Integrators

Create AtomicSolution for DAE.

Create AtomicSolution for ODE.

Create AtomicSolution for partitioned DAE.

Create AtomicSolution for partitioned ODE.

Create AtomicSolution for partitioned SDE.

Create AtomicSolution for SDE.

Holds the information for the various methods' tableaus.

Create a vector of S solution matrices of type DT to store the solution of S internal stages for a problem with DxM dimensions.

Create a vector of S solution vectors of type DT to store the solution of S internal stages for a problem with D dimensions.

Create a vector of (S,M+1) solution vectors of type DT to store the solution of S internal stages and M random processes for a problem with D dimensions.

Create a vector of S+1 solution vectors of type DT to store the solution of S internal stages and the solution of the previous timestep for a problem with D dimensions.

Create nonlinear solver object for a system of N equations with data type DT. The function $f(x)=0$ to be solved for is determined by a julia function function_stages!(x, b, params), where x is the current solution and b is the output vector, s.th. $b = f(x)$. params are a set of parameters depending on the equation and integrator that is used. The solver type is obtained from the config dictionary (:nls_solver).

Print error for integrators not implemented, yet.

Create integrator for special partitioned additive Runge-Kutta tableau.

Create integrator for variational partitioned additive Runge-Kutta tableau.

Create integrator for variational special partitioned additive Runge-Kutta tableau.

Create integrator for variational special partitioned additive Runge-Kutta tableau with projection on primary constraint.

Create integrator for Projected Gauss-Legendre Runge-Kutta tableau.

Create integrator for variational partitioned Runge-Kutta tableau.

Create integrator for diagonally implicit Runge-Kutta tableau.

Create integrator for explicit Runge-Kutta tableau.

Create integrator for fully implicit Runge-Kutta tableau.

Create integrator for Hamiltonian partitioned additive Runge-Kutta tableau.

Create integrator for Hamiltonian special partitioned additive Runge-Kutta tableau.

Create integrator for Hamiltonian special partitioned additive Runge-Kutta tableau with projection on primary constraint.

Create integrator for explicit partitioned Runge-Kutta tableau.

Create integrator for implicit partitioned Runge-Kutta tableau.

Create integrator for stochastic fully implicit partitioned Runge-Kutta tableau.

Create integrator for stochastic explicit Runge-Kutta tableau.

Create integrator for stochastic fully implicit Runge-Kutta tableau.

Create integrator for weak explicit Runge-Kutta tableau.

Create integrator for weak fully implicit Runge-Kutta tableau.

Create integrator for splitting tableau.

Create integrator for stochastic fully implicit split partitioned Runge-Kutta tableau.

Create integrator for variational special partitioned additive Runge-Kutta tableau with projection on secondary constraint.

Create integrator for formal Lagrangian Runge-Kutta tableau.

Create integrator constructor for diagonally implicit Runge-Kutta tableau.

Create integrator constructor for explicit Runge-Kutta tableau.

Create integrator constructor for fully implicit Runge-Kutta tableau.

Create integrator constructor for exact solution.

Integrate ODE for initial conditions m with m₁ ≤ m ≤ m₂.

Integrate equation for all initial conditions.

Integrate ODE for initial conditions m with m₁ ≤ m ≤ m₂ for time steps n with n₁ ≤ n ≤ n₂.

Integrate given equation with given tableau for ntime time steps and return solution.

Apply integrator for ntime time steps and return solution.

Integrate ODE specified by vector field and initial condition with given tableau for ntime time steps and return solution.

Integrate PODE specified by two vector fields and initial conditions with given tableau for ntime time steps and return solution.

Compute p(x) where p is the unique polynomial of degree length(xi), such that p(x[i]) = y[i]) for all i.

ti: interpolation nodes
xi: interpolation values
t:  evaluation point
x:  evaluation value

Euler extrapolation method with arbitrary order p.

v:  function to compute vector field
t₀: initial time
t₁: final   time
x₀: initial value
x₁: final   value
s:  number of interpolations (order p=s+1)

TODO This is probably broken!

Midpoint extrapolation method with arbitrary order p.

v:  function to compute vector field
f:  function to compute force  field
t₀: initial time
t₁: final   time
q₀: initial positions
p₀: initial momenta
q₁: final   positions
p₁: final   momenta
s:  number of interpolations (order p=2s+2)

Midpoint extrapolation method with arbitrary order p.

v:  function to compute vector field
t₀: initial time
t₁: final   time
x₀: initial value
x₁: final   value
s:  number of interpolations (order p=2s+2)

InitialGuessODE: Initial guess for ordinary differential equations

Fields

• int: interpolation structure
• v: vector field
• Δt: time step
• s: number of extrapolation stages (for initialisation)

Initialise initial guess, i.e., given t₀, t₁, q₁ compute q₀, v₀, v₁.

compute vector field of new solution

InitialGuessPODE: Initial guess for partitioned ordinary differential equations

Fields

• int: interpolation structure
• v: vector field for q
• f: vector field for p
• Δt: time step
• s: number of extrapolation stages (for initialisation)

Initialise initial guess, i.e., given t₀, t₁, q₁ compute q₀, v₀, v₁.

Splitting integrator.

Construct splitting integrator.

Integrate ODE with splitting integrator.

Holds the coefficients of a Runge-Kutta method.

Print Runge-Kutta coefficients.

Holds the tableau of an explicit Runge-Kutta method.

Holds the tableau of an implicit Runge-Kutta method.

Holds the tableau of a partitioned Runge-Kutta method.

Holds the tableau of a Runge-Kutta method.

Reads and parses Tableau metadata from file.

Write Runge-Kutta tableau to file.

Explicit Runge-Kutta integrator cache.

Explicit Runge-Kutta integrator.

Parameters for right-hand side function of explicit Runge-Kutta methods.

Holds the tableau of an explicit Runge-Kutta method.

Integrate ODE with explicit Runge-Kutta integrator.

Read explicit Runge-Kutta tableau from file.

Diagonally implicit Runge-Kutta integrator cache.

Diagonally implicit Runge-Kutta integrator.

Parameters for right-hand side function of diagonally implicit Runge-Kutta methods.

Holds the tableau of a diagonally implicit Runge-Kutta method.

Compute stages of fully implicit Runge-Kutta methods.

Compute initial guess for internal stages.

Initialise initial guess

Integrate ODE with diagonally implicit Runge-Kutta integrator.

Fully implicit Runge-Kutta integrator cache.

Fields

• q̃: initial guess of solution
• ṽ: initial guess of vector field
• s̃: holds shift due to periodicity of solution
• Q: internal stages of solution
• V: internal stages of vector field
• Y: vector field of internal stages

Fully implicit Runge-Kutta integrator.

Parameters for right-hand side function of fully implicit Runge-Kutta methods.

Holds the tableau of a fully implicit Runge-Kutta method.

Compute stages of fully implicit Runge-Kutta methods.

Integrate ODE with fully implicit Runge-Kutta integrator.

Compute stages of fully implicit Runge-Kutta methods.

Explicit Runge-Kutta integrator cache.

Explicit partitioned Runge-Kutta integrator.

Parameters for right-hand side function of explicit partitioned Runge-Kutta methods.

TableauEPRK: Tableau of an Explicit Partitioned Runge-Kutta method

\[\begin{aligned} V_{n,i} &= \hphantom{-} \dfrac{\partial H}{\partial p} (Q_{n,i}, P_{n,i}) , & Q_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} a_{ij} \, V_{n,j} , & q_{n+1} &= q_{n} + h \sum \limits_{i=1}^{s} b_{i} \, V_{n,i} , \\ F_{n,i} &= - \dfrac{\partial H}{\partial q} (Q_{n,i}, P_{n,i}) , & P_{n,i} &= p_{n} + h \sum \limits_{i=1}^{s} \bar{a}_{ij} \, F_{n,j} , & p_{n+1} &= p_{n} + h \sum \limits_{i=1}^{s} \bar{b}_{i} \, F_{n,i} , \end{aligned}\]

usually satisfying the symplecticity conditions

\[\begin{aligned} b_{i} \bar{a}_{ij} + b_{j} a_{ji} &= b_{i} b_{j} , & \bar{b}_i &= b_i . \end{aligned}\]

Compute P stages of explicit partitioned Runge-Kutta methods.

Compute Q stages of explicit partitioned Runge-Kutta methods.

Integrate partitioned ODE with explicit partitioned Runge-Kutta integrator.

Implicit partitioned Runge-Kutta integrator cache.

Fields

• q̃: initial guess of q
• p̃: initial guess of p
• ṽ: initial guess of v
• f̃: initial guess of f
• s̃: holds shift due to periodicity of solution
• Q: internal stages of q
• P: internal stages of p
• V: internal stages of v
• F: internal stages of f
• Y: vector field of internal stages of q
• Z: vector field of internal stages of p

Implicit partitioned Runge-Kutta integrator.

Parameters for right-hand side function of implicit partitioned Runge-Kutta methods.

