Tableaus

Tableau for the explicit 4-stage CL method due to K. Burrage and P. Burrage Method cited in Eq. (56) in K. Burrage, P. Burrage (1996) "High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations". According to the paper, the method has strong order 1.5 for one-dimensional Brownian motion. Reduces to the classical R-K method of order 4 when noise is zero.

Tableau for the explicit 4-stage E1 method due to K. Burrage and P. Burrage Method cited in Eq. (4.2)-(4.3) in K. Burrage, P. Burrage (2000) "Order conditions for stochastic Runge-Kutta methods by B-series". According to the paper, the method has strong order 1.0 for one-dimensional Brownian motion.

Tableau for the explicit 5-stage G5 method due to K. Burrage and P. Burrage Method cited in Section 4 of K. Burrage, P. Burrage (2000) "Order conditions for stochastic Runge-Kutta methods by B-series". According to the paper, the method has strong order 1.5 for one-dimensional Brownian motion.

Tableau for the explicit 2-stage R2 method due to K. Burrage and P. Burrage Method cited in Eq. (51) in K. Burrage, P. Burrage (1996) "High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations". According to the paper, the method has strong order 1.0 for one-dimensional Brownian motion

Tableau for explicit Runge-Kutta method of order four (1/6 rule)

Tableau for explicit Runge-Kutta method of order four (3/8 rule)

Tableau for explicit Euler method

Tableau for explicit midpoint method

Tableau for Gauss-Legendre method with s stages and symplectic projection.

Tableau for Gauss-Legendre method with s stages and symplectic projection.

Tableau for Heun's method

Implicit Euler

Implicit Midpoint

Tableau for Kutta's method of order three

Gauss-Lobatto-IIIA Runge-Kutta, s=2

Gauss-Lobatto-IIIA Runge-Kutta, s=3

Gauss-Lobatto-IIIA Runge-Kutta, s=4

Tableau for Gauss-Lobatto IIIA-IIIB method with two stages and symmetric projection.

Tableau for Gauss-Lobatto IIIA-IIIB method with two stages and symplectic projection.

Tableau for Gauss-Lobatto IIIA-IIIB method with three stages and symmetric projection.

Tableau for Gauss-Lobatto IIIA-IIIB method with three stages and symplectic projection.

Gauss-Lobatto-IIIB Runge-Kutta, s=2

Gauss-Lobatto-IIIB Runge-Kutta, s=3

Gauss-Lobatto-IIIB Runge-Kutta, s=4

Gauss-Lobatto-IIIC Runge-Kutta, s=2

Gauss-Lobatto-IIIC Runge-Kutta, s=3

Gauss-Lobatto-IIIC Runge-Kutta, s=4

Gauss-Lobatto-IIID Runge-Kutta, s=2

Gauss-Lobatto-IIID Runge-Kutta, s=3

Gauss-Lobatto-IIID Runge-Kutta, s=4

Gauss-Lobatto-IIIE Runge-Kutta, s=2

Gauss-Lobatto-IIIE Runge-Kutta, s=3

Gauss-Lobatto-IIIE Runge-Kutta, s=4

Gauss-Lobatto-IIIF Runge-Kutta, s=2

Gauss-Lobatto-IIIF Runge-Kutta, s=3

Gauss-Lobatto-IIIF Runge-Kutta, s=4

Gauss-Lobatto-IIIG Runge-Kutta, s=2

Gauss-Lobatto-IIIG Runge-Kutta, s=3

Gauss-Lobatto-IIIG Runge-Kutta, s=4

Tableau for Gauss-Lobatto IIIAIIIB method with s=2 stages

Tableau for Gauss-Lobatto IIIBIIIA method with s=2 stages

Tableau for the 2-stage modified stochastic LobattoIIIA-IIIB method Tableau for the 2-stage modified stochastic LobattoIIIA-IIIB method Satisfies the conditions for Lagrange-d'Alembert integrators and the conditions for convergence of order 1.0 for one Wiener process

Tableau for the explicit Platen method Platen's method cited in Eq. (52) in K. Burrage, P. Burrage (1996) "High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations". According to the paper, the method has strong order 1.0 for one-dimensional Brownian motion. Appears to have a rather poor long-time performance.

