OrdinaryDiffEqNordsieck

Nordsieck form multistep methods represent an alternative approach to traditional multistep algorithms. Instead of storing past solution values, these methods maintain a vector of scaled derivatives (similar to Taylor series coefficients) to advance the solution. This representation was pioneered in classic codes like LSODE, VODE, and CVODE.

Key Properties

Nordsieck methods provide:

• Derivative-based representation instead of solution history
• Improved restartability after discontinuities using derivative information
• Variable order and stepsize capabilities
• Alternative to history-based multistep methods
• Research and experimental implementations

When to Use Nordsieck Methods

These methods are recommended for:

• Research applications exploring alternative multistep representations
• Problems with discontinuities where restartability is important
• Experimental comparisons with traditional multistep methods
• Development of discontinuity-aware algorithms

Important Limitations

Experimental Status

• Considered experimental and inferior to modern BDF implementations
• Generally recommend FBDF instead for production use
• Maintained for research purposes and future development
• Numerical instabilities can arise from higher derivative representations

Performance Considerations

• Less robust than fixed-leading coefficient BDF methods
• Higher computational overhead for derivative maintenance
• Potential stability issues with derivative representations

Mathematical Background

The Nordsieck form represents the solution using scaled derivatives: y_n = [y, h*y', h²*y''/2!, h³*y'''/3!, ...]

This representation allows reconstruction of the solution and its derivatives, enabling restarts after discontinuities without losing accuracy.

Solver Selection Guide

Nordsieck implementations

• AN5: Fifth-order Adams method with fixed leading coefficient
• JVODE: Variable order Adams/BDF method (experimental LSODE-style)
• JVODE_Adams: JVODE configured for Adams methods
• JVODE_BDF: JVODE configured for BDF methods

Recommended alternatives

• For most applications: Use QNDF or FBDF instead
• For stiff problems: Prefer modern BDF implementations
• For research: These methods are appropriate for experimental work

Research and Development

These implementations serve as:

• Experimental testbed for Nordsieck form algorithms
• Research platform for discontinuity-aware methods
• Development basis for future improved BDF implementations
• Educational examples of alternative multistep representations

Usage Guidelines

• Not recommended for production applications
• Use FBDF or QNDF for reliable multistep integration
• Consider these methods only for research or experimental purposes
• Expect potentially lower performance compared to modern alternatives

Installation

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

using Pkg
Pkg.add("OrdinaryDiffEqNordsieck")

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 OrdinaryDiffEqNordsieck

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, AN5())

Full list of solvers

AN5()