 10.1007/s40324-024-00351-x

SSP2322()

A second-order, three-stage type I IMEX method developed by Pareschi and Russo (2005). The explicit part is strong stability preserving (SSP), the implicit part is stiffly accurate (SA) and thus L-stable.

References

• Lorenzo Pareschi and Giovanni Russo (2005) Implicit–Explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation. Journal of Computational Physics 203(2):469–491. DOI: 10.1007/s10915-004-4636-4

SSP2332()

A second-order, three-stage type I IMEX method developed by Pareschi and Russo (2005). The explicit part is strong stability preserving (SSP), the implicit part is stiffly accurate (SA) and thus L-stable.

References

• Lorenzo Pareschi and Giovanni Russo (2005) Implicit–Explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation. Journal of Computational Physics 203(2):469–491. DOI: 10.1007/s10915-004-4636-4
• Sebastiano Boscarino and Giovanni Russo (2024) Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review. DOI: 10.1007/s40324-024-00351-x

SSP3332()

A second-order, three-stage type I IMEX method developed by Pareschi and Russo (2005). The explicit part is strong stability preserving (SSP) and third-order accurate, the implicit part is L-stable.

References

• Lorenzo Pareschi and Giovanni Russo (2005) Implicit–Explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation. Journal of Computational Physics 203(2):469–491. DOI: 10.1007/s10915-004-4636-4

SSP3433()

A third-order, four-stage type I IMEX method developed by Pareschi and Russo (2005). The explicit part is strong stability preserving (SSP) and third-order accurate, the implicit part is L-stable.

References

• Lorenzo Pareschi and Giovanni Russo (2005) Implicit–Explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation. Journal of Computational Physics 203(2):469–491. DOI: 10.1007/s10915-004-4636-4
• Sebastiano Boscarino and Giovanni Russo (2024) Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review. DOI: 10.1007/s40324-024-00351-x

HT222()

A second-order, two-stage type II IMEX method. The explicit part is strong stability preserving (SSP), the implicit part is A-stable but not L-stable.

References

• Sebastiano Boscarino, Lorenzo Pareschi, and Giovanni Russo (2025) Implicit-explicit methods for evolutionary partial differential equations. DOI: 10.1137/1.9781611978209

ARS111()

A first-order, effectively one-stage, globally stiffly accurate (GSA) type II IMEX method developed by Ascher, Ruuth, and Spiteri (1997).

References

• Uri M. Ascher, Steven J. Ruuth, and Raymond J Spiteri (1997) Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. DOI: 10.1016/S0168-9274(97)00056-1
• Sebastiano Boscarino and Giovanni Russo (2024) Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review. DOI: 10.1007/s40324-024-00351-x
• Sebastiano Boscarino, Lorenzo Pareschi, and Giovanni Russo (2025) Implicit-explicit methods for evolutionary partial differential equations. DOI: 10.1137/1.9781611978209

ARS222()

A second-order, effectively two-stage, globally stiffly accurate (GSA) type II IMEX method developed by Ascher, Ruuth, and Spiteri (1997).

References

• Uri M. Ascher, Steven J. Ruuth, and Raymond J Spiteri (1997) Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. DOI: 10.1016/S0168-9274(97)00056-1
• Sebastiano Boscarino and Giovanni Russo (2024) Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review. DOI: 10.1007/s40324-024-00351-x
• Sebastiano Boscarino, Lorenzo Pareschi, and Giovanni Russo (2025) Implicit-explicit methods for evolutionary partial differential equations. DOI: 10.1137/1.9781611978209

ARS233()

A third-order, effectively three-stage type II IMEX method developed by Ascher, Ruuth, and Spiteri (1997). The implicit part is A-stable but not L-stable.

References

• Uri M. Ascher, Steven J. Ruuth, and Raymond J Spiteri (1997) Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. DOI: 10.1016/S0168-9274(97)00056-1
• Sebastiano Boscarino, Lorenzo Pareschi, and Giovanni Russo (2025) Implicit-explicit methods for evolutionary partial differential equations. DOI: 10.1137/1.9781611978209

ARS443()

A third-order, effectively four-stage, globally stiffly accurate (GSA) type II IMEX method developed by Ascher, Ruuth, and Spiteri (1997).

References

• Uri M. Ascher, Steven J. Ruuth, and Raymond J Spiteri (1997) Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. DOI: 10.1016/S0168-9274(97)00056-1
• Sebastiano Boscarino and Giovanni Russo (2024) Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review. DOI: 10.1007/s40324-024-00351-x
• Sebastiano Boscarino, Lorenzo Pareschi, and Giovanni Russo (2025) Implicit-explicit methods for evolutionary partial differential equations. DOI: 10.1137/1.9781611978209

SSPKnoth()

A three-stage, second-order strong-stability preserving (SSP) Rosenbrock-W method by Knoth and Wolke.

References

• O. Knoth and R. Wolke (1998) Implicit-explicit coupled multirate methods for reactive flow with stiff chemistry. Atmospheric Environment, 32(3):507–519.

ROS2()

A two-stage, second-order Rosenbrock-W method with diagonal parameter $\gamma = (1 + 1/\sqrt{3})/2$.

References

• Ernst Hairer and Gerhard Wanner (1996) Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics, 2nd edition. DOI: 10.1007/978-3-642-05221-7