TableauIPRK: Tableau of an Implicit Partitioned Runge-Kutta method

\[\begin{aligned} P_{n,i} &= \dfrac{\partial L}{\partial v} (Q_{n,i}, V_{n,i}) , & Q_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} a_{ij} \, V_{n,j} , & q_{n+1} &= q_{n} + h \sum \limits_{i=1}^{s} b_{i} \, V_{n,i} , \\ F_{n,i} &= \dfrac{\partial L}{\partial q} (Q_{n,i}, V_{n,i}) , & P_{n,i} &= p_{n} + h \sum \limits_{i=1}^{s} \bar{a}_{ij} \, F_{n,j} , & p_{n+1} &= p_{n} + h \sum \limits_{i=1}^{s} \bar{b}_{i} \, F_{n,i} , \end{aligned}\]

usually satisfying the symplecticity conditions

\[\begin{aligned} b_{i} \bar{a}_{ij} + b_{j} a_{ji} &= b_{i} b_{j} , & \bar{b}_i &= b_i . \end{aligned}\]

Compute stages of implicit partitioned Runge-Kutta methods.

Integrate ODE with implicit partitioned Runge-Kutta integrator.

Formal Lagrangian Runge-Kutta integrator.

Parameters for right-hand side function of formal Lagrangian Runge-Kutta methods.

Compute stages of formal Lagrangian Runge-Kutta methods.

Holds the coefficients of a projected Gauss-Legendre Runge-Kutta method.

Projected Gauss-Legendre Runge-Kutta integrator.

Reference: LUIGI BRUGNANO, FELICE IAVERNARO, AND DONATO TRIGIANTE.
    ENERGY- AND QUADRATIC INVARIANTS–PRESERVING INTEGRATORS BASED
    UPON GAUSS COLLOCATION FORMULAE.
    SIAM J. NUMER. ANAL. Vol. 50, No. 6, pp. 2897–2916, 2012.

Parameters for right-hand side function of projected Gauss-Legendre Runge-Kutta methods.

Print Runge-Kutta coefficients.

Compute stages of projected Gauss-Legendre Runge-Kutta methods.

Integrate ODE with projected Gauss-Legendre Runge-Kutta integrator.

Parameters for right-hand side function of Specialised Partitioned Additive Runge-Kutta methods.

Holds the tableau of an Specialised Partitioned Additive Runge-Kutta method for Variational systems.

Holds the coefficients of an additive Runge-Kutta method.

Holds the coefficients of an additive Runge-Kutta method.

Holds the multiplier Runge-Kutta coefficients.

Holds the coefficients of a projective Runge-Kutta method.

Holds the coefficients of a SPARK method.

Cache of a Specialised Partitioned Additive Runge-Kutta integrator.

Fields

• n: time step number
• t: time of current time step
• t̅: time of previous time step
• q: current solution of q
• q̅: previous solution of q
• p: current solution of p
• p̅: previous solution of p
• v: vector field of q
• v̅: vector field of q̅
• f: vector field of p
• f̅: vector field of p̅
• q̃: initial guess of q
• p̃: initial guess of p
• ṽ: initial guess of v
• f̃: initial guess of f
• s̃: holds shift due to periodicity of solution
• Q: internal stages of q
• P: internal stages of p
• V: internal stages of v
• F: internal stages of f
• Y: vector field of internal stages of q
• Z: vector field of internal stages of p

Partitioned Additive Runge-Kutta integrator for Hamiltonian systems subject to Dirac constraints EXPERIMENTAL.

This integrator solves the following system of equations for the internal stages,

\[\begin{aligned} Q_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} a_{ij} V_{n,j} + h \sum \limits_{j=1}^{r} \alpha_{ij} U_{n,j} , & i &= 1, ..., s , \\ P_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} a_{ij} F_{n,j} + h \sum \limits_{j=1}^{r} \alpha_{ij} G_{n,j} , & i &= 1, ..., s , \\ \tilde{Q}_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} \tilde{a}_{ij} V_{n,j} + h \sum \limits_{j=1}^{r} \tilde{\alpha}_{ij} U_{n,j} , & i &= 1, ..., r , \\ \tilde{P}_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} \tilde{a}_{ij} F_{n,j} + h \sum \limits_{j=1}^{r} \tilde{\alpha}_{ij} G_{n,j} , & i &= 1, ..., r , \\ \tilde{\Phi}_{n,i} &= 0 , & i &= 1, ..., r , \end{aligned}\]

with definitions

\[\begin{aligned} V_{n,i} &= \hphantom{-} \frac{\partial H}{\partial p} (Q_{n,i}, P_{n,i}) , & i &= 1, ..., s , \\ F_{n,i} &= - \frac{\partial H}{\partial q} (Q_{n,i}, P_{n,i}) , & i &= 1, ..., s , \\ U_{n,i} &= \hphantom{-} \frac{\partial \phi}{\partial p} (\tilde{Q}_{n,i}, \tilde{P}_{n,i})^{T} \Lambda_{n,i} , & i &= 1, ..., r , \\ G_{n,i} &= - \frac{\partial \phi}{\partial q} (\tilde{Q}_{n,i}, \tilde{P}_{n,i})^{T} \Lambda_{n,i} , & i &= 1, ..., r , \\ \tilde{\Phi}_{n,i} &= \phi(\tilde{Q}_{n,i}, \tilde{P}_{n,i}) , & i &= 1, ..., r , \end{aligned}\]

and update rule

\[\begin{aligned} q_{n+1} &= q_{n} + h \sum \limits_{i=1}^{s} b_{i} V_{n,i} + h \sum \limits_{i=1}^{r} \beta_{i} U_{n,i} , \\ p_{n+1} &= p_{n} + h \sum \limits_{i=1}^{s} b_{i} F_{n,i} + h \sum \limits_{i=1}^{r} \beta_{i} G_{n,i} . \end{aligned}\]

Specialised Partitioned Additive Runge-Kutta integrator for Hamiltonian systems EXPERIMENTAL.

This integrator solves the following system of equations for the internal stages,

\[\begin{aligned} Q_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} a_{ij} V_{n,j} + h \sum \limits_{j=1}^{r} \alpha_{ij} U_{n,j} , & i &= 1, ..., s , \\ P_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} a_{ij} F_{n,j} + h \sum \limits_{j=1}^{r} \alpha_{ij} G_{n,j} , & i &= 1, ..., s , \\ \tilde{Q}_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} \tilde{a}_{ij} V_{n,j} + h \sum \limits_{j=1}^{r} \tilde{\alpha}_{ij} U_{n,j} , & i &= 1, ..., r , \\ \tilde{P}_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} \tilde{a}_{ij} F_{n,j} + h \sum \limits_{j=1}^{r} \tilde{\alpha}_{ij} G_{n,j} , & i &= 1, ..., r , \\ 0 &= \sum \limits_{j=1}^{r} \omega_{ij} \tilde{\Phi}_{n,j} , & i &= 1, ..., r-1 , \end{aligned}\]

with definitions

\[\begin{aligned} V_{n,i} &= \hphantom{-} \frac{\partial H}{\partial p} (Q_{n,i}, P_{n,i}) , & i &= 1, ..., s , \\ F_{n,i} &= - \frac{\partial H}{\partial q} (Q_{n,i}, P_{n,i}) , & i &= 1, ..., s , \\ U_{n,i} &= \hphantom{-} \frac{\partial \phi}{\partial p} (\tilde{Q}_{n,i}, \tilde{P}_{n,i})^{T} \Lambda_{n,i} , & i &= 1, ..., r , \\ G_{n,i} &= - \frac{\partial \phi}{\partial q} (\tilde{Q}_{n,i}, \tilde{P}_{n,i})^{T} \Lambda_{n,i} , & i &= 1, ..., r , \\ \tilde{\Phi}_{n,i} &= \phi(\tilde{Q}_{n,i}, \tilde{P}_{n,i}) , & i &= 1, ..., r , \end{aligned}\]

and update rule

\[\begin{aligned} q_{n+1} &= q_{n} + h \sum \limits_{i=1}^{s} b_{i} V_{n,i} + h \sum \limits_{i=1}^{r} \beta_{i} U_{n,i} , \\ p_{n+1} &= p_{n} + h \sum \limits_{i=1}^{s} b_{i} F_{n,i} + h \sum \limits_{i=1}^{r} \beta_{i} G_{n,i} , \\ 0 &= \phi (q_{n+1}, p_{n+1}) . \end{aligned}\]

Specialised Partitioned Additive Runge-Kutta integrator for Hamiltonian systems EXPERIMENTAL.

