Split ODE Solvers

# Split ODE Solvers

The solvers which are available for a SplitODEProblem depend on the input linearity and number of components. Each solver has functional form (or many) that it allows.

## Implicit-Explicit (IMEX) ODE

The Implicit-Explicit (IMEX) ODE is a SplitODEProblem with two functions:

$\frac{du}{dt} = f_1(t,u) + f_2(t,u)$

where the first function is the stiff part and the second function is the non-stiff part (implicit integration on f1, explicit integration on f2).

### Recommended Methods

The recommended method in most cases is KenCarp4. In cases of extreme stiffness or for high tolerances, KenCarp3 can be a good choice. The ARKODE methods are generally inefficient and diverge unless the options are tweaked to match the problem, though for large enough PDEs the ARKODE method with linear_solver=:GMRES is a good choice.

### OrdinaryDiffEq.jl

• SplitEuler: 1st order fully explicit method. Used for testing accuracy of splits.
• IMEXEuler : 1st order explicit Euler mixed with implicit Euler. Fixed time step only.
• CNAB2: Crank-Nicholson Adams Bashforth Order 2. Fixed time step only.
• CNLF: Crank-Nicholson Leapfrog of Order 2. Fixed time step only.
• SBDF2 : 2nd order IMEX BDF method. Fixed time step only.
• SBDF3 : 3rd order IMEX BDF method. Fixed time step only. In development.
• SBDF4 : 4th order IMEX BDF method. Fixed time step only. In development.
• KenCarp3: An A-L stable stiffly-accurate 3rd order ESDIRK method.
• KenCarp4: An A-L stable stiffly-accurate 4rd order ESDIRK method.
• KenCarp5: An A-L stable stiffly-accurate 5rd order ESDIRK method.

### Sundials.jl

• ARKODE: An additive Runge-Kutta method. Order between 3rd and 5th. For a list of available options, please see its ODE solver page

## Semilinear ODE

The Semilinear ODE is a SplitODEProblem with one linear operator and one nonlinear function:

$\frac{du}{dt} = Au + f(t,u)$

See the documentation page for DiffEqOperator for details about how to define linear operators from a matrix or finite difference discretization of derivative operators.

The appropriate algorithms for this form are:

### OrdinaryDiffEq.jl

• GenericIIF1 - First order Implicit Integrating Factor method. Fixed timestepping only. Doesn't support Krylov approximation.
• GenericIIF2 - Second order Implicit Integrating Factor method. Fixed timestepping only. Doesn't support Krylov approximation.
• LawsonEuler - First order exponential Euler scheme. Fixed timestepping only.
• NorsettEuler - First order exponential-RK scheme. Fixed timestepping only. Alias: ETD1.
• ETD2 - Second order Exponential Time Differencing method (in development). Fixed timestepping only. Doesn't support Krylov approximation.
• ETDRK2 - 2nd order exponential-RK scheme. Fixed timestepping only.
• ETDRK3 - 3rd order exponential-RK scheme. Fixed timestepping only.
• ETDRK4 - 4th order exponential-RK scheme. Fixed timestepping only.
• HochOst4 - 4th order exponential-RK scheme with stiff order 4. Fixed timestepping only.
• Exprb32 - 3rd order adaptive Exponential Rosenbrock scheme (in development).
• Exprb43 - 4th order adaptive Exponential Rosenbrock scheme (in development).

Note that the generic algorithms GenericIIF1 and GenericIIF2 allow for a choice of nlsolve.

By default, the exponential methods cache matrix functions such as exp(dt*A) to accelerate the time stepping for small systems. For large systems, using Krylov-based versions of the methods can allow for lazy calculation of exp(dt*A)*v and similar entities, and thus improve performance.

To tell a solver to use Krylov methods, pass krylov=true to its constructor. You can also manually set the size of the Krylov subspace by setting the m parameter, which defaults to 30. For example

LawsonEuler(krylov=true, m=50)

constructs a Lawson-Euler method which uses a size-50 Krylov subspace. Note that m only sets an upper bound to the Krylov subspace size. If a convergence criterion is met (determined by the reltol of the integrator), "happy breakdown" will occur and the Krylov subspace will only be constructed partially.

For more advanced control over the Krylov algorithms, you can change the length of the incomplete orthogonalization procedure (IOP) [1] by setting the iop parameter in the constructor. By default, IOP is turned off and full Arnoldi iteration is used. Note that if the linear operator is hermitian, then the Lanczos algorithm will always be used and IOP setting is ignored.

[1]

Koskela, A. (2015). Approximating the matrix exponential of an advection-diffusion operator using the incomplete orthogonalization method. In Numerical Mathematics and Advanced Applications-ENUMATH 2013 (pp. 345-353). Springer, Cham.