# DAE Solvers

For medium to low accuracy small numbers of DAEs in constant mass matrices form, the Rosenbrock23 and Rodas4 methods are good choices which will get good efficiency if the mass matrix is constant. Rosenbrock23 is better for low accuracy (error tolerance <1e-4) and Rodas4 is better for high accuracy. Another choice at high accuracy is RadauIIA5.

Non-constant mass matrices are not directly supported: users are advised to transform their problem through substitution to a DAE with constant mass matrices.

If the problem cannot be defined in mass matrix form, the recommended method for performance is IDA from the Sundials.jl package if you are solving problems with Float64. If Julia types are required, currently DABDF2 is the best method.

## Full List of Methods

#### Initialization Schemes

For all OrdinaryDiffEq.jl methods, an initialization scheme can be set with a common keyword argument initializealg. The choices are:

• BrownFullBasicInit: For Index-1 DAEs implicit DAEs and and semi-explicit DAEs in mass matrix form. Keeps the differential variables constant. Requires du0 when used on a DAEProblem.
• ShampineCollocationInit: For Index-1 DAEs implicit DAEs and and semi-explicit DAEs in mass matrix form. Changes both the differential and algebraic variables.
• NoInit: Explicitly opts-out of DAE initialization.

### OrdinaryDiffEq.jl (Implicit ODE)

These methods from OrdinaryDiffEq are for DAEProblem specifications.

• DImplicitEuler - 1st order A-L and stiffly stable adaptive implicit Euler
• DABDF2 - 2nd order A-L stable adaptive BDF method.

### OrdinaryDiffEq.jl (Mass Matrix)

These methods require the DAE to be an ODEProblem in mass matrix form. For extra options for the solvers, see the ODE solver page.

#### Rosenbrock Methods

• ROS3P - 3rd order A-stable and stiffly stable Rosenbrock method. Keeps high accuracy on discretizations of nonlinear parabolic PDEs.
• Rodas3 - 3rd order A-stable and stiffly stable Rosenbrock method.
• RosShamp4- An A-stable 4th order Rosenbrock method.
• Veldd4 - A 4th order D-stable Rosenbrock method.
• Velds4 - A 4th order A-stable Rosenbrock method.
• GRK4T - An efficient 4th order Rosenbrock method.
• GRK4A - An A-stable 4th order Rosenbrock method. Essentially "anti-L-stable" but efficient.
• Ros4LStab - A 4th order L-stable Rosenbrock method.
• Rodas4 - A 4th order A-stable stiffly stable Rosenbrock method with a stiff-aware 3rd order interpolant
• Rodas42 - A 4th order A-stable stiffly stable Rosenbrock method with a stiff-aware 3rd order interpolant
• Rodas4P - A 4th order A-stable stiffly stable Rosenbrock method with a stiff-aware 3rd order interpolant. 4th order on linear parabolic problems and 3rd order accurate on nonlinear parabolic problems (as opposed to lower if not corrected).
• Rodas4P2 - A 4th order L-stable stiffly stable Rosenbrock method with a stiff-aware 3rd order interpolant. 4th order on linear parabolic problems and 3rd order accurate on nonlinear parabolic problems. It is an improvement of Roadas4P and in case of inexact Jacobians a second order W method.
• Rodas5 - A 5th order A-stable stiffly stable Rosenbrock method. Currently has a Hermite interpolant because its stiff-aware 3rd order interpolant is not yet implemented. This means the interpolation is unstable on algebraic variables, meaning this algorithm should not be used with saveat or post-solution interpolation on DAEs.

#### Rosenbrock-W Methods

• Rosenbrock23 - An Order 2/3 L-Stable Rosenbrock-W method which is good for very stiff equations with oscillations at low tolerances. 2nd order stiff-aware interpolation.
• Rosenbrock32 - An Order 3/2 A-Stable Rosenbrock-W method which is good for mildy stiff equations without oscillations at low tolerances. Note that this method is prone to instability in the presence of oscillations, so use with caution. 2nd order stiff-aware interpolation.
• RosenbrockW6S4OS - A 4th order L-stable Rosenbrock-W method (fixed step only).
• ROS34PW1a - A 4th order L-stable Rosenbrock-W method.
• ROS34PW1b - A 4th order L-stable Rosenbrock-W method.
• ROS34PW2 - A 4th order stiffy accurate Rosenbrock-W method for PDAEs.
• ROS34PW3 - A 4th order strongly A-stable (Rinf~0.63) Rosenbrock-W method.

#### FIRK Methods

• RadauIIA5 - An A-B-L stable fully implicit Runge-Kutta method with internal tableau complex basis transform for efficiency.

#### SDIRK Methods

• ImplicitEuler - Stage order 1. A-B-L-stable. Adaptive timestepping through a divided differences estimate via memory. Strong-stability preserving (SSP).
• ImplicitMidpoint - Stage order 1. Symplectic. Good for when symplectic integration is required.
• Trapezoid - A second order A-stable symmetric ESDIRK method. "Almost symplectic" without numerical dampening. Also known as Crank-Nicolson when applied to PDEs. Adaptive timestepping via divided differences on the memory. Good for highly stiff equations which are non-oscillatory.

#### Multistep Methods

Quasi-constant stepping is the time stepping strategy which matches the classic GEAR, LSODE, and ode15s integrators. The variable-coefficient methods match the ideas of the classic EPISODE integrator and early VODE designs. The Fixed Leading Coefficient (FLC) methods match the behavior of the classic VODE and Sundials CVODE integrator.

