OrdinaryDiffEqLinear

Specialized methods for linear and semi-linear differential equations where the system can be written in matrix form du/dt = A(t,u) * u or du/dt = A(t) * u. These methods exploit the linear structure to provide exact solutions or highly accurate integration.

Key Properties

Linear ODE methods provide:

Exact solutions for time-independent linear systems
Geometric integration preserving Lie group structure
High-order Magnus expansions for time-dependent linear systems
Lie group methods for matrix differential equations
Excellent stability for a wide range of linear systems
Specialized algorithms for different types of linear operators

When to Use Linear Methods

These methods are essential for:

Linear systemsdu/dt = A * u with constant or time-dependent matrices
Matrix differential equations on Lie groups (rotation matrices, etc.)
Quantum dynamics with Hamiltonian evolution
Linear oscillators and harmonic systems
Time-dependent linear systems with periodic or smooth coefficients
Geometric mechanics requiring preservation of group structure

Mathematical Background

For linear systems du/dt = A(t) * u, the exact solution is u(t) = exp(∫A(s)ds) * u₀. Linear methods approximate this matrix exponential using various mathematical techniques like Magnus expansions, Lie group integrators, and specialized exponential methods.

Solver Selection Guide

Time and state-independent (constant A)

LinearExponential: Exact solution for du/dt = A * u with constant matrix A

Time-dependent, state-independent (A(t))

MagnusMidpoint: Second-order Magnus method
MagnusLeapfrog: Second-order Magnus leapfrog scheme
MagnusGauss4: Fourth-order with Gauss quadrature
MagnusGL4: Fourth-order Gauss-Legendre Magnus method
MagnusGL6: Sixth-order Gauss-Legendre Magnus method
MagnusGL8: Eighth-order Gauss-Legendre Magnus method
MagnusNC6: Sixth-order Newton-Cotes Magnus method
MagnusNC8: Eighth-order Newton-Cotes Magnus method

State-dependent (A(u))

LieEuler: First-order Lie group method
RKMK2: Second-order Runge-Kutta-Munthe-Kaas method
RKMK4: Fourth-order Runge-Kutta-Munthe-Kaas method
LieRK4: Fourth-order Lie Runge-Kutta method
CG2: Second-order Crouch-Grossman method
CG4a: Fourth-order Crouch-Grossman method
CayleyEuler: First-order method using Cayley transformations

Adaptive methods

MagnusAdapt4: Fourth-order adaptive Magnus method

Time and state-dependent (A(t,u))

CG3: Third-order Crouch-Grossman method for most general case

Method Selection Guidelines

For constant linear systems: LinearExponential (exact)
For time-dependent systems: Magnus methods based on desired order
For matrix Lie groups: Lie group methods (RKMK, LieRK4, CG)
For high accuracy: Higher-order Magnus methods (GL6, GL8)
For adaptive integration: MagnusAdapt4

Special Considerations

These methods require:

Proper problem formulation with identified linear structure
Matrix operator interface for operator-based problems
Understanding of Lie group structure for geometric problems

Installation

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

using Pkg
Pkg.add("OrdinaryDiffEqLinear")

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 OrdinaryDiffEqLinear, SciMLOperators
function update_func(A, u, p, t)
    A[1, 1] = 0
    A[2, 1] = sin(u[1])
    A[1, 2] = -1
    A[2, 2] = 0
end
A0 = ones(2, 2)
A = MatrixOperator(A0, update_func = update_func)
u0 = ones(2)
tspan = (0.0, 30.0)
prob = ODEProblem(A, u0, tspan)
sol = solve(prob, LieRK4(), dt = 1 / 4)

Full list of solvers

Time and State-Independent Solvers

OrdinaryDiffEqLinear.LinearExponential — Type

LinearExponential(; krylov = :off,
                    m = 10,
                    iop = 0)

Semilinear ODE solver Exact solution formula for linear, time-independent problems.

Keyword Arguments

krylov:
- :off: cache the operator beforehand. Requires Matrix(A) method defined for the operator A.
- :simple: uses simple Krylov approximations with fixed subspace size m.
- :adaptive: uses adaptive Krylov approximations with internal timestepping.
m: Controls the size of Krylov subspace if krylov=:simple, and the initial subspace size if krylov=:adaptive.
iop: If not zero, determines the length of the incomplete orthogonalization procedure Note that if the linear operator/Jacobian is hermitian, then the Lanczos algorithm will always be used and the IOP setting is ignored.

References

@book{strogatz2018nonlinear, title={Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering}, author={Strogatz, Steven H}, year={2018}, publisher={CRC press}}