OrdinaryDiffEqExponentialRK

Exponential Runge-Kutta methods are specialized integrators for semi-linear differential equations of the form du/dt = Au + f(u,t), where A is a linear operator (often representing diffusion or dispersion) and f represents nonlinear terms. These methods are particularly effective for stiff linear parts combined with non-stiff nonlinear terms. Important: The nonlinear term f(u,t) must be non-stiff for these methods to be effective.

Key Properties

Exponential RK methods provide:

  • Exact treatment of linear parts using matrix exponential functions
  • High-order accuracy for both linear and nonlinear components
  • Excellent stability properties for problems with stiff linear operators
  • Efficient handling of semi-linear PDEs after spatial discretization
  • Reduced timestep restrictions compared to traditional explicit methods
  • Preservation of qualitative behavior for many physical systems

When to Use Exponential RK Methods

These methods are recommended for:

  • Semi-linear PDEs with stiff diffusion/dispersion and moderate non-stiff nonlinearity
  • Reaction-diffusion systems with fast diffusion and slower non-stiff reactions
  • Nonlinear Schrödinger equations and other dispersive wave equations with non-stiff nonlinear terms
  • Pattern formation problems (Turing patterns, phase field models) where nonlinearity is non-stiff
  • Quantum dynamics with linear Hamiltonian and non-stiff nonlinear interactions
  • Problems with strong linear damping or oscillatory linear parts combined with non-stiff nonlinear terms
  • Spatially discretized PDEs where the linear part dominates stiffness but the nonlinear part remains non-stiff

Mathematical Background

For problems du/dt = Au + f(u,t), exponential methods compute the exact solution of the linear part Au using exp(A*dt) and treat the nonlinear part f(u,t) with Runge-Kutta-like stages. This approach is particularly effective when A represents well-understood physics (diffusion, dispersion, linear oscillations).

Solver Selection Guide

Basic exponential time differencing (ETD)

  • LawsonEuler: First-order exponential Euler method
  • NorsettEuler / ETD1: Alternative first-order scheme
  • ETDRK2: Second-order exponential RK
  • ETDRK3: Third-order exponential RK
  • ETDRK4: Fourth-order exponential RK, popular choice
  • ETD2: Second-order exponential time differencing (in development)

High-order specialized methods

  • HochOst4: Fourth-order exponential RK with enhanced stability
  • Exp4: Fourth-order EPIRK scheme

Adaptive exponential Rosenbrock

  • Exprb32: Third-order adaptive method with error control
  • Exprb43: Fourth-order adaptive method

EPIRK (Exponential Propagation Iterative RK) methods

  • EPIRK4s3A: Fourth-order with stiff order 4
  • EPIRK4s3B: Alternative fourth-order variant
  • EPIRK5s3: Fifth-order method (note: marked as broken)
  • EXPRB53s3: Fifth-order with stiff order 5
  • EPIRK5P1, EPIRK5P2: Fifth-order variants

Performance Recommendations

  • For most semi-linear problems: ETDRK4
  • For adaptive stepsize: Exprb43
  • For high stiffness in linear part: EPIRK4s3A or EPIRK4s3B
  • For maximum accuracy: EXPRB53s3

Implementation Requirements

These methods require:

  • Computation of matrix exponentialsexp(A*dt) and related functions
  • Krylov subspace methods for large systems (automatic in most cases)
  • Proper problem formulation with identified linear and nonlinear parts

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

Installation

To be able to access the solvers in OrdinaryDiffEqExponentialRK, 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

function lorenz!(du, u, p, t)
    du[1] = 10.0 * (u[2] - u[1])
    du[2] = u[1] * (28.0 - u[3]) - u[2]
    du[3] = u[1] * u[2] - (8 / 3) * u[3]
end
u0 = [1.0; 0.0; 0.0]
tspan = (0.0, 100.0)
prob = ODEProblem(lorenz!, u0, tspan)
sol = solve(prob, EPIRK5s3())

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.

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.

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.

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.

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.

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.

Adaptive Exponential Rosenbrock Methods

OrdinaryDiffEqExponentialRK.Exprb32 — Type
Exprb32(; m = 30,
          iop = 0)

Semilinear ODE solver 3rd order adaptive Exponential-Rosenbrock scheme.

Keyword Arguments

  • 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, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286. (https://doi.org/10.1017/S0962492910000048)

OrdinaryDiffEqExponentialRK.Exprb43 — Type
Exprb43(; m = 30,
          iop = 0)

Semilinear ODE solver 4th order adaptive Exponential-Rosenbrock scheme.

Keyword Arguments

  • 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, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286. (https://doi.org/10.1017/S0962492910000048)

Exponential Propagation Iterative Runge-Kutta Methods (EPIRK)

Missing docstring.

Missing docstring for Exp4. Check Documenter's build log for details.

Missing docstring.

Missing docstring for EPIRK4s3A. Check Documenter's build log for details.

Missing docstring.

Missing docstring for EPIRK4s3B. Check Documenter's build log for details.

Missing docstring.

Missing docstring for EPIRK5s3. Check Documenter's build log for details.

Missing docstring.

Missing docstring for EXPRB53s3. Check Documenter's build log for details.

Missing docstring.

Missing docstring for EPIRK5P1. Check Documenter's build log for details.

Missing docstring.

Missing docstring for EPIRK5P2. Check Documenter's build log for details.