Gauss-Radau-IIA Runge-Kutta, s=2

Gauss-Radau-IIA Runge-Kutta, s=3

Tableau for the explicit 4-stage RS1 method due to Andreas Rossler Method cited in Table 5.2 in Andreas Rossler, "Second order Runge-Kutta methods for Stratonovich stochastic differential equations", BIT Numerical Mathematics (2007) 47 According to the paper, the method has weak order 2.0.

Tableau for the explicit 4-stage RS2 method due to Andreas Rossler Method cited in Table 5.3 in Andreas Rossler, "Second order Runge-Kutta methods for Stratonovich stochastic differential equations", BIT Numerical Mathematics (2007) 47 According to the paper, the method has weak order 2.0.

Gauss-Legendre Runge-Kutta, s=3

Tableau for the 1-stage SRKw1 method due to Wang, Hong & Xu Method cited in Wang, Hong, Xu, "Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems", Commun. Comput. Phys. 21(1), 2017 According to the paper, the method has weak order 1.0.

Tableau for the 4-stage SRKw2 method due to Wang, Hong & Xu Method cited in Wang, Hong, Xu, "Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems", Commun. Comput. Phys. 21(1), 2017 According to the paper, the method has weak order 2.0 when applied to systems driven by one-dimensional noise.

Tableau for the 2-stage stochastic symplectic DIRK method Tableau for the stochastic symplectic DIRK method Satisfies the conditions for Lagrange-d'Alembert integrators. Satisfies the conditions for strong convergence of order 1.0 for one Wiener process

Tableau for the s-stage Gauss-Lobatto SFIRK method

Tableau for the explicit 2-stage stochastic Heun method

Tableau for the 2-stage stochastic LobattoIIIA-IIIB-IIID method Tableau for the 2-stage stochastic LobattoIIIA-IIIB-IIID method (based on the deterministic LobattoIIIA-IIIB-IIID due to L. Jay) It satisfies the conditions for convergence of order 1.0 for one Wiener process, but it doesn't satisfy the conditions for Lagrange-d'Alembert integrators

Tableau for the 2-stage stochastic LobattoIIA-IIB method (Stormer-Verlet)

Tableau for the stochastic symplectic Euler method Tableau for the stochastic symplectic Euler method Satisfies the conditions for Lagrange-d'Alembert integrators. Satisfies the conditions for strong convergence of order 1.0 for one Wiener process for special choices of the stochastic Hamiltonians and forces, e.g., h=h(q), f=0.

Tableau for symplectic Euler-A method

Tableau for symplectic Euler-B method

Tableau for variational Gauss-Legendre method with s stages

Tableau for variational Gauss-Lobatto IIIA-IIIB method with two stages

Tableau for variational Gauss-Lobatto IIIA-IIIB method with three stages

Tableau for variational Gauss-Lobatto IIIA-IIIB method with four stages

Tableau for Gauss-Lobatto IIIA-IIIA method with two stages

Tableau for Gauss-Lobatto IIIA-IIIA method with three stages

Tableau for Gauss-Lobatto IIIA-IIIA method with four stages

Tableau for variational Gauss-Lobatto IIIA-IIIB method with two stages

Tableau for variational Gauss-Lobatto IIIA-IIIB method with three stages

Tableau for variational Gauss-Lobatto IIIA-IIIB method with four stages

Tableau for variational Gauss-Lobatto IIIC-III method with two stages

Tableau for variational Gauss-Lobatto IIIC-III method with three stages

Tableau for variational Gauss-Lobatto IIIC-III method with four stages

Tableau for variational Gauss-Lobatto IIID method with two stages

Tableau for variational Gauss-Lobatto IIID method with three stages

Tableau for variational Gauss-Lobatto IIID method with four stages

Tableau for variational Gauss-Lobatto IIIE method with two stages

Tableau for variational Gauss-Lobatto IIIE method with three stages

Tableau for variational Gauss-Lobatto IIIE method with four stages

Tableau for variational Gauss-Lobatto IIIF method with two stages

Tableau for variational Gauss-Lobatto IIIF method with three stages

Tableau for variational Gauss-Lobatto IIIF method with four stages

Tableau for variational Gauss-Lobatto IIIG method with two stages

Tableau for variational Gauss-Lobatto IIIG method with three stages

Tableau for variational Gauss-Lobatto IIIG method with four stages

Tableau for Gauss-Radau IIA-IIA method with two stages

Tableau for Gauss-Radau IIA-IIA method with three stages

Tableau for variational symmetric Runge-Kutta method with 3 stages