• QNDF1 - An adaptive order 1 quasi-constant timestep L-stable numerical differentiation function (NDF) method. Optional parameter kappa defaults to Shampine's accuracy-optimal -0.1850.
• QBDF1 - An adaptive order 1 L-stable BDF method. This is equivalent to implicit Euler but using the BDF error estimator.
• ABDF2 - An adaptive order 2 L-stable fixed leading coefficient multistep BDF method.
• QNDF2 - An adaptive order 2 quasi-constant timestep L-stable numerical differentiation function (NDF) method.
• QBDF2 - An adaptive order 2 L-stable BDF method using quasi-constant timesteps.
• QNDF - An adaptive order quasi-constant timestep NDF method. Utilizes Shampine's accuracy-optimal kappa values as defaults (has a keyword argument for a tuple of kappa coefficients).
• QBDF - An adaptive order quasi-constant timestep BDF method.

### Sundials.jl

Note that this setup is not automatically included with DifferentialEquations.jl. To use the following algorithms, you must install and use Sundials.jl:

]add Sundials
using Sundials
• IDA - This is the IDA method from the Sundials.jl package.

Note that the constructors for the Sundials algorithms take a main argument:

• linearsolver - This is the linear solver which is used in the Newton iterations. The choices are:

• :Dense - A dense linear solver.
• :Band - A solver specialized for banded Jacobians. If used, you must set the position of the upper and lower non-zero diagonals via jac_upper and jac_lower.
• :LapackDense - A version of the dense linear solver that uses the Julia-provided OpenBLAS-linked LAPACK for multithreaded operations. This will be faster than :Dense on larger systems but has noticable overhead on smaller (<100 ODE) systems.
• :LapackBand - A version of the banded linear solver that uses the Julia-provided OpenBLAS-linked LAPACK for multithreaded operations. This will be faster than :Band on larger systems but has noticable overhead on smaller (<100 ODE) systems.
• :GMRES - A GMRES method. Recommended first choice Krylov method
• :BCG - A Biconjugate gradient method.
• :PCG - A preconditioned conjugate gradient method. Only for symmetric linear systems.
• :TFQMR - A TFQMR method.
• :KLU - A sparse factorization method. Requires that the user specifies a Jacobian. The Jacobian must be set as a sparse matrix in the ODEProblem type.

Example:

IDA() # Newton + Dense solver
IDA(linear_solver=:Band,jac_upper=3,jac_lower=3) # Banded solver with nonzero diagonals 3 up and 3 down
IDA(linear_solver=:BCG) # Biconjugate gradient method                                   

All of the additional options are available. The constructor is:

IDA(;linear_solver=:Dense,jac_upper=0,jac_lower=0,krylov_dim=0,
max_order = 5,
max_error_test_failures = 7,
max_nonlinear_iters = 3,
nonlinear_convergence_coefficient = 0.33,
nonlinear_convergence_coefficient_ic = 0.0033,
max_num_steps_ic = 5,
max_num_jacs_ic = 4,
max_num_iters_ic = 10,
max_num_backs_ic = 100,
use_linesearch_ic = true,
max_convergence_failures = 10,
init_all = false,
prec = nothing, psetup = nothing)

See the Sundials manual for details on the additional options. The option init_all controls the initial condition consistency routine. If the initial conditions are inconsistant (i.e. they do not satisfy the implicit equation), init_all=false means that the algebraic variables and derivatives will be modified in order to satisfy the DAE. If init_all=true, all initial conditions will be modified to satify the DAE.

Note that here prec is a (left) preconditioner function prec(z,r,p,t,y,fy,gamma,delta) where:

• z: the computed output vector
• r: the right-hand side vector of the linear system
• p: the parameters
• t: the current independent variable
• du: the current value of f(u,p,t)
• gamma: the gamma of W = M - gamma*J
• delta: the iterative method tolerance

and psetup is the preconditioner setup function for pre-computing Jacobian information. Where:

• p: the parameters
• t: the current independent variable
• resid: the current residual
• u: the current state
• du: the current derivative of the state
• gamma: the gamma of W = M - gamma*J

psetup is optional when prec is set.

DASKR.jl is not automatically included by DifferentialEquations.jl. To use this algorithm, you will need to install and use the package:

]add DASKR
using DASKR
• daskr - This is a wrapper for the well-known DASKR algorithm.

All additional options are available. The constructor is:

function daskr(;linear_solver=:Dense,
jac_upper=0,jac_lower=0,max_order = 5,
non_negativity_enforcement = 0,
non_negativity_enforcement_array = nothing,
max_krylov_iters = nothing,
num_krylov_vectors = nothing,
max_number_krylov_restarts = 5,
krylov_convergence_test_constant = 0.05,
exclude_algebraic_errors = false)

Choices for the linear solver are:

• :Dense
• :Banded
• :SPIGMR, a Krylov method

### DASSL.jl

• dassl - A native Julia implementation of the DASSL algorithm.

### ODEInterfaceDiffEq.jl

These methods require the DAE to be an ODEProblem in mass matrix form. For extra options for the solvers, see the ODE solver page.

• seulex - Extrapolation-algorithm based on the linear implicit Euler method.
• radau - Implicit Runge-Kutta (Radau IIA) of variable order between 5 and 13.
• radau5 - Implicit Runge-Kutta method (Radau IIA) of order 5.
• rodas - Rosenbrock 4(3) method.