This integrator solves the following system of equations for the internal stages,

\[\begin{aligned} Q_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} a_{ij} V_{n,j} + h \sum \limits_{j=1}^{r} \alpha_{ij} U_{n,j} , & i &= 1, ..., s , \\ P_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} a_{ij} F_{n,j} + h \sum \limits_{j=1}^{r} \alpha_{ij} G_{n,j} , & i &= 1, ..., s , \\ \tilde{Q}_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} \tilde{a}_{ij} V_{n,j} + h \sum \limits_{j=1}^{r} \tilde{\alpha}_{ij} U_{n,j} , & i &= 1, ..., r , \\ \tilde{P}_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} \tilde{a}_{ij} F_{n,j} + h \sum \limits_{j=1}^{r} \tilde{\alpha}_{ij} G_{n,j} , & i &= 1, ..., r , \\ 0 &= \sum \limits_{j=1}^{r} \omega_{ij} \tilde{\Phi}_{n,j} , & i &= 1, ..., r-1 , \end{aligned}\]

with definitions

\[\begin{aligned} V_{n,i} &= \hphantom{-} \frac{\partial H}{\partial p} (Q_{n,i}, P_{n,i}) , & i &= 1, ..., s , \\ F_{n,i} &= - \frac{\partial H}{\partial q} (Q_{n,i}, P_{n,i}) , & i &= 1, ..., s , \\ U_{n,i} &= \hphantom{-} \frac{\partial \phi}{\partial p} (\tilde{Q}_{n,i}, \tilde{P}_{n,i})^{T} \Lambda_{n,i} , & i &= 1, ..., r , \\ G_{n,i} &= - \frac{\partial \phi}{\partial q} (\tilde{Q}_{n,i}, \tilde{P}_{n,i})^{T} \Lambda_{n,i} , & i &= 1, ..., r , \\ \tilde{\Phi}_{n,i} &= \phi(\tilde{Q}_{n,i}, \tilde{P}_{n,i}) , & i &= 1, ..., r , \end{aligned}\]

and update rule

\[\begin{aligned} q_{n+1} &= q_{n} + h \sum \limits_{i=1}^{s} b_{i} V_{n,i} + h \sum \limits_{i=1}^{r} \beta_{i} U_{n,i} , \\ p_{n+1} &= p_{n} + h \sum \limits_{i=1}^{s} b_{i} F_{n,i} + h \sum \limits_{i=1}^{r} \beta_{i} G_{n,i} , \\ 0 &= \phi (q_{n+1}, p_{n+1}) . \end{aligned}\]

Specialised Partitioned Additive Runge-Kutta integrator for Hamiltonian systems subject to Dirac constraints with projection on secondary constraint EXPERIMENTAL.

This integrator solves the following system of equations for the internal stages,

\[\begin{aligned} Q_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} a_{ij} V_{n,j} + h \sum \limits_{j=1}^{r} \alpha_{ij} U_{n,j} , & i &= 1, ..., s , \\ P_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} a_{ij} F_{n,j} + h \sum \limits_{j=1}^{r} \alpha_{ij} G_{n,j} , & i &= 1, ..., s , \\ \tilde{Q}_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} \tilde{a}_{ij} V_{n,j} + h \sum \limits_{j=1}^{r} \tilde{\alpha}_{ij} U_{n,j} , & i &= 1, ..., r , \\ \tilde{P}_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} \tilde{a}_{ij} F_{n,j} + h \sum \limits_{j=1}^{r} \tilde{\alpha}_{ij} G_{n,j} , & i &= 1, ..., r , \\ 0 &= \sum \limits_{j=1}^{r} \omega_{ij} \tilde{\Phi}_{n,j} , & i &= 1, ..., r-1 , \end{aligned}\]

with definitions

\[\begin{aligned} V_{n,i} &= \hphantom{-} \frac{\partial H}{\partial p} (Q_{n,i}, P_{n,i}) , & i &= 1, ..., s , \\ F_{n,i} &= - \frac{\partial H}{\partial q} (Q_{n,i}, P_{n,i}) , & i &= 1, ..., s , \\ U_{n,i} &= \hphantom{-} \frac{\partial \phi}{\partial p} (\tilde{Q}_{n,i}, \tilde{P}_{n,i})^{T} \Lambda_{n,i} , & i &= 1, ..., r , \\ G_{n,i} &= - \frac{\partial \phi}{\partial q} (\tilde{Q}_{n,i}, \tilde{P}_{n,i})^{T} \Lambda_{n,i} , & i &= 1, ..., r , \\ \tilde{\Phi}_{n,i} &= \phi(\tilde{Q}_{n,i}, \tilde{P}_{n,i}) , & i &= 1, ..., r , \end{aligned}\]

and update rule

\[\begin{aligned} q_{n+1} &= q_{n} + h \sum \limits_{i=1}^{s} b_{i} V_{n,i} + h \sum \limits_{i=1}^{r} \beta_{i} U_{n,i} , \\ p_{n+1} &= p_{n} + h \sum \limits_{i=1}^{s} b_{i} F_{n,i} + h \sum \limits_{i=1}^{r} \beta_{i} G_{n,i} , \\ 0 &= \phi (q_{n+1}, p_{n+1}) . \end{aligned}\]

Specialised Partitioned Additive Runge-Kutta integrator for degenerate variational systems with projection on secondary constraint.

This integrator solves the following system of equations for the internal stages,

\[\begin{aligned} Q_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} a^1_{ij} V_{n,j} + h \sum \limits_{j=1}^{\sigma} a^2_{ij} \tilde{\Lambda}_{n,j} , & i &= 1, ..., s , \\ P_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} \bar{a}^1_{ij} F^1_{n,j} + h \sum \limits_{j=1}^{s} \bar{a}^2_{ij} F^2_{n,j} + h \sum \limits_{j=1}^{\sigma} \bar{a}^3_{ij} \tilde{F}^3_{n,j} , & i &= 1, ..., s , \\ 0 &= \Phi_{n,i} , & i &= 1, ..., s , \end{aligned}\]

the projective stages

\[\begin{aligned} \tilde{Q}_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} \alpha^1_{ij} \tilde{V}_{n,j} + h \sum \limits_{j=1}^{\sigma} \alpha^2_{ij} \tilde{\Lambda}_{n,j} , & i &= 1, ..., \sigma , \\ \tilde{P}_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} \bar{\alpha}^1_{ij} F^1_{n,j} + h \sum \limits_{j=1}^{s} \bar{\alpha}^2_{ij} \tilde{F}^2_{n,j} + h \sum \limits_{j=1}^{\sigma} \bar{\alpha}^3_{ij} \tilde{F}^3_{n,j} , & i &= 1, ..., \sigma , \\ 0 &= \tilde{\Phi}_{n,i} , & i &= 1, ..., \sigma , \\ 0 &= \sum \limits_{j=1}^{\sigma} \omega_{ij} \tilde{\Psi}_{n,i} , & i &= 1, ..., \sigma-1 , \end{aligned}\]

and update rule

\[\begin{aligned} q_{n+1} &= q_{n} + h \sum \limits_{i=1}^{s} b^1_{i} V_{n,i} + h \sum \limits_{i=1}^{\sigma} b^2_{i} \tilde{\Lambda}_{n,i} , \\ p_{n+1} &= p_{n} + h \sum \limits_{i=1}^{s} b^1_{i} F^1_{n,i} + h \sum \limits_{i=1}^{s} b^2_{i} F^2_{n,i} + h \sum \limits_{i=1}^{\sigma} b^3_{i} \tilde{F}^3_{n,i} , \\ 0 &= \phi (q_{n+1}, p_{n+1}) , \end{aligned}\]

with definitions

\[\begin{aligned} F^1_{n,i} &= \nabla H (Q_{n,i}) , & i &= 1, ..., s , \\ F^2_{n,i} &= \nabla \vartheta (Q_{n,i}) \cdot V_{n,i} , & i &= 1, ..., s , \\ \tilde{F}^3_{n,i} &= \nabla \phi (\tilde{Q}_{n,i}, \tilde{P}_{n,i}) \cdot \tilde{\Lambda}_{n,i} , & i &= 1, ..., \sigma , \\ \Phi_{n,i} &= \phi (Q_{n,i}, P_{n,i}) = P_{n,i} - \vartheta (Q_{n,i}) , & i &= 1, ..., s , \\ \tilde{\Phi}_{n,i} &= \phi (\tilde{Q}_{n,i}, \tilde{P}_{n,i}) = \tilde{P}_{n,i} - \vartheta (\tilde{Q}_{n,i}) , & i &= 1, ..., \sigma , \\ \tilde{\Psi}_{n,i} &= \psi (\tilde{Q}_{n,i}, \tilde{V}_{n,i}, \tilde{P}_{n,i}, \tilde{F}_{n,i}) = \tilde{F}_{n,i} - \tilde{V}_{n,i} \cdot \nabla \vartheta (\tilde{Q}_{n,i}) , & i &= 1, ..., \sigma , \end{aligned}\]

so that

\[\begin{aligned} F^1_{n,i} + F^2_{n,i} &= \frac{\partial L}{\partial q} (Q_{n,i}, V_{n,i}) , & i &= 1, ..., s , \\ \phi (Q_{n,i}, P_{n,i}) &= P_{n,i} - \frac{\partial L}{\partial v} (Q_{n,i}, V_{n,i}) , & i &= 1, ..., s , \\ \psi (\tilde{Q}_{n,i}, \tilde{V}_{n,i}, \tilde{P}_{n,i}, \tilde{F}_{n,i}) &= \tilde{F}_{n,i} - \tilde{V}_{n,i} \cdot \nabla \frac{\partial L}{\partial v} (\tilde{Q}_{n,i}, \tilde{V}_{n,i}) , & i &= 1, ..., \sigma . \end{aligned}\]

Specialised Partitioned Additive Runge-Kutta integrator for index-two DAE systems.

