Numerical-Solvers-Edge

Ordinary differential equation (ODE) integrators evaluated on Hamiltonian systems under 32-bit floating-point precision.

Theoretical Focus

Classical Runge-Kutta algorithms produce low truncation errors per cycle but fail to preserve the symplectic structure of Hamiltonian phase space. Over long integration windows, artificial dissipation occurs. Symplectic Euler preserves invariant manifolds, holding total energy error strictly bounded.

Governing Equation: Undamped Linear Oscillator

$$\frac{dx}{dt} = v, \quad \frac{dv}{dt} = -\omega^2 x$$

$$\mathcal{H}(x, v) = \frac{1}{2}v^2 + \frac{1}{2}\omega^2 x^2$$

Computational Performance

Method	Order	Force Evaluations / Step	Phase Space Invariant	Memory Overhead
Symplectic Euler	1	1	Preserved	8 Bytes
Runge-Kutta 4 (RK4)	4	4	Dissipative / Drifting	8 Bytes

Build and Execution

cmake -B build
cmake --build build
./build/bench_solvers
python scripts/plot_drift.py

On Windows with the Visual Studio generator, use:

cmake -S . -B build
cmake --build build --config Release
.\build\Release\bench_solvers.exe
python .\scripts\plot_drift.py

The benchmark writes drift_data.csv, and the plotting script saves drift_plot.png.

Install the plotting dependency once with:

python -m pip install matplotlib