OrdinaryDiffEqExponentialRK

Methods for semi-linear differential equations.

Installation

To be able to access the solvers in OrdinaryDiffEqLinear, you must first install them use the Julia package manager:

using Pkg
Pkg.add("OrdinaryDiffEqExponentialRK")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqExponentialRK, SciMLOperators
A = [2.0 -1.0; -1.0 2.0]
linnonlin_f1 = MatrixOperator(A)
linnonlin_f2 = (du, u, p, t) -> du .= 1.01 .* u
linnonlin_fun_iip = SplitFunction(linnonlin_f1, linnonlin_f2)
tspan = (0.0, 1.0)
u0 = [0.1, 0.1]
prob = SplitODEProblem(linnonlin_fun_iip, u0, tspan)
sol = solve(prob, ETDRK4(), dt = 1 / 4)

Full list of solvers

OrdinaryDiffEqExponentialRK.LawsonEuler — Type

LawsonEuler(; krylov = false,
              m = 30,
              iop = 0)

Semilinear ODE solver First order exponential Euler scheme (fixed timestepping)

Keyword Arguments

krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
m: Controls the size of Krylov subspace.
iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Hochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.

source

OrdinaryDiffEqExponentialRK.NorsettEuler — Type

NorsettEuler(; krylov = false,
               m = 30,
               iop = 0)

Semilinear ODE solver First order exponential-RK scheme. Alias: ETD1

Keyword Arguments

krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
m: Controls the size of Krylov subspace.
iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Hochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.

source

OrdinaryDiffEqExponentialRK.ETD1 — Type

ETD1(;
    krylov = false, m = 30, iop = 0, autodiff = AutoForwardDiff(),
    concrete_jac = nothing, step_limiter = trivial_limiter!
)

ETD1: First Order Exponential Time Differencing (Semilinear ODE Solver)

Alias for NorsettEuler: ETD1 === NorsettEuler. ETD1 is the name this method carries in the exponential time differencing literature, NorsettEuler the name from the exponential-Runge-Kutta literature.

The linear part of a semilinear problem u' = Au + f(u,t) is integrated exactly with the matrix exponential, and the nonlinear part with a first order quadrature.

Method Properties

Order: 1
Time stepping: Fixed
Problem type: Semilinear ODEs (SplitODEProblem, or a supplied Jacobian)

Keyword Arguments

Identical to NorsettEuler; krylov, m, and iop control whether and how the matrix exponential actions are approximated in a Krylov subspace.

References

Hochbruck, Marlis, and Alexander Ostermann. "Exponential Integrators." Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.

source

OrdinaryDiffEqExponentialRK.ETD2 — Type

ETD2: Exponential Runge-Kutta Method Second order Exponential Time Differencing method (in development).

source

OrdinaryDiffEqExponentialRK.ETDRK2 — Type

ETDRK2(; krylov = false,
         m = 30,
         iop = 0)

Semilinear ODE solver 2nd order exponential-RK scheme.

Keyword Arguments

krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
m: Controls the size of Krylov subspace.
iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Hochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.

source

OrdinaryDiffEqExponentialRK.ETDRK3 — Type

ETDRK3(; krylov = false,
         m = 30,
         iop = 0)

Semilinear ODE solver 3rd order exponential-RK scheme.

Keyword Arguments

krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
m: Controls the size of Krylov subspace.
iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Hochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.

source

OrdinaryDiffEqExponentialRK.ETDRK4 — Type

ETDRK4(; krylov = false,
         m = 30,
         iop = 0)

Semilinear ODE solver 4th order exponential-RK scheme (fixed timestepping)

Keyword Arguments

krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
m: Controls the size of Krylov subspace.
iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Hochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.

source

OrdinaryDiffEqExponentialRK.HochOst4 — Type

HochOst4(; krylov = false,
           m = 30,
           iop = 0)

Semilinear ODE solver 4th order exponential-RK scheme with stiff order 4.

Keyword Arguments

krylov: Determines whether Krylov approximation or operator caching is used, the latter only available for semilinear problems. krylov=true is much faster for larger systems and is thus recommended whenever there are >100 ODEs.
m: Controls the size of Krylov subspace.
iop: If not zero, determines the length of the incomplete orthogonalization procedure (IOP). Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

Hochbruck, Marlis, and Alexander Ostermann. “Exponential Integrators.” Acta Numerica 19 (2010): 209–286. doi:10.1017/S0962492910000048.

source