This integrator solves the following system of equations for the internal stages,

\[\begin{aligned} Q_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} a_{ij} V_{n,j} + h \sum \limits_{j=1}^{r} \alpha_{ij} U_{n,j} , & i &= 1, ..., s , \\ P_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} a_{ij} F_{n,j} + h \sum \limits_{j=1}^{r} \alpha_{ij} G_{n,j} , & i &= 1, ..., s , \\ \tilde{Q}_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} \tilde{a}_{ij} V_{n,j} + h \sum \limits_{j=1}^{r} \tilde{\alpha}_{ij} U_{n,j} , & i &= 1, ..., r , \\ \tilde{P}_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} \tilde{a}_{ij} F_{n,j} + h \sum \limits_{j=1}^{r} \tilde{\alpha}_{ij} G_{n,j} , & i &= 1, ..., r , \\ 0 &= \sum \limits_{j=1}^{r} \omega_{ij} \tilde{\Phi}_{n,j} , & i &= 1, ..., r-1 , \end{aligned}\]

with definitions

\[\begin{aligned} P_{n,i} &= \frac{\partial L}{\partial v} (Q_{n,i}, V_{n,i}) , & i &= 1, ..., s , \\ F_{n,i} &= \frac{\partial L}{\partial q} (Q_{n,i}, V_{n,i}) , & i &= 1, ..., s , \\ U_{n,i} &= \hphantom{-} \frac{\partial \phi}{\partial p} (\tilde{Q}_{n,i}, \tilde{P}_{n,i})^{T} \Lambda_{n,i} , & i &= 1, ..., r , \\ G_{n,i} &= - \frac{\partial \phi}{\partial q} (\tilde{Q}_{n,i}, \tilde{P}_{n,i})^{T} \Lambda_{n,i} , & i &= 1, ..., r , \\ \tilde{\Phi}_{n,i} &= \phi(\tilde{Q}_{n,i}, \tilde{P}_{n,i}) , & i &= 1, ..., r , \end{aligned}\]

and update rule

\[\begin{aligned} q_{n+1} &= q_{n} + h \sum \limits_{i=1}^{s} b_{i} V_{n,i} + h \sum \limits_{i=1}^{r} \beta_{i} U_{n,i} , \\ p_{n+1} &= p_{n} + h \sum \limits_{i=1}^{s} b_{i} F_{n,i} + h \sum \limits_{i=1}^{r} \beta_{i} G_{n,i} , \\ 0 &= \phi (q_{n+1}, p_{n+1}) . \end{aligned}\]

Variational partitioned additive Runge-Kutta integrator.

This integrator solves the following system of equations for the internal stages,

\[\begin{aligned} Q_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} a_{ij} V_{n,j} + h \sum \limits_{j=1}^{r} \alpha_{ij} U_{n,j} , & i &= 1, ..., s , \\ P_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} a_{ij} F_{n,j} + h \sum \limits_{j=1}^{r} \alpha_{ij} G_{n,j} , & i &= 1, ..., s , \\ \tilde{Q}_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} \tilde{a}_{ij} V_{n,j} + h \sum \limits_{j=1}^{r} \tilde{\alpha}_{ij} U_{n,j} , & i &= 1, ..., r , \\ \tilde{P}_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} \tilde{a}_{ij} F_{n,j} + h \sum \limits_{j=1}^{r} \tilde{\alpha}_{ij} G_{n,j} , & i &= 1, ..., r , \\ \tilde{\Phi}_{n,i} &= 0 , & i &= 1, ..., r , \end{aligned}\]

with definitions

\[\begin{aligned} P_{n,i} &= \frac{\partial L}{\partial v} (Q_{n,i}, V_{n,i}) , & i &= 1, ..., s , \\ F_{n,i} &= \frac{\partial L}{\partial q} (Q_{n,i}, V_{n,i}) , & i &= 1, ..., s , \\ U_{n,i} &= \hphantom{-} \frac{\partial \phi}{\partial p} (\tilde{Q}_{n,i}, \tilde{P}_{n,i})^{T} \Lambda_{n,i} , & i &= 1, ..., r , \\ G_{n,i} &= - \frac{\partial \phi}{\partial q} (\tilde{Q}_{n,i}, \tilde{P}_{n,i})^{T} \Lambda_{n,i} , & i &= 1, ..., r , \\ \tilde{\Phi}_{n,i} &= \phi(\tilde{Q}_{n,i}, \tilde{P}_{n,i}) , & i &= 1, ..., r , \end{aligned}\]

and update rule

\[\begin{aligned} q_{n+1} &= q_{n} + h \sum \limits_{i=1}^{s} b_{i} V_{n,i} + h \sum \limits_{i=1}^{r} \beta_{i} U_{n,i} , \\ p_{n+1} &= p_{n} + h \sum \limits_{i=1}^{s} b_{i} F_{n,i} + h \sum \limits_{i=1}^{r} \beta_{i} G_{n,i} . \end{aligned}\]

Specialised Partitioned Additive Runge-Kutta integrator for Variational systems.

This integrator solves the following system of equations for the internal stages,

\[\begin{aligned} Q_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} a_{ij} V_{n,j} + h \sum \limits_{j=1}^{r} \alpha_{ij} U_{n,j} , & i &= 1, ..., s , \\ P_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} a_{ij} F_{n,j} + h \sum \limits_{j=1}^{r} \alpha_{ij} G_{n,j} , & i &= 1, ..., s , \\ \tilde{Q}_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} \tilde{a}_{ij} V_{n,j} + h \sum \limits_{j=1}^{r} \tilde{\alpha}_{ij} U_{n,j} , & i &= 1, ..., r , \\ \tilde{P}_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} \tilde{a}_{ij} F_{n,j} + h \sum \limits_{j=1}^{r} \tilde{\alpha}_{ij} G_{n,j} , & i &= 1, ..., r , \\ 0 &= \sum \limits_{j=1}^{r} \omega_{ij} \tilde{\Phi}_{n,j} , & i &= 1, ..., r-1 , \end{aligned}\]

with definitions

\[\begin{aligned} P_{n,i} &= \frac{\partial L}{\partial v} (Q_{n,i}, V_{n,i}) , & i &= 1, ..., s , \\ F_{n,i} &= \frac{\partial L}{\partial q} (Q_{n,i}, V_{n,i}) , & i &= 1, ..., s , \\ U_{n,i} &= \hphantom{-} \frac{\partial \phi}{\partial p} (\tilde{Q}_{n,i}, \tilde{P}_{n,i})^{T} \Lambda_{n,i} , & i &= 1, ..., r , \\ G_{n,i} &= - \frac{\partial \phi}{\partial q} (\tilde{Q}_{n,i}, \tilde{P}_{n,i})^{T} \Lambda_{n,i} , & i &= 1, ..., r , \\ \tilde{\Phi}_{n,i} &= \phi(\tilde{Q}_{n,i}, \tilde{P}_{n,i}) , & i &= 1, ..., r , \end{aligned}\]

and update rule

\[\begin{aligned} q_{n+1} &= q_{n} + h \sum \limits_{i=1}^{s} b_{i} V_{n,i} + h \sum \limits_{i=1}^{r} \beta_{i} U_{n,i} , \\ p_{n+1} &= p_{n} + h \sum \limits_{i=1}^{s} b_{i} F_{n,i} + h \sum \limits_{i=1}^{r} \beta_{i} G_{n,i} , \\ 0 &= \phi (q_{n+1}, p_{n+1}) . \end{aligned}\]

Specialised Partitioned Additive Runge-Kutta integrator for Variational systems.

This integrator solves the following system of equations for the internal stages,

\[\begin{aligned} Q_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} a_{ij} V_{n,j} + h \sum \limits_{j=1}^{r} \alpha_{ij} U_{n,j} , & i &= 1, ..., s , \\ P_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} a_{ij} F_{n,j} + h \sum \limits_{j=1}^{r} \alpha_{ij} G_{n,j} , & i &= 1, ..., s , \\ \tilde{Q}_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} \tilde{a}_{ij} V_{n,j} + h \sum \limits_{j=1}^{r} \tilde{\alpha}_{ij} U_{n,j} , & i &= 1, ..., r , \\ \tilde{P}_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} \tilde{a}_{ij} F_{n,j} + h \sum \limits_{j=1}^{r} \tilde{\alpha}_{ij} G_{n,j} , & i &= 1, ..., r , \\ 0 &= \sum \limits_{j=1}^{r} \omega_{ij} \tilde{\Phi}_{n,j} , & i &= 1, ..., r-1 , \\ 0 &= \sum \limits_{i=1}^{r} \tilde{d}_i \, \Lambda_{n,i} , \end{aligned}\]

with definitions

\[\begin{aligned} P_{n,i} &= \frac{\partial L}{\partial v} (Q_{n,i}, V_{n,i}) , & i &= 1, ..., s , \\ F_{n,i} &= \frac{\partial L}{\partial q} (Q_{n,i}, V_{n,i}) , & i &= 1, ..., s , \\ U_{n,i} &= \hphantom{-} \frac{\partial \phi}{\partial p} (\tilde{Q}_{n,i}, \tilde{P}_{n,i})^{T} \Lambda_{n,i} , & i &= 1, ..., r , \\ G_{n,i} &= - \frac{\partial \phi}{\partial q} (\tilde{Q}_{n,i}, \tilde{P}_{n,i})^{T} \Lambda_{n,i} , & i &= 1, ..., r , \\ \tilde{\Phi}_{n,i} &= \phi(\tilde{Q}_{n,i}, \tilde{P}_{n,i}) , & i &= 1, ..., r , \end{aligned}\]

and update rule

\[\begin{aligned} q_{n+1} &= q_{n} + h \sum \limits_{i=1}^{s} b_{i} V_{n,i} + h \sum \limits_{i=1}^{r} \beta_{i} U_{n,i} , \\ p_{n+1} &= p_{n} + h \sum \limits_{i=1}^{s} b_{i} F_{n,i} + h \sum \limits_{i=1}^{r} \beta_{i} G_{n,i} . \end{aligned}\]

Specialised Partitioned Additive Runge-Kutta integrator for degenerate variational systems with projection on secondary constraint.

This integrator solves the following system of equations for the internal stages,

\[\begin{aligned} Q_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} a^1_{ij} V_{n,j} + h \sum \limits_{j=1}^{\sigma} a^2_{ij} \tilde{\Lambda}_{n,j} , & i &= 1, ..., s , \\ P_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} \bar{a}^1_{ij} F^1_{n,j} + h \sum \limits_{j=1}^{s} \bar{a}^2_{ij} F^2_{n,j} + h \sum \limits_{j=1}^{\sigma} \bar{a}^3_{ij} \tilde{F}^3_{n,j} , & i &= 1, ..., s , \\ 0 &= \Phi_{n,i} , & i &= 1, ..., s , \end{aligned}\]

the projective stages

\[\begin{aligned} \tilde{Q}_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} \alpha^1_{ij} \tilde{V}_{n,j} + h \sum \limits_{j=1}^{\sigma} \alpha^2_{ij} \tilde{\Lambda}_{n,j} , & i &= 1, ..., \sigma , \\ \tilde{P}_{n,i} &= p_{n} + h \sum \limits_{j=1}^{s} \bar{\alpha}^1_{ij} F^1_{n,j} + h \sum \limits_{j=1}^{s} \bar{\alpha}^2_{ij} \tilde{F}^2_{n,j} + h \sum \limits_{j=1}^{\sigma} \bar{\alpha}^3_{ij} \tilde{F}^3_{n,j} , & i &= 1, ..., \sigma , \\ 0 &= \tilde{\Phi}_{n,i} , & i &= 1, ..., \sigma , \\ 0 &= \sum \limits_{j=1}^{\sigma} \omega_{ij} \tilde{\Psi}_{n,i} , & i &= 1, ..., \sigma-1 , \end{aligned}\]

and update rule

\[\begin{aligned} q_{n+1} &= q_{n} + h \sum \limits_{i=1}^{s} b^1_{i} V_{n,i} + h \sum \limits_{i=1}^{\sigma} b^2_{i} \tilde{\Lambda}_{n,i} , \\ p_{n+1} &= p_{n} + h \sum \limits_{i=1}^{s} b^1_{i} F^1_{n,i} + h \sum \limits_{i=1}^{s} b^2_{i} F^2_{n,i} + h \sum \limits_{i=1}^{\sigma} b^3_{i} \tilde{F}^3_{n,i} , \\ 0 &= \phi (q_{n+1}, p_{n+1}) , \end{aligned}\]

with definitions

\[\begin{aligned} F^1_{n,i} &= \nabla H (Q_{n,i}) , & i &= 1, ..., s , \\ F^2_{n,i} &= \nabla \vartheta (Q_{n,i}) \cdot V_{n,i} , & i &= 1, ..., s , \\ \tilde{F}^3_{n,i} &= \nabla \phi (\tilde{Q}_{n,i}, \tilde{P}_{n,i}) \cdot \tilde{\Lambda}_{n,i} , & i &= 1, ..., \sigma , \\ \Phi_{n,i} &= \phi (Q_{n,i}, P_{n,i}) = P_{n,i} - \vartheta (Q_{n,i}) , & i &= 1, ..., s , \\ \tilde{\Phi}_{n,i} &= \phi (\tilde{Q}_{n,i}, \tilde{P}_{n,i}) = \tilde{P}_{n,i} - \vartheta (\tilde{Q}_{n,i}) , & i &= 1, ..., \sigma , \\ \tilde{\Psi}_{n,i} &= \psi (\tilde{Q}_{n,i}, \tilde{V}_{n,i}, \tilde{P}_{n,i}, \tilde{F}_{n,i}) = \tilde{F}_{n,i} - \tilde{V}_{n,i} \cdot \nabla \vartheta (\tilde{Q}_{n,i}) , & i &= 1, ..., \sigma , \end{aligned}\]

so that

\[\begin{aligned} F^1_{n,i} + F^2_{n,i} &= \frac{\partial L}{\partial q} (Q_{n,i}, V_{n,i}) , & i &= 1, ..., s , \\ \phi (Q_{n,i}, P_{n,i}) &= P_{n,i} - \frac{\partial L}{\partial v} (Q_{n,i}, V_{n,i}) , & i &= 1, ..., s , \\ \psi (\tilde{Q}_{n,i}, \tilde{V}_{n,i}, \tilde{P}_{n,i}, \tilde{F}_{n,i}) &= \tilde{F}_{n,i} - \tilde{V}_{n,i} \cdot \nabla \frac{\partial L}{\partial v} (\tilde{Q}_{n,i}, \tilde{V}_{n,i}) , & i &= 1, ..., \sigma . \end{aligned}\]

Parameters for right-hand side function of Hamiltonian Partitioned Additive Runge-Kutta methods.

Parameters for right-hand side function of Specialised Partitioned Additive Runge-Kutta methods for Variational systems.

Parameters for right-hand side function of Variational Partitioned Additive Runge-Kutta methods.

Parameters for right-hand side function of Specialised Partitioned Additive Runge-Kutta methods for Variational systems.

Holds the tableau of a Hamiltonian Partitioned Additive Runge-Kutta methods.

Holds the tableau of an Specialised Partitioned Additive Runge-Kutta method for Variational systems.

Holds the tableau of an Specialised Partitioned Additive Runge-Kutta method.

Holds the tableau of an Variational Partitioned Additive Runge-Kutta method.

Holds the tableau of an Specialised Partitioned Additive Runge-Kutta method for Variational systems.

Print additive Runge-Kutta coefficients.

Print additive Runge-Kutta coefficients.

Print multiplier Runge-Kutta coefficients.

Print projective Runge-Kutta coefficients.

Print SPARK coefficients.

Compute stages of specialised partitioned additive Runge-Kutta methods for variational systems.

Compute stages of specialised partitioned additive Runge-Kutta methods for variational systems.

Compute stages of specialised partitioned additive Runge-Kutta methods for variational systems.

Compute stages of specialised partitioned additive Runge-Kutta methods for variational systems.

Compute stages of specialised partitioned additive Runge-Kutta methods for variational systems.

Compute stages of variational partitioned additive Runge-Kutta methods.

Compute stages of specialised partitioned additive Runge-Kutta methods for variational systems.

Compute stages of variational partitioned additive Runge-Kutta methods.

Compute stages of specialised partitioned additive Runge-Kutta methods for variational systems.

Compute stages of specialised partitioned additive Runge-Kutta methods for variational systems.

Integrate an implicit DAE with a specialised partitioned additive Runge-Kutta integrator.

Parameters for right-hand side function of variational partitioned Runge-Kutta methods.

Variational partitioned Runge-Kutta integrator cache.

Fields

• q̃: initial guess of q
• p̃: initial guess of p
• ṽ: initial guess of v
• f̃: initial guess of f
• s̃: holds shift due to periodicity of solution
• Q: internal stages of q
• P: internal stages of p
• V: internal stages of v
• F: internal stages of f
• Y: integral of vector field of internal stages of q
• Z: integral of vector field of internal stages of p

Variational partitioned Runge-Kutta integrator.

Variational partitioned Runge-Kutta integrator.

Variational partitioned Runge-Kutta integrator.

Variational special partitioned additive Runge-Kutta integrator.

Variational partitioned Runge-Kutta integrator.

Variational partitioned Runge-Kutta integrator with projection on secondary constraint.

The VPRK integrator solves the following system of equations for the internal stages,

\[\begin{aligned} Q_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} a_{ij} \, \big( V_{n,j} + \Lambda_{n,j} \big) , \\ P_{n,i} &= p_{n} + h \sum \limits_{i=1}^{s} \bar{a}_{ij} \, \big( F_{n,j} + \nabla \vartheta (Q_{n,j})^{T} \cdot \Lambda_{n,j} \big) - d_i \lambda , \\ 0 &= \sum \limits_{i=1}^{s} d_i V_i , \\ 0 &= \sum \limits_{j=1}^{s} \omega_{ij} \Psi_{n,j} , \end{aligned}\]

with definitions

\[\begin{aligned} P_{n,i} &= \dfrac{\partial L}{\partial v} (Q_{n,i}, V_{n,i}) , \\ F_{n,i} &= \dfrac{\partial L}{\partial q} (Q_{n,i}, V_{n,i}) , \\ \Psi_{n,i} &= \psi(Q_{n,i}, V_{n,i}, P_{n,i}, F_{n,i}) , \end{aligned}\]

and update rule

\[\begin{aligned} q_{n+1} &= q_{n} + h \sum \limits_{i=1}^{s} b_{i} \, \big( V_{n,i} + \Lambda_{n,i} \big) , \\ p_{n+1} &= p_{n} + h \sum \limits_{i=1}^{s} \bar{b}_{i} \, \big( F_{n,i} + \nabla \vartheta (Q_{n,j})^{T} \cdot \Lambda_{n,j} \big) , \\ 0 &= \phi (q_{n+1}, p_{n+1}) , \end{aligned}\]

satisfying the symplecticity conditions

\[\begin{aligned} b_{i} \bar{a}_{ij} + \bar{b}_{j} a_{ji} &= b_{i} \bar{b}_{j} , & \bar{b}_i &= b_i , \end{aligned}\]

the primary constraint,

\[\begin{aligned} \phi(q,p) = p - \vartheta (q) = 0 , \end{aligned}\]

at the final solution $(q_{n+1}, p_{n+1})$, and super positions of the secondary constraints,

\[\begin{aligned} \psi(q,\dot{q},p,\dot{p}) = \dot{p} - \dot{q} \cdot \nabla \vartheta (q) = \big( \nabla \vartheta (q) - \nabla \vartheta^{T} (q) \big) \cdot \dot{q} - \nabla H (q) = 0, \end{aligned}\]

which, evaluated at the internal stages, read

\[\begin{aligned} \Psi_{n,j} = \big( \nabla \vartheta (Q_{n,j}) - \nabla \vartheta^{T} (Q_{n,j}) \big) \cdot V_{n,j} - \nabla H (Q_{n,j}) . \end{aligned}\]

Here, $\omega$ is a $(s-1) \times s$ matrix, chosen such that the resulting method has optimal order. The vector $d$ is zero for Gauss-Legendre methods and needs to be chosen appropriately for Gauss-Lobatto methods (for details see documentation of VPRK methods).

Variational partitioned Runge-Kutta integrator.

Variational partitioned Runge-Kutta integrator with standard projection.

Variational partitioned Runge-Kutta integrator.

Projected Variational Gauss-Legendre Runge-Kutta integrator.

Variational partitioned Runge-Kutta integrator.

Parameters for right-hand side function of Variational Partitioned Runge-Kutta methods.

Parameters for right-hand side function of Variational Partitioned Runge-Kutta methods.

Parameters for right-hand side function of Variational Partitioned Runge-Kutta methods.

Parameters for right-hand side function of Variational Partitioned Runge-Kutta methods.

Parameters for right-hand side function of Variational Partitioned Runge-Kutta methods.

Parameters for right-hand side function of Variational Partitioned Runge-Kutta methods.

Parameters for right-hand side function of Variational Partitioned Runge-Kutta methods.

Parameters for right-hand side function of Variational Partitioned Runge-Kutta methods.

Parameters for right-hand side function of projected Gauss-Legendre Runge-Kutta methods.

Parameters for right-hand side function of Variational Partitioned Runge-Kutta methods.

TableauVPRK: Tableau of a Variational Partitioned Runge-Kutta method

\[\begin{aligned} P_{n,i} &= \dfrac{\partial L}{\partial v} (Q_{n,i}, V_{n,i}) , & Q_{n,i} &= q_{n} + h \sum \limits_{j=1}^{s} a_{ij} \, V_{n,j} , & q_{n+1} &= q_{n} + h \sum \limits_{i=1}^{s} b_{i} \, V_{n,i} , \\ F_{n,i} &= \dfrac{\partial L}{\partial q} (Q_{n,i}, V_{n,i}) , & P_{n,i} &= p_{n} + h \sum \limits_{i=1}^{s} \bar{a}_{ij} \, F_{n,j} - d_i \lambda , & p_{n+1} &= p_{n} + h \sum \limits_{i=1}^{s} \bar{b}_{i} \, F_{n,i} , \\ && 0 &= \sum \limits_{i=1}^{s} d_i V_i , && \end{aligned}\]

satisfying the symplecticity conditions

\[\begin{aligned} b_{i} \bar{a}_{ij} + b_{j} a_{ji} &= b_{i} b_{j} , & \bar{b}_i &= b_i . \end{aligned}\]

Variational partitioned Runge-Kutta integrator with symplectic projection.

Variational partitioned Runge-Kutta integrator with variational projection on $(p_{n}, q_{n+1})$.

Variational partitioned Runge-Kutta integrator with variational projection on $(q_{n}, p_{n+1})$.

Compute stages of variational partitioned Runge-Kutta methods.

Compute solution of degenerate symplectic partitioned Runge-Kutta methods.

Compute stages of variational partitioned Runge-Kutta methods.

Compute stages of variational special partitioned additive Runge-Kutta methods.

Compute stages of variational partitioned Runge-Kutta methods.

Compute stages of variational partitioned Runge-Kutta methods.

Compute stages of projected variational partitioned Runge-Kutta methods.

Compute stages of variational partitioned Runge-Kutta methods.

Compute stages of projected variational partitioned Runge-Kutta methods.

Compute stages of projected Gauss-Legendre Runge-Kutta methods.

Integrate ODE with variational partitioned Runge-Kutta integrator.

Integrate ODE with variational partitioned Runge-Kutta integrator.

Integrate DAE with variational special partitioned additive Runge-Kutta integrator.

Integrate ODE with variational partitioned Runge-Kutta integrator.

Integrate ODE with variational partitioned Runge-Kutta integrator.

Integrate ODE with variational partitioned Runge-Kutta integrator.

Integrate ODE with variational partitioned Runge-Kutta integrator.

Integrate ODE with projected Gauss-Legendre Runge-Kutta integrator.

Integrate ODE with variational partitioned Runge-Kutta integrator.

Integrate ODE with variational partitioned Runge-Kutta integrator.

Continuous Galerkin Variational Integrator.

ParametersCGVI: Parameters for right-hand side function of continuous Galerkin variational Integrator.

Parameters

• Θ: function of the noncanonical one-form (∂L/∂v)
• f: function of the force (∂L/∂q)
• Δt: time step
• b: weights of the quadrature rule
• c: nodes of the quadrature rule
• x: nodes of the basis
• m: mass matrix
• a: derivative matrix
• r₀: reconstruction coefficients at the beginning of the interval
• r₁: reconstruction coefficients at the end of the interval
• t: initial time
• q: solution of q at time t
• p: solution of p at time t

Compute stages of variational partitioned Runge-Kutta methods.

Integrate ODE with variational partitioned Runge-Kutta integrator.

Nonlinear function cache for Discontinuous Galerkin Variational Integrator.

Parameters

• ST: data type
• D: number of dimensions
• S: number of degrees of freedom
• R: number of nodes of quadrature formula

Fields

• X: degrees of freedom
• Q: solution at quadrature nodes
• V: velocity at quadrature nodes
• P: one-form at quadrature nodes
• F: forces at quadrature nodes
• q: current solution of $q_{n}$
• q⁻: current solution of $q_{n}^{-}$
• q⁺: current solution of $q_{n}^{+}$
• q̅: current solution of $q_{n+1}$
• q̅⁻: current solution of $q_{n+1}^{-}$
• q̅⁺: current solution of $q_{n+1}^{+}$
• ϕ: average of the solution at $t_{n}$
• ϕ̅: average of the solution at $t_{n+1}$
• λ: jump of the solution at $t_{n}$
• λ̅: jump of the solution at $t_{n+1}$
• θ: one-form evaluated across at $t_{n}$
• Θ̅: one-form evaluated across at $t_{n+1}$
• g: projection evaluated across at $t_{n}$
• g̅: projection evaluated across at $t_{n+1}$

IntegratorDGVI: Discontinuous Galerkin Variational Integrator.

The DGVI integrators arise from the discretization of the action integral

\[\mathcal{A} [q] = \int \limits_{0}^{T} L(q(t), \dot{q}(t)) \, dt ,\]

with $L$ a fully degenerate Lagrangian of the form

\[L(q, \dot{q}) = \vartheta (q) \cdot \dot{q} - H(q) ,\]

where $\vartheta (q)$ denotes the Cartan one-form and $H(q)$ the Hamiltonian, which is usually given by the total energy of the system.

Discretization

Within each interval $(t_{n}, t_{n+1})$ a piecewise-polynomial approximation $q_h$ of the trajectory $q$ is constructed using $S$ basis functions $\varphi_{i}$,

\[q_h(t) \vert_{(t_{n}, t_{n+1})} = \sum \limits_{i=1}^{S} x_{n,i} \, \bar{\varphi}_{n,i} (t) ,\]

where $\bar{\varphi}_{n,i} (t)$ is a rescaled basis function, defined by

\[\bar{\varphi}_{n,i} (t) = \varphi_{i} \bigg( \frac{t - t_{n}}{t_{n+1} - t_{n}} \bigg) ,\]

and it is assumed that $\varphi_{i}$ is compactly supported on $[0,1]$. These approximations $q_h(t)$ are not assumed to be continuous across interval boundaries $t_{n}$ but usuaslly have jumps.

The integral over $(t_{n}, t_{n+1})$ is approximated by a quadrature rule with $R$ nodes $c_i$ and weights $b_i$. Denote by $m$ and $a$ mass and derivative matrices, respectively, whose elements are given by

\[m_{ij} = \varphi_j (c_i) , \qquad a_{ij} = \varphi_j' (c_i) , \qquad i = 1, ..., R , \; j = 1, ..., S .\]

With that, the solution and its time derivative at the quadrature points can be written as

\[Q_{n,i} \equiv q_h(t_n + c_i h) = m_{ij} x_{n,j} , \qquad V_{n,i} \equiv \dot{q}_h (t_n + c_i h) = a_{ij} x_{n,j} ,\]

where

\[x_{n} = ( x_{n,1}, ..., x_{n,S} )^T\]

is the vector containing the degrees of freedom of $q_h \vert_{[t_{n}, t_{n+1}]}$. The limits of $q_h(t)$ at $t_{n}$ and $t_{n+1}$ are given by

\[q_{n}^{+} = \lim \limits_{t \downarrow t_{n}} q_h(t) = \sum \limits_{j=1}^{S} r^{+}_{j} \delta x_{n,j} , \hspace{3em} q_{n+1}^{-} = \lim \limits_{t \uparrow t_{n+1}} q_h(t) = \sum \limits_{j=1}^{S} r^{-}_{j} \delta x_{n,j} .\]

The discrete action reads

\[\mathcal{A}_d [x_d] = h \sum \limits_{n=0}^{N-1} \bigg[ \sum \limits_{i=1}^{R} b_i \big[ \vartheta (Q_{n,i}) \cdot V_{n,i} - H(Q_{n,i}) \big] \\ + \frac{\vartheta (q_n) + \vartheta (q_n^+)}{2} \cdot (q_n^+ - q_n) + \frac{\vartheta (q_{n+1}^-) + \vartheta (q_{n+1})}{2} \cdot (q_{n+1} - q_{n+1}^-) \bigg] ,\]

so that using the relations

\[\delta Q_{n,i} = m_{ij} \delta x_{n,j} , \qquad \delta V_{n,i} = \frac{a_{ij}}{h} \delta x_{n,j} , \qquad \delta q_{n}^- = r^{-}_{j} \delta x_{n-1,j} , \qquad \delta q_{n}^+ = r^{+}_{j} \delta x_{n,j} ,\]

the discrete action principle leads to the discrete equations of motion,

\[0 = \sum \limits_{i=1}^{R} b_i \big[ h m_{ij} \nabla \vartheta (Q_{n,i}) \cdot V_{n,i} + a_{ij} \vartheta (Q_{n,i}) - h m_{ij} \nabla H(Q_{n,i}) \big] \\ + r^{+}_{j} \, \frac{ \vartheta ( q_{n } ) + \vartheta( q_{n }^+ ) }{2} - r^{-}_{j} \, \frac{ \vartheta ( q_{n+1}^- ) + \vartheta( q_{n+1} ) }{2} \big] \\ + h r^{+}_{j} \, \nabla \vartheta (q_{n }^+) \cdot (q_{n }^+ - q_{n } ) + h r^{-}_{j} \, \nabla \vartheta (q_{n+1}^-) \cdot (q_{n+1} - q_{n+1}^-) ,\]

and

\[\vartheta(q_{n}^+) = \vartheta (q_{n}^-) + \nabla \vartheta (q_{n}) \cdot (q_{n}^+ - q_{n}^-) ,\]

for all $n$ and all $j$. Let us introduce the variable $p_n$ as

\[p_{n} = \vartheta (q_{n}^-) + \nabla \vartheta (q_{n}) \cdot (q_{n} - q_{n}^-) ,\]

so that

\[\vartheta(q_{n}^+) = p_{n} + \nabla \vartheta (q_{n}) \cdot (q_{n}^+ - q_{n}) .\]

Then the above equations provide a map $(q_{n}, p_{n}) \mapsto (q_{n+1}, p_{n+1})$. In order to solve these equations, initial conditions $q_{0}$ and $p_{0} = \vartheta(q_{0})$ have to be prescribed.

Fields

• equation: Implicit Ordinary Differential Equation
• basis: piecewise polynomial basis
• quadrature: numerical quadrature rule
• Δt: time step
• params: ParametersDGVI
• solver: nonlinear solver
• iguess: initial guess
• q: current solution vector for trajectory
• p: current solution vector for one-form
• cache: temporary variables for nonlinear solver

ParametersDGVI: Parameters for right-hand side function of Discontinuous Galerkin Variational Integrator.

Parameters

• DT: data type
• TT: parameter type
• D: dimension of the system
• S: number of basis nodes
• R: number of quadrature nodes

Fields

• Θ: function of the noncanonical one-form (∂L/∂v)
• f: function of the force (∂L/∂q)
• g: function of the projection ∇ϑ(q)⋅v
• Δt: time step
• b: quadrature weights
• c: quadrature nodes
• m: mass matrix
• a: derivative matrix
• r⁻: reconstruction coefficients, jump lhs value
• r⁺: reconstruction coefficients, jump rhs value
• t: initial time
• q: solution of q at time t
• q⁻: solution of q⁻ at time t
• q⁺: solution of q⁺ at time t

Compute one-form and forces at quadrature nodes.

Compute solution at quadrature nodes and across jump.

Compute velocities at quadrature nodes.

Compute stages of variational partitioned Runge-Kutta methods.

Stochastic Explicit Runge-Kutta integrator cache.

Structure for holding the internal stages Q, the values of the drift vector and the diffusion matrix evaluated at the internal stages V=v(Q), B=B(Q), and the increments Y = Δt*a_drift*v(Q) + a_diff*B(Q)*ΔW.

Structure for holding the internal stages Q, the values of the drift vector and the diffusion matrix evaluated at the internal stages V=v(Q), B=B(Q), and the increments Y = a_drift*v(Q)*Δt + a_diff*B(Q)*ΔW.

Structure for holding the internal stages Q, the values of the drift vector and the diffusion matrix evaluated at the internal stages V=v(Q), B=B(Q), and the increments Y = Δt*a_drift*v(Q) + a_diff*B(Q)*ΔW.

Weak Stochastic Explicit Runge-Kutta integrator cache.

Structure for holding the internal stages Q0, and Q1 the values of the drift vector and the diffusion matrix evaluated at the internal stages V=v(Q0), B=B(Q1), and the increments Y = Δt*a_drift*v(Q) + a_diff*B(Q)*ΔW.

Stochastic Explicit Runge-Kutta integrator.

Stochastic implicit partitioned Runge-Kutta integrator.

Stochastic implicit Runge-Kutta integrator.

Stochastic implicit partitioned Runge-Kutta integrator.

Weak Stochastic Explicit Runge-Kutta integrator.

Stochastic implicit Runge-Kutta integrator.

Parameters for stochastic explicit Runge-Kutta methods.

Parameters for right-hand side function of implicit Runge-Kutta methods.

A - if positive, the upper bound of the Wiener process increments; if A=0.0, no truncation

Parameters for right-hand side function of implicit Runge-Kutta methods.

A - if positive, the upper bound of the Wiener process increments; if A=0.0, no truncation

Parameters for right-hand side function of implicit Runge-Kutta methods.

A - if positive, the upper bound of the Wiener process increments; if A=0.0, no truncation.

Parameters for weak stochastic explicit Runge-Kutta methods.

Parameters for right-hand side function of weak implicit Runge-Kutta methods.

Holds the tableau of a stochastic explicit Runge-Kutta method.

Order of the tableau is not included, because unlike in the deterministic setting, it depends on the properties of the noise (e.g., the dimension of the Wiener process and the commutativity properties of the diffusion matrix).

Orders stored in qdrift, qdiff and qdiff2 are understood as the classical orders of these methods.

Holds the tableau of a stochastic implicit partitioned Runge-Kutta method. qdrift, pdrift hold the RK coefficients for the drift part, and qdiff, pdiff hold the RK coefficients for the diffusion part of the SDE.

Order of the tableau is not included, because unlike in the deterministic setting, it depends on the properties of the noise (e.g., the dimension of the Wiener process and the commutativity properties of the diffusion matrix).

Orders stored in qdrift and qdiff are understood as the classical orders of these methods.

Holds the tableau of a stochastic implicit Runge-Kutta method.

qdrift holds the RK coefficients for the drift part, qdiff holds the RK coefficients for the diffusion part of the SDE.

Order of the tableau is not included, because unlike in the deterministic setting, it depends on the properties of the noise (e.g., the dimension of the Wiener process and the commutativity properties of the diffusion matrix).

Orders stored in qdrift and qdiff are understood as the classical orders of these methods.

Holds the tableau of a stochastic implicit split partitioned Runge-Kutta method.

qdrift, pdrift1, pdrift2 hold the RK coefficients for the drift parts, and qdiff, pdiff1, pdiff2 hold the RK coefficients for the diffusion part of the SDE.

Order of the tableau is not included, because unlike in the deterministic setting, it depends on the properties of the noise (e.g., the dimension of the Wiener process and the commutativity properties of the diffusion matrix)

Orders stored in qdrift and qdiff are understood as the classical orders of these methods.

Holds the tableau of a weak explicit Runge-Kutta method.

Reference: Andreas Rossler, "Second order Runge-Kutta methods for Stratonovich stochastic differential equations",
BIT Numerical Mathematics (2007) 47, equation (5.1).

Order of the tableau is not included, because unlike in the deterministic setting, it depends on the properties of the noise (e.g., the dimension of the Wiener process and the commutativity properties of the diffusion matrix)

Orders stored in qdrift, qdiff are irrelevant and set to 0.

qdrift0, qdrift1, qdrift2 correspond to A0, A1, A2 in the paper qdiff0, qdiff1, qdiff2, qdiff3 correspond to B0, B1, B2, B3 qdrift0.b = alpha qdiff0.b = beta1 qdiff3.b = beta2 qdrift0.c = c0 qdrift1.c = c1 qdrift2.c = c2

Holds the tableau of a weak implicit Runge-Kutta method.

Reference: Wang, Hong, Xu, "Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems",
Commun. Comput. Phys. 21(1), 2017.

Order of the tableau is not included, because unlike in the deterministic setting, it depends on the properties of the noise (e.g., the dimension of the Wiener process and the commutativity properties of the diffusion matrix)

Orders stored in qdrift and qdiff are understood as the classical orders of these methods.

qdrift0, qdrift1 correspond to A0, A1 in the paper qdiff0, qdiff1, qdiff3 correspond to B0, B1, B3 qdrift0.b = alpha qdiff0.b = beta qdrift0.c = c0 qdrift1.c = c1

Unpacks the data stored in x = (Y[1][1], Y[1][2], ... Y[1][D], Y[2][1], ...) into Y, calculates the internal stages Q, the values of the RHS of the SDE ( v(Q) and B(Q) ), and assigns them to V and B. Unlike for FIRK, here Y = a v(Q) Δt + â B(Q) ΔW

Unpacks the data stored in x = (Y0[1][1], Y0[1][2]], ... Y0[1][D], ... Y0[S][D], Y1[1][1,1], Y1[1][2,1], ... Y1[1][D,1], Y1[1][1,2], Y1[1][2,2], ... Y1[1][D,2], ... Y1[S][D,M] ) into Y0 and Y1, calculates the internal stages Q0 and Q1, the values of the RHS of the SDE ( v(Q0) and B(Q1) ), and assigns them to V and B. Unlike for FIRK, here Y = Δt a v(Q) + â B(Q) ΔW.

Unpacks the data stored in x = (Y[1][1], Y[1][2], ... Y[1][D], Y[2][1], ..., Z[1][1], Z[1][2], ... Z[1][D], Z[2][1], ...) into Y, Z, calculates the internal stages Q, P, the values of the RHS of the SDE ( v(Q,P), f(Q,P), B(Q,P) and G(Q,P) ), and assigns them to V, F, B and G. Unlike for FIRK, here Y = Δt a_drift v(Q,P) + a_diff B(Q,P) ΔW, Z = Δt â_drift v(Q,P) + â_diff B(Q,P) ΔW.

Unpacks the data stored in x = (Y[1][1], Y[1][2], ... Y[1][D], Y[2][1], ..., Z[1][1], Z[1][2], ... Z[1][D], Z[2][1], ...) into Y, Z, calculates the internal stages Q, P, the values of the RHS of the SDE ( vi(Q,P), fi(Q,P), Bi(Q,P) and Gi(Q,P) ), and assigns them to V[i], F[i], B[i] and G[i]. Unlike for FIRK, here Y = Δt a_drift v(Q,P) + a_diff B(Q,P) ΔW, Z = Δt â1_drift f1(Q,P) + Δt â2_drift f2(Q,P) + â1_diff G1(Q,P) ΔW + â2_diff G2(Q,P) ΔW.

This function computes initial guesses for Y, Z and assigns them to int.solver.x The prediction is calculated using an explicit integrator.

SIMPLE SOLUTION The simplest initial guess for Y, Z is 0: int.solver.x .= zeros(eltype(int), 2*tableau(int).s*ndims(int))

USING AN EXPLICIT INTEGRATOR TO COMPUTE AN INITIAL GUESS Below we use the R2 method of Burrage & Burrage to calculate the internal stages at the times c[1]...c[s]. This approach seems to give very good approximations if the time step and magnitude of noise are not too large. If the noise intensity is too big, one may have to perform a few iterations of the explicit method with a smaller time step, use a higher-order explicit method (e.g. CL or G5), or use the simple solution above.

When calling this function, int.params should contain the data: int.params.q - the q solution at the previous time step int.params.p - the p solution at the previous time step int.params.t - the time of the previous step int.params.ΔW- the increment of the Brownian motion for the current step

This function computes initial guesses for Y and assigns them to int.solver.x. The prediction is calculated using an explicit integrator.

SIMPLE SOLUTION The simplest initial guess for Y is 0: int.solver.x .= zeros(eltype(int), tableau(int).s*ndims(int))

USING AN EXPLICIT INTEGRATOR TO COMPUTE AN INITIAL GUESS Below we use the R2 method of Burrage & Burrage to calculate the internal stages at the times c[1]...c[s]. This approach seems to give very good approximations if the time step and magnitude of noise are not too large. If the noise intensity is too big, one may have to perform a few iterations of the explicit method with a smaller time step, use a higher-order explicit method (e.g. CL or G5), or use the simple solution above.

When calling this function, int.params should contain the data: int.params.q - the solution at the previous time step int.params.t - the time of the previous step int.params.ΔW - the increment of the Brownian motion for the current step

This function computes initial guesses for Y, Z and assigns them to int.solver.x.

For SISPRK we are NOT IMPLEMENTING an InitialGuess.

SIMPLE SOLUTION The simplest initial guess for Y, Z is 0.

This function computes initial guesses for Y and assigns them to int.solver.x For WIRK we are NOT IMPLEMENTING an InitialGuess.

Using an explicit integrator to predict the next step's value (like in SIRK) does not seem to be a good idea here, because the integrators are convergent in the weak sense only, and there is no guarantee that the explicit integrator will produce anything close to the desired solution...

The simplest initial guess for Y is 0.

Update solution for stochastic split partitioned Runge-Kutta methods

• q, p: the solution vector to be updated
• V, F1, F2: the matrix containing the drift vectors evaluated at the internal stages v(Qi), fi(Qi)
• B, G1, G2: the array containing the diffusion matrices evaluated at the internal stages B(Qi), Gi(Qi)
• bqdrift, bpdrift1, bpdrift2: the Runge-Kutta coefficients for the drift parts of the q and p equations
• bqdiff, bpdiff1, bpdiff2: the Runge-Kutta coefficients for the diffusion parts of the q and p equations
• Δt: the time step
• ΔW: the increments of the Brownian motion

Update solution for stochastic partitioned Runge-Kutta methods

• q, p: the solution vector to be updated
• V, F: the matrix containing the drift vectors evaluated at the internal stages v(Qi), f(Qi)
• B, G: the array containing the diffusion matrices evaluated at the internal stages B(Qi), G(Qi)
• bqdrift, bpdrift: the Runge-Kutta coefficients for the drift parts of the q and p equations
• bqdiff, bpdiff: the Runge-Kutta coefficients for the diffusion parts of the q and p equations
• Δt: the time step
• ΔW: the increments of the Brownian motion

Update solution for weak Runge-Kutta methods WERK

• x: the solution vector to be updated
• V: the matrix containing the drift vector evaluated at the internal stages v(Q_i)
• B1: the array containing the diffusion matrix evaluated at the internal stages H^(l)i, such that B1[:,l,i] is evaluated at H^(l)i
• B2: the array containing the diffusion matrix evaluated at the internal stages Ĥ^(l)i, such that B2[:,l,i] is evaluated at Ĥ^(l)i
• α: the Runge-Kutta coefficients for the drift part
• β1: the Runge-Kutta coefficients for the diffusion term with the random increments
• β2: the Runge-Kutta coefficients for the second diffusion term
• Δt: the time step
• ΔW: the increments of the Brownian motion represented by the random variables Î^(k)

Update solution for stochastic Runge-Kutta methods (SERK)

• x: the solution vector to be updated
• V: the matrix containing the drift vector evaluated at the internal stages v(Q_i)
• B: the array containing the diffusion matrix evaluated at the internal stages B(Q_i)
• bdrift: the Runge-Kutta coefficients for the drift part
• bdiff: the Runge-Kutta coefficients for the ΔW terms of the diffusion part
• bdiff2: the Runge-Kutta coefficients for the ΔZ terms of the diffusion part
• Δt: the time step
• ΔW: the increments of the Brownian motion
• ΔZ: the integrals of the increments of the Brownian motion

Update solution for stochastic Runge-Kutta methods (SIRK and WIRK)

• x: the solution vector to be updated
• V: the matrix containing the drift vector evaluated at the internal stages v(Qi) (SIRK) or v(Q0i) (WIRK)
• B: the array containing the diffusion matrix evaluated at the internal stages B(Qi) (SIRK) or B(Q1^(l)i) (WIRK)
• bdrift: the Runge-Kutta coefficients for the drift part
• bdiff: the Runge-Kutta coefficients for the diffusion part
• Δt: the time step
• ΔW: the increments of the Brownian motion (SFIRK) or the increments represented by the random variables Î^(k) (WFIRK)

Compute stages of implicit Runge-Kutta methods.

Compute stages of stochastic implicit Runge-Kutta methods.

Compute stages of stochastic implicit split partitioned Runge-Kutta methods.

Compute stages of weak implicit Runge-Kutta methods.

Integrate SDE with explicit Runge-Kutta integrator.

Integrate PSDE with a stochastic implicit partitioned Runge-Kutta integrator.

Integrate SDE with a stochastic implicit Runge-Kutta integrator.

Integrate PSDE with a stochastic implicit partitioned Runge-Kutta integrator.

Integrate SDE with explicit Runge-Kutta integrator.

Integrate SDE with a stochastic implicit Runge-Kutta integrator.