Computational Physics Portfolio

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Nels Buhrley Physics Student — Brigham Young University–Idaho

Table of Contents

About

I created this repository as a collection of thirteen numerical simulation and computational mathematics projects developed during my physics coursework and independent study. The work spans classical mechanics, electrostatics, statistical mechanics, quantum mechanics and number theory — all implemented in C++17 with Python 3 visualization pipelines.

Core Competencies

Area Details
Languages C++17, Python 3
Numerical Methods Runge-Kutta (RK4), Euler-Cromer, Störmer-Verlet, Monte Carlo (Metropolis), Finite Differences, Successive Over-Relaxation, FFT spectral analysis
Parallelism OpenMP multi-threadin, manual thread allocation, and parrellel logic implimentation in tightly coupled simulations
High-Performance Computing BYU Supercomputer — SLURM batch scheduling with up to 128 CPU cores for intensive simulations (Projects 4, 6, 7)
I/O & Visualization CSV, NPZ (via cnpy/zlib), Matplotlib (3D surfaces, contour maps, animations, phase-space portraits), Plotly (interactive HTML plots)
Build Systems GNU Make with multi-target builds (debug, release, profile-guided optimization)

Academic Projects

# Project Physical System Numerical Method OpenMP HPC
1 Realistic Projectile Motion 3D ballistics with drag, spin (Magnus), wind 4th-order Runge-Kutta — —
2 Driven Damped Oscillations Nonlinear pendulum → periodic & chaotic regimes RK4 + Euler-Cromer; Poincaré sections — —
3 Celestial Dynamics N-body gravitational orbits (full Solar System) RK4, Euler-Cromer, Störmer-Verlet — —
4 Overrelaxation 3D Laplace’s equation (electrostatics) Red-Black SOR with optimal $\omega$ ✓ ✓
5 Oscillations on a String Damped stiff wave equation + spectral analysis Finite differences (2nd + 4th order) + KissFFT ✓ parallel spatial —
6 Diffusion 3D Brownian random-walk ensemble Monte Carlo with reflective BCs ✓ parallel particles ✓
7 The Ising Model 3D ferromagnetic phase transition Metropolis MCMC, checkerboard sweep ✓ multifactor parallel ✓ 128 CPUs
8 Molecular Dynamics 2D Lennard-Jones fluid (400 particles) Velocity Verlet, O(N²) pair forces ✓ thread-local accumulators ✓ 8 CPUs
9 Quantum Mechanics 1D Schrödinger equation, symmetric polynomial potentials (even degrees: 2, 4, 6, 8, 10) Numerov 4th-order integration, nodal bracketing, dual shooting/matching bisection — —

Progression

The projects follow a deliberate arc of increasing computational sophistication:

Personal Projects

# Project Domain Key Challenge Parallelism
1 Collatz Conjecture (3n+1) Number theory Exhaustive sequence analysis for $n \leq 10^6$; recursive max-value chaining —
2 Euler’s Idoneal Numbers Number theory Sieve over triple loop $a < b < c$ up to $5 \times 10^7$; thread-safe monotonic writes ✓ OpenMP dynamic
3 Monty Hall Problem Probability & Statistics Monte Carlo simulation to empirically verify counterintuitive theoretical predictions —

Selected Results

Project 7: The Ising Model

View Simulation Fullscreen ↗️

Ising Model — 3D magnetization surface Ising Model — contour map

Magnetization surface and contour map of the 3D Ising model, revealing the ferromagnetic phase transition at $T_c \approx 4.51\,J/k_B$. Monte Carlo Metropolis algorithm with checkerboard sweep optimization across 2D temperature × magnetic field parameter space.

→ View detailed results

Project 8: Molecular Dynamics


Heating/cooling cycle simulation of 400 particles under Lennard-Jones potential. Particle trajectories show transition from ordered grid to gas-like disorder during heating, then re-ordering during cooling. A shock wave propagates down from the top and reflects back up after the particles expand to the container boundary.

→ View detailed results

Project 4: Overrelaxation (Electrostatics)

View Simulation Fullscreen ↗️

Interactive 3D visualization of electrostatic potential field solved via Successive Over-Relaxation (SOR) with optimal relaxation parameter $\omega \approx 1.84$. Navigate through z-axis slices to explore the full 3D solution space ($N=1000^3$ grid points).

→ View detailed results

Project 3: Celestial Dynamics

videos/

Celestial Dynamics — 3D orbits
N-body orbital dynamics of the Solar System computed with 4th-order Runge-Kutta integration. Left: Static plot of solar system. Right: Video of the solar system with the mass of Jupeter equil to about 0.94 Solar Masses.

→ View detailed results

Project 9: Quantum Mechanics

View Eigenstates Fullscreen ↗️

Interactive visualization of bound states for symmetric polynomial potential wells (even degrees: 2, 4, 6, 8, 10). Each tab displays eigenstates with 3D wavefunction plots overlaid with the potential well shape. Eigenstates computed via Numerov 4th-order integration, energy quantization by nodal counting, and dual-method bisection refinement (selectable shooting method with forward integration or matching method with origin-centric convergence) achieving high precