Methods

StochasticMethod <: GeometricMethod

Root of the stochastic method hierarchy.

GeometricIntegratorsBase splits GeometricMethod into DeterministicMethod and nothing else; the stochastic branch lives here. A stochastic method differs from a deterministic one in that each step consumes a realisation of the driving noise, drawn according to convergence from the AbstractStochasticProcess carried by the problem.

Stochastic method for a SDEProblem.

Stochastic method for a PSDEProblem.

Stochastic method for a SPSDEProblem.

convergence(method)

Whether a method needs increments that approximate the driving Wiener process in the strong (mean-square) or the weak sense — :strong or :weak.

This is a property of the scheme, not a choice the caller makes. A strong scheme is derived against the true increments $\Delta W^r = \int_{t_n}^{t_{n+1}} dW^r$, and needs them drawn from the correct Gaussian law, because it is asked to track an individual sample path. A weak scheme is derived only against the moments of the increments, and deliberately replaces them with a cheap discrete random variable that matches those moments — which is valid for computing expectations and wrong for tracking a path.

Feeding a scheme the other kind of increment produces plausible-looking numbers of the wrong accuracy, silently, which is why this is derived from the method rather than left as an option. See sample_noise! for the two laws.

julia> convergence(BurrageE1())
:strong

julia> convergence(StochasticStoermerVerlet())
:strong

julia> convergence(SRKw1())
:weak

issdemethod(method)

Whether method integrates an SDEProblem. Accepts an instance or a type, and answers false for any other GeometricMethod, so it can be asked of a method of unknown kind.

ispsdemethod(method)

Whether method integrates a PSDEProblem. See issdemethod.

isspsdemethod(method)

Whether method integrates an SPSDEProblem, the split form in which the forcing is given as two separate terms. See issdemethod.

Number of internal stages of a stochastic Runge-Kutta method.

SERK(tableau; rng = Random.default_rng())

Stochastic explicit Runge-Kutta method for a SDEProblem, of strong (mean-square) convergence.

Each internal stage is built from the stages before it, so a step costs one evaluation of the drift and the diffusion per stage and involves no solve:

\[Q_i = q_n + \Delta t \sum_{j<i} a^{v}_{ij} V_j + \sum_r \Delta W^r \sum_{j<i} a^{B}_{ij} B_j^{\cdot r} + \frac{1}{\Delta t} \sum_r \Delta Z^r \sum_{j<i} a^{B_2}_{ij} B_j^{\cdot r} .\]

Being explicit it is cheap per step but has a bounded stability region, which for a stiff drift or a large noise intensity forces a much smaller time step than the implicit methods.

Constructed through one of the named schemes: StochasticEuler, StochasticHeun, Platen, BurrageR2, BurrageCL, BurrageE1, BurrageG5.

TableauSERK{T}

Tableau of a stochastic explicit Runge-Kutta method.

Three Butcher tableaus, all strictly lower triangular: qdrift for the drift, qdiff for the diffusion terms carrying the Wiener increments $\Delta W$, and qdiff2 for the terms carrying the iterated integrals $\Delta Z$. A method that uses only $\Delta W$ leaves qdiff2 empty, which is recorded by giving it the name :NULL; hasdiffusion2 is the predicate.

No overall order is stored. Unlike in the deterministic case the order of a stochastic Runge-Kutta method is not a property of the coefficients alone — it depends on the dimension of the driving Wiener process and on whether the diffusion matrix satisfies the commutativity conditions. The o fields of the three tableaus are the classical orders of the underlying deterministic schemes and nothing more.

Each Butcher tableau it holds gets its own type parameter, so that the fields are concrete: RungeKutta.Tableau is Tableau{T,S,R∞,L}, and a field declared ::Tableau{T} would be abstract and make every coefficient access allocate. The constructors infer them.

Whether a tableau carries a second diffusion tableau, applied to the iterated integrals ΔZ.

StochasticEuler(; rng = Random.default_rng())

Stochastic Euler method — the one-stage explicit scheme

\[q_{n+1} = q_n + \Delta t \, v(t_n, q_n) + \sum_r \Delta W^r \, B^{\cdot r}(t_n, q_n) .\]

Strong order 0.5 in general. The cheapest thing that works, and the natural baseline: any scheme worth using should beat it at equal cost.

StochasticHeun(; rng = Random.default_rng())

Stochastic Heun method — the two-stage explicit trapezoidal scheme, with the same tableau applied to the drift and the diffusion. Strong order 0.5, weak order 1.0 in general.

Platen(; rng = Random.default_rng())

Platen's two-stage explicit method, Eq. (52) of Burrage & Burrage (1996), High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations.

Strong order 1.0 for a one-dimensional Wiener process. Its long-time behaviour is poor — it is not structure-preserving, and the energy of an oscillator drifts visibly over long runs — so it is mainly of interest for short integrations and for comparison.

BurrageR2(; rng = Random.default_rng())

Two-stage explicit R2 method of Burrage & Burrage, Eq. (51) of High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations (1996).

Strong order 1.0 for a one-dimensional Wiener process. This is also the predictor used for the initial guess of the implicit SIRK methods.

BurrageCL(; rng = Random.default_rng())

Four-stage explicit CL method of Burrage & Burrage, Eq. (56) of High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations (1996). Strong order 1.5 for a one-dimensional Wiener process.

BurrageE1(; rng = Random.default_rng())

Four-stage explicit E1 method of Burrage & Burrage, Order conditions for stochastic Runge-Kutta methods by B-series (2000).

Strong order 1.0 for a one-dimensional Wiener process, and measured at 1.05 on the Kubo oscillator by scripts/convergence_order.jl.

It carries a second diffusion tableau against the iterated integrals $\Delta Z$. Those terms are necessary to exceed the order-1.0 ceiling that schemes built from $\Delta W$ alone are subject to, but they are not sufficient on their own — this method uses them and is still of order 1.0, whereas BurrageCL and BurrageG5 reach 1.5.

BurrageG5(; rng = Random.default_rng())

Five-stage explicit G5 method of Burrage & Burrage, Order conditions for stochastic Runge-Kutta methods by B-series (2000). Like BurrageE1 it uses the $\Delta Z$ terms.

SIRK(tableau; K = 0, rng = Random.default_rng())

Stochastic implicit Runge-Kutta method for a SDEProblem, of strong (mean-square) convergence.

The internal stages are coupled, so each step solves

\[Y_i = \Delta t \sum_j a^{v}_{ij} \, v(Q_j) + \sum_r \Delta W^r \sum_j a^{B}_{ij} \, B^{\cdot r}(Q_j), \qquad Q_i = q_n + Y_i\]

for the stage increments $Y_i$ by Newton's method, in $D \times S$ unknowns. Note that the unknowns are the increments themselves rather than the stage derivatives: unlike the deterministic case the drift and the diffusion enter with different coefficients, so there is no single derivative to solve for.

K sets the Milstein-Tretyakov truncation of the Wiener increments — see truncation. It defaults to 0, no truncation.

Constructed through StochasticGLRK or StochasticDIRK.

TableauSIRK{T}

Tableau of a stochastic implicit Runge-Kutta method: qdrift for the drift, qdiff for the diffusion.

As for TableauSERK no overall order is stored, because the order of a stochastic Runge-Kutta method depends on the noise as well as on the coefficients. The o fields are the classical orders of the underlying deterministic schemes.

Each Butcher tableau it holds gets its own type parameter, so that the fields are concrete: RungeKutta.Tableau is Tableau{T,S,R∞,L}, and a field declared ::Tableau{T} would be abstract and make every coefficient access allocate. The constructors infer them.

StochasticGLRK(s; K = 0, rng = Random.default_rng())

s-stage stochastic Gauss-Legendre Runge-Kutta method: the Gauss collocation tableau applied to both the drift and the diffusion.

For s = 1 this is the stochastic implicit midpoint rule, which is the scheme Kraus & Tyranowski list first among their examples and the simplest Lagrange-d'Alembert integrator. Because drift and diffusion share one tableau, the symplecticity conditions reduce to the classical $\alpha_i a_{ij} + \alpha_j a_{ji} = \alpha_i \alpha_j$ that Gauss satisfies by construction, so the whole family is symplectic for a stochastic Hamiltonian system.

Setting the noise to zero recovers the deterministic Gauss method of the same number of stages, which is a useful check on an implementation.

StochasticDIRK(c = 0.5; K = 0, rng = Random.default_rng())

Two-stage diagonally implicit stochastic Runge-Kutta method.

This is the family of Kraus & Tyranowski, Variational integrators for stochastic dissipative Hamiltonian systems, §3.3.3, where it is shown to be the most general two-stage DIRK satisfying both the Lagrange-d'Alembert conditions and the conditions for mean-square order 1.0. Every one of the six coefficient matrices is

\[a = \begin{pmatrix} c/2 & 0 \\ c & (1-c)/2 \end{pmatrix}, \qquad \alpha = (c, \; 1-c) ,\]

with c free. c = 0 and c = 1 collapse it to the stochastic midpoint method.

Being diagonally implicit, its stages can be solved one after another rather than all at once, which is what makes it cheaper than StochasticGLRK at equal stage count. The paper reports it as remaining accurate at time steps where explicit schemes have long since failed.

SIPRK(tableau; K = 0, rng = Random.default_rng())

Stochastic implicit partitioned Runge-Kutta method for a PSDEProblem, of strong (mean-square) convergence.

For a stochastic Hamiltonian system

\[\begin{aligned} d q &= \frac{\partial H}{\partial p} \, dt + \sum_r \frac{\partial h_r}{\partial p} \circ dW^r, \\ d p &= -\frac{\partial H}{\partial q} \, dt - \sum_r \frac{\partial h_r}{\partial q} \circ dW^r, \end{aligned}\]

a partitioned method applies different coefficients to the $q$ and $p$ equations, which is what makes it possible to be symplectic without being fully implicit in both. When the coefficients satisfy the Lagrange-d'Alembert conditions

\[\alpha_i \bar a_{ij} + \alpha_j a_{ji} = \alpha_i \alpha_j, \qquad \beta_i \bar b_{ij} + \beta_j b_{ji} = \beta_i \beta_j, \qquad \ldots\]

the scheme is a variational integrator: it arises from a discrete Lagrange-d'Alembert principle and preserves the symplectic structure pathwise. In practice that is what stops the energy of an oscillator from drifting over long integrations, which is the whole reason to prefer these methods to a general-purpose stochastic solver.

The solve is in $2 D S$ unknowns — the stage increments of both $q$ and $p$.

Constructed through StochasticSymplecticEuler or StochasticStoermerVerlet.

TableauSIPRK{T}

Tableau of a stochastic implicit partitioned Runge-Kutta method: qdrift, pdrift for the drift parts of the $q$ and $p$ equations, qdiff, pdiff for their diffusion parts.

In the notation of Kraus & Tyranowski, Variational integrators for stochastic dissipative Hamiltonian systems, qdrift $= (a, \alpha)$, qdiff $= (b, \beta)$, pdrift $= (\bar a, \alpha)$ and pdiff $= (\bar b, \beta)$. Note that the $q$ and $p$ tableaus of a pair share their weight vectors — only their coefficient matrices differ. That is a requirement of the Lagrange-d'Alembert conditions, not a convention.

As elsewhere, no overall order is stored: the order of a stochastic Runge-Kutta method depends on the noise as well as on the coefficients.

Each Butcher tableau it holds gets its own type parameter, so that the fields are concrete: RungeKutta.Tableau is Tableau{T,S,R∞,L}, and a field declared ::Tableau{T} would be abstract and make every coefficient access allocate. The constructors infer them.

StochasticSymplecticEuler(; K = 0, rng = Random.default_rng())

One-stage stochastic symplectic Euler method — implicit in $q$, explicit in $p$.

Satisfies all eight Lagrange-d'Alembert conditions, so it is a variational integrator. Its mean-square order conditions, however, are not satisfied in general: $\beta^T b e = 1 \neq 1/2$. It attains strong order 1.0 only for special choices of the stochastic Hamiltonians and forces — for instance $h = h(q)$ with $f = 0$, for which every term of the order conditions carries a vanishing factor and the conditions hold trivially. Outside such cases expect order 0.5.

Verified against the conditions of Kraus & Tyranowski by scripts/tableau_conditions.jl; the method itself does not appear in that paper.

StochasticStoermerVerlet(; K = 0, rng = Random.default_rng())

Two-stage stochastic Störmer-Verlet method, built from the Lobatto IIIA-IIIB pair.

This is the scheme of Kraus & Tyranowski §3.3.3, Example 2, and its tableau matches theirs exactly: $(a, b) =$ Lobatto IIIA(2), $(\bar a, \bar b) =$ Lobatto IIIB(2), all weights $(1/2, 1/2)$. It satisfies both the Lagrange-d'Alembert conditions and the conditions for mean-square order 1.0.

It is the efficient choice among the strong partitioned methods. The stage equations decouple: the first is solved on its own, the second becomes explicit when the Hamiltonians are separable, and the last is an explicit update. If the forcing is independent of $p$ the whole method is explicit.

SISPRK(tableau; K = 0, rng = Random.default_rng())

Stochastic implicit split partitioned Runge-Kutta method for a SPSDEProblem, of strong (mean-square) convergence.

This is the method of Kraus & Tyranowski for stochastic forced (dissipative) Hamiltonian systems

\[\begin{aligned} d q &= \frac{\partial H}{\partial p} dt + \sum_r \frac{\partial h_r}{\partial p} \circ dW^r , \\ d p &= \left[ -\frac{\partial H}{\partial q} + F \right] dt + \sum_r \left[ -\frac{\partial h_r}{\partial q} + f_r \right] \circ dW^r . \end{aligned}\]

The point of the split is that the Hamiltonian and the forcing terms of the $p$ equation get different coefficients — $(\bar a, \bar b)$ against $(\hat a, \hat b)$. That extra freedom is what allows the scheme to satisfy the Lagrange-d'Alembert conditions in the presence of forcing, making it a discrete Lagrange-d'Alembert variational integrator rather than merely a symplectic method with a dissipative term bolted on. It is the reason such a scheme tracks the correct energy decay of a damped system over long integrations instead of accumulating its own spurious drift.

Constructed through StochasticLobattoIIIABD2 or ModifiedStochasticStoermerVerlet.

TableauSISPRK{T}

Tableau of a stochastic implicit split partitioned Runge-Kutta method.

Six Butcher tableaus. qdrift, qdiff act on the $q$ equation; the $p$ equation is split, with pdrift1, pdiff1 acting on the Hamiltonian part and pdrift2, pdiff2 on the forcing part. In the notation of Kraus & Tyranowski, Variational integrators for stochastic dissipative Hamiltonian systems, Eq. (SPRK):

The paper's tableau shares weight vectors: $\alpha$ is used for both qdrift and pdrift1, and $\beta$ for both qdiff and pdiff1; only the forcing tableaus carry independent weights $\hat\alpha$, $\hat\beta$. This is a consequence of the Lagrange-d'Alembert conditions, not a convention, so the constructor checks it — a tableau that violates it cannot be a variational integrator whatever else it satisfies.

Each Butcher tableau it holds gets its own type parameter, so that the fields are concrete: RungeKutta.Tableau is Tableau{T,S,R∞,L}, and a field declared ::Tableau{T} would be abstract and make every coefficient access allocate. The constructors infer them.

StochasticLobattoIIIABD2(; K = 0, rng = Random.default_rng())

Two-stage stochastic Lobatto IIIA-IIIB-IIID method, with the Lobatto IIID pair of L. Jay in the forcing slots $\hat a$, $\hat b$.

Satisfies the conditions for mean-square order 1.0 for one Wiener process, but not the Lagrange-d'Alembert conditions — so it is not a variational integrator. Concretely, conditions (6) and (8) of Kraus & Tyranowski fail at $i = j = 1$: with $\hat b =$ Lobatto IIID(2) $= \begin{pmatrix} 1/4 & -1/4 \\ 3/4 & 1/4 \end{pmatrix}$,

\[\beta_i \hat b_{ij} + \hat\beta_j b_{ji} = \tfrac{1}{2}\cdot\tfrac{1}{4} + \tfrac{1}{2}\cdot 0 = \tfrac{1}{8} \neq \tfrac{1}{4} = \beta_i \hat\beta_j .\]

Conditions (1)–(4), which involve only the IIIA/IIIB pair, do hold. This is checked in scripts/tableau_conditions.jl, which asserts the failure rather than merely tolerating it.

Included for comparison: it shows what is lost when the forcing tableaus are chosen for deterministic accuracy rather than for the variational structure.

ModifiedStochasticStoermerVerlet(c = 0.0; K = 0, rng = Random.default_rng())

Two-stage modified stochastic Störmer-Verlet method: the Lobatto IIIA-IIIB pair with a one-parameter family

\[\hat a = \begin{pmatrix} c & 0 \\ c & 0 \end{pmatrix}, \qquad \hat\alpha = (c, \; 1-c)\]

in the forcing drift slot.

Satisfies both the Lagrange-d'Alembert conditions and the conditions for mean-square order 1.0. Neither set constrains c at all: $\hat a$ and $\hat\alpha$ enter the order conditions only through $\sum_i \hat\alpha_i = c + (1-c) = 1$, and the symplecticity residuals are identically zero in c. The constructor nevertheless restricts c to $[0,1]$, which keeps the collocation nodes $(c, c)$ inside the time step — a restriction on the method rather than one the conditions impose.

At $c = 1/2$ it reduces to the plain stochastic Störmer-Verlet method of Kraus & Tyranowski, of which this is a generalisation to forced systems with a free parameter.

All of the above is checked by scripts/tableau_conditions.jl.

WERK(tableau; rng = Random.default_rng())

Weak explicit Runge-Kutta method for a SDEProblem.

A weak method approximates expectations $E[\varphi(q(T))]$ rather than individual sample paths, which is what one actually wants for computing statistics — mean energy, ergodic limits, distributions. That freedom is what makes it cheap: the true Gaussian increments are replaced by the discrete random variables of sample_noise!, which match the moments the scheme was derived against and cost a single uniform draw apiece.

Structurally it is more elaborate than a strong method. Beyond the drift stages $H^{(0)}_i$ it carries a separate family of stages $H^{(l)}_i$ and $\hat H^{(l)}_i$ for each noise dimension $l$, and the diffusion matrix is evaluated at a different stage vector for each of its columns. That is what buys second-order weak accuracy for multi-dimensional noise.

Constructed through RoesslerRS1 or RoesslerRS2.

TableauWERK{T}

Tableau of a weak explicit Runge-Kutta method, in the layout of Rößler, Second order Runge-Kutta methods for Stratonovich stochastic differential equations, BIT 47 (2007), Eq. (5.1).

Seven strictly lower triangular Butcher tableaus. In the notation of the paper,

with weights qdrift0.b $= \alpha$, qdiff0.b $= \beta_1$, qdiff3.b $= \beta_2$ and nodes qdrift0.c, qdrift1.c, qdrift2.c $= c^{(0)}, c^{(1)}, c^{(2)}$. The o fields are meaningless here and set to zero.

Each Butcher tableau it holds gets its own type parameter, so that the fields are concrete: RungeKutta.Tableau is Tableau{T,S,R∞,L}, and a field declared ::Tableau{T} would be abstract and make every coefficient access allocate. The constructors infer them.

WIRK(tableau; rng = Random.default_rng())

Weak implicit Runge-Kutta method for a SDEProblem.

Like WERK it targets expectations rather than sample paths, and like SIRK it solves for its stage increments. It carries $m+1$ families of internal stages — one, $Q^{(0)}$, against which the drift is evaluated, and one $Q^{(l)}$ per noise dimension, against which column $l$ of the diffusion is evaluated. The solver system is correspondingly larger: $D \times S \times (1 + m)$ unknowns rather than $D \times S$.

The methods of Wang, Hong & Xu implemented here are symplectic, which for a stochastic Hamiltonian system is what keeps the energy from drifting over long runs — the property that makes a weak method usable for ergodic averages in the first place.

Constructed through SRKw1 or SRKw2.

TableauWIRK{T}

Tableau of a weak implicit Runge-Kutta method, in the layout of Wang, Hong & Xu, Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems, Commun. Comput. Phys. 21(1),

In the notation of that paper qdrift0 $= A^{(0)}$, qdrift1 $= A^{(1)}$, qdiff0 $= B^{(0)}$, qdiff1 $= B^{(1)}$, qdiff3 $= B^{(3)}$, with weights qdrift0.b $= \alpha$ and qdiff0.b $= \beta$, and nodes qdrift0.c, qdrift1.c.

Kraus & Tyranowski, §3.4, show that the symplecticity conditions of that paper are equivalent to their weak Lagrange-d'Alembert conditions, so these methods remain variational integrators when a forcing term is added.

Each Butcher tableau it holds gets its own type parameter, so that the fields are concrete: RungeKutta.Tableau is Tableau{T,S,R∞,L}, and a field declared ::Tableau{T} would be abstract and make every coefficient access allocate. The constructors infer them.

RoesslerRS1(; rng = Random.default_rng())

Weak explicit Runge-Kutta method RS1 of Rößler, Second order Runge-Kutta methods for Stratonovich stochastic differential equations, BIT 47 (2007). Weak order 2.0.

RoesslerRS2(; rng = Random.default_rng())

Weak explicit Runge-Kutta method RS2 of Rößler, Second order Runge-Kutta methods for Stratonovich stochastic differential equations, BIT 47 (2007). Weak order 2.0.

SRKw1(λ = 0.0; rng = Random.default_rng())

One-stage weak implicit Runge-Kutta method SRKw1 of Wang, Hong & Xu, Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems, Commun. Comput. Phys. 21(1), 2017. Weak order 1.0.

λ is a free parameter of the family; λ = 1/2 reduces the method to the implicit midpoint rule with the Wiener increments replaced by the discrete weak variables.

Reproduced in Kraus & Tyranowski §3.4.3, where it is shown that the symplecticity conditions of Wang, Hong & Xu are equivalent to the weak Lagrange-d'Alembert conditions — so this is a variational integrator for forced systems too.

SRKw2(λ₁ = 0.0, λ₂ = 0.0, λ₃ = 0.0; rng = Random.default_rng())

Four-stage weak implicit Runge-Kutta method SRKw2 of Wang, Hong & Xu (2017). Weak order 2.0 for systems driven by one-dimensional noise.

The three parameters are free and, as the paper notes, have no effect on $q_{n+1}$ or $p_{n+1}$; they default to zero for convenience. Of the weak methods here this is the most accurate.

Tableau for the explicit 1-stage stochastic Euler method

Tableau for the explicit 2-stage stochastic Heun method

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.

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 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 s-stage Gauss-Lobatto SFIRK method

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

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 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 4-stage RS1 method due to Andreas Rössler 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 Rössler 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.

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.