DAE Solvers

Recomended Methods

The recommended method for performance is IDA from the Sundials.jl package if you are solving problems with Float64. It's a very well-optimized method, and allows you to have a little bit of control over the linear solver to better tailor it to your problem. A similar algorithm is daskr. Which one is more efficient is problem-dependent.

If your problem requires special Julia types like arbitrary precision numbers, then dassl is the method of choice.

Full List of Methods

Sundials.jl

• 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.

  • :Diagonal - This method is specialized for diagonal Jacobians.

  • BCG - A Biconjugate gradient method.

  • TFQMR - A TFQMR method.

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)

See the Sundials manual for details on the additional options.

DASKR.jl

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

Pkg.add("DASKR")
using DASKR

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

DASSL.jl

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