Differential Equation Integrator Benchmarks in MATLAB

A MATLAB benchmark repository for implementing and comparing numerical integrators on classical ordinary differential equation (ODE) test problems.

The project evaluates explicit, implicit, adaptive, multistep, stiff, symplectic, splitting, and energy-oriented methods on six benchmark problems:

1. Linear decay / Dahlquist test equation
2. Harmonic oscillator
3. Van der Pol oscillator
4. Lorenz system
5. Robertson stiff chemical kinetics problem
6. Kepler two-body problem

The repository is designed for numerical-analysis education, control engineering, robotics simulation, computational physics, and scientific-computing benchmarking.

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What this repository does

For each benchmark problem, the framework:

• runs all applicable integrators,
• computes a high-accuracy reference trajectory,
• measures numerical error,
• measures CPU time,
• counts function evaluations,
• detects blow-up / NaN / solver failure,
• evaluates conservation-law errors when available,
• produces tables in .csv format,
• produces comparison plots in .png format.

The benchmark output is generated under:

results/<BenchmarkName>/

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Repository structure

.
├── README.md
├── LICENSE
├── CONTRIBUTING.md
├── CITATION.cff
├── .gitignore
├── run_all.m
├── run_one_benchmark.m
├── src/
│   ├── benchmarks/       # Benchmark differential equations
│   ├── config/           # Method and benchmark registries
│   ├── core/             # Shared integration, metrics, reference, Newton utilities
│   ├── methods/          # MATLAB implementation of each integrator
│   └── plotting/         # Plot generation utilities
├── docs/
│   ├── methods/          # One Markdown file per integrator
│   ├── benchmarks/       # One Markdown file per benchmark equation
│   ├── tutorials/        # How to add methods/problems
│   └── references.md
├── tests/
│   └── test_smoke.m
└── results/              # Generated result tables and plots

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Quick start

Open MATLAB in the repository root and run:

run_all

To run only one benchmark:

run_one_benchmark('HarmonicOscillator')
run_one_benchmark('RobertsonKinetics')
run_one_benchmark('KeplerTwoBody')

Results are written to:

results/<BenchmarkName>/

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Differential equations: short overview

Differential equations relate unknown functions to their derivatives. Major types include:

Type	Example	Typical use
ODE	dx/dt = f(t,x)	control, mechanics, circuits, biology
PDE	u_t = α u_xx	heat, waves, fluids, electromagnetics
DAE	F(t,x,x') = 0	constrained mechanics, circuits, process systems
SDE	dX = a(X)dt + b(X)dW	finance, noise-driven physics, biology
Delay differential equation	x'(t)=f(x(t),x(t-τ))	population, control with delay
Hybrid differential equation	continuous dynamics + switching	robotics contact, relays, digital control

This repository focuses on initial-value ODEs, including first-order systems and second-order mechanical systems rewritten in first-order form.

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Integrator families implemented

Category	Implemented methods	Documentation
Basic explicit	Forward Euler, Explicit Midpoint, Heun, Ralston	docs/methods/ForwardEuler.md, ExplicitMidpoint.md, Heun.md, Ralston.md
Runge-Kutta	Kutta RK3, Classical RK4, RKF45, Dormand-Prince RK45	KuttaRK3.md, RK4.md, RKF45.md, DormandPrinceRK45.md
Implicit one-step	Backward Euler, Implicit Midpoint, Trapezoidal Rule	BackwardEuler.md, ImplicitMidpoint.md, TrapezoidalRule.md
Linear multistep	Adams-Bashforth 2, Adams-Bashforth 4, Adams-Moulton 2, ABM4 predictor-corrector, BDF2	AdamsBashforth2.md, AdamsBashforth4.md, AdamsMoulton2.md, ABM4PredictorCorrector.md, BDF2.md
Stiff / linearly implicit	Rosenbrock-Euler, ROS2, Exponential Euler / exponential Rosenbrock-Euler	RosenbrockEuler.md, ROS2.md, ExponentialEuler.md
Geometric / mechanical	Symplectic Euler, Verlet, Velocity Verlet, Leapfrog, Yoshida 4, Strang splitting, Newmark-beta	SymplecticEuler.md, Verlet.md, VelocityVerlet.md, Leapfrog.md, Yoshida4.md, StrangSplitting.md, NewmarkBeta.md
Energy-oriented	Discrete Gradient Oscillator	DiscreteGradientOscillator.md

Not every method is meaningful for every benchmark. For example, Velocity Verlet is only run on separable conservative mechanical systems such as the harmonic oscillator and Kepler problem. Stiff solvers are especially meaningful for Robertson and high-parameter Van der Pol.

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Benchmark equations

Benchmark	Main property tested	Documentation
Linear decay / Dahlquist test equation	absolute stability, order, scalar accuracy	LinearTestEquation.md
Harmonic oscillator	energy drift, phase error, phase portrait	HarmonicOscillator.md
Van der Pol oscillator	nonlinear oscillation, stiffness transition	VanDerPolOscillator.md
Lorenz system	chaos, short-time accuracy, qualitative attractor behavior	LorenzSystem.md
Robertson kinetics	severe stiffness, positivity, mass conservation	RobertsonKinetics.md
Kepler two-body problem	orbital drift, energy and angular momentum preservation	KeplerTwoBody.md

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Performance metrics

The framework computes the following metrics whenever applicable:

Metric	Meaning
final_error	norm of final-time error against exact/reference solution
max_error	maximum trajectory error over the metric window
rmse_error	root-mean-square trajectory error
cpu_time	MATLAB wall-clock time measured with tic/toc
n_steps	number of accepted output steps
nfev	number of right-hand-side evaluations
n_rejected	rejected adaptive steps
n_jacobian	number of numerical Jacobian evaluations
n_newton	Newton iterations used by implicit methods
max_invariant_error	maximum drift in energy, mass, or angular momentum
status	success, failure, blow-up, or non-applicable

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Generated plots

For every benchmark, the plotting scripts generate:

• solution trajectory comparison,
• error versus time,
• final error bar chart,
• CPU-time bar chart,
• function-evaluation bar chart,
• work-precision diagram,
• invariant error plot when invariants exist,
• phase portrait when the benchmark has a useful phase-space representation.

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Important interpretation rule

There is no universal best integrator.

A good conclusion should be problem-dependent:

• RK4 and adaptive RK methods are strong for smooth non-stiff problems.
• DOPRI/RK45-style methods are excellent when automatic step control is needed.
• BDF, Radau-type, Rosenbrock, and MATLAB ode15s-style solvers are usually better for stiff problems.
• Symplectic and splitting methods are better for long-term conservative mechanics.
• Energy-preserving methods are useful when invariant drift is more important than pointwise trajectory error.

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MATLAB version

The code is written in plain MATLAB style and avoids Simulink dependencies. It should run on recent MATLAB releases. Some plotting details may need minor adjustment on very old releases.

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Simulation video

Below is a link to the simulation video on YouTube.

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License

This project is released under the MIT License. See LICENSE.