GPU-Accelerated Integrators — Benchmark Experiment Documentation

Overview

This document describes the benchmarking methodology used to evaluate numerical integrators for the Hénon–Heiles system on CPU and GPU implementations.

The goal is not just performance measurement, but a scientifically meaningful comparison across:

• Numerical accuracy
• Energy conservation
• Long-term stability
• Computational efficiency

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1. Objective

The benchmark is designed to answer three core questions:

1.1 Numerical Accuracy vs Physical Fidelity

• How accurately does each integrator approximate trajectories?
• Does higher-order accuracy (e.g. RK4) imply better physical correctness?

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1.2 Long-Term Stability

• Does the method preserve invariants such as energy?
• Does error drift or remain bounded?

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1.3 Performance and Scalability

• How does GPU performance scale with number of trajectories?
• What is the speedup over CPU implementations?

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2. Systems Under Study

2.1 Dynamical System

The Hénon–Heiles Hamiltonian:

[ H = \frac{1}{2}(p_x^2 + p_y^2) + \frac{1}{2}(x^2 + y^2) + x^2 y - \frac{1}{3} y^3 ]

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2.2 State Representation

Each trajectory is defined by:

[ z = (x, y, p_x, p_y) ]

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2.3 Integrators Compared

Method	Order	Symplectic	Expected Behavior
Euler	1	❌	Fast divergence
RK4	4	❌	Accurate short-term, energy drift
Leapfrog	2	✅	Long-term stable

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3. Experimental Design

3.1 Initial Conditions

• Multiple trajectories initialized with:

  • Fixed energy level ( H \approx