# SDAE Solvers

The recommendations for SDAEs are the same recommended implicit SDE methods for stiff equations when the SDAE is specified in mass matrix form.

#### Mass Matrix Form

• ImplicitEM - An order 0.5 Ito drift-implicit method. This is a theta method which defaults to theta=1 or the Trapezoid method on the drift term. This method defaults to symplectic=false, but when true and theta=1/2 this is the implicit Midpoint method on the drift term and is symplectic in distribution. Can handle all forms of noise, including non-diagonal, scalar, and colored noise. Uses a 1.0/1.5 heuristic for adaptive time stepping.
• STrapezoid - An alias for ImplicitEM with theta=1/2
• SImplicitMidpoint - An alias for ImplicitEM with theta=1/2 and symplectic=true
• ImplicitEulerHeun - An order 0.5 Stratonovich drift-implicit method. This is a theta method which defaults to theta=1/2 or the Trapezoid method on the drift term. This method defaults to symplectic=false, but when true and theta=1 this is the implicit Midpoint method on the drift term and is symplectic in distribution. Can handle all forms of noise, including non-diagonal, scalar, and colored noise. Uses a 1.0/1.5 heuristic for adaptive time stepping.
• ImplicitRKMil - An order 1.0 drift-implicit method. This is a theta method which defaults to theta=1 or the Trapezoid method on the drift term. Defaults to solving the Ito problem, but ImplicitRKMil(interpretation=:Stratonovich) makes it solve the Stratonovich problem. This method defaults to symplectic=false, but when true and theta=1/2 this is the implicit Midpoint method on the drift term and is symplectic in distribution. Handles diagonal and scalar noise. Uses a 1.5/2.0 heuristic for adaptive time stepping.
• ISSEM - An order 0.5 split-step Ito implicit method. It is fully implicit, meaning it can handle stiffness in the noise term. This is a theta method which defaults to theta=1 or the Trapezoid method on the drift term. This method defaults to symplectic=false, but when true and theta=1/2 this is the implicit Midpoint method on the drift term and is symplectic in distribution. Can handle all forms of noise, including non-diagonal, scalar, and colored noise. Uses a 1.0/1.5 heuristic for adaptive time stepping.
• ISSEulerHeun - An order 0.5 split-step Stratonovich implicit method. It is fully implicit, meaning it can handle stiffness in the noise term. This is a theta method which defaults to theta=1 or the Trapezoid method on the drift term. This method defaults to symplectic=false, but when true and theta=1/2 this is the implicit Midpoint method on the drift term and is symplectic in distribution. Can handle all forms of noise, including non-diagonal,Q scalar, and colored noise. Uses a 1.0/1.5 heuristic for adaptive time stepping.
• SKenCarp - Adaptive L-stable drift-implicit strong order 1.5 for additive Ito and Stratonovich SDEs with weak order 2. Can handle diagonal, non-diagonal and scalar additive noise.*†

## Notes

†: Does not step to the interval endpoint. This can cause issues with discontinuity detection, and discrete variables need to be updated appropriately.

*: Note that although SKenCarp uses the same table as KenCarp3, solving a ODE problem using SKenCarp by setting g(du,u,p,t) = du .= 0 will take much more steps than KenCarp3 because error estimator of SKenCarp is different (because of noise terms) and default value of qmax (maximum permissible ratio of relaxing/tightening dt for adaptive steps) is smaller for StochasticDiffEq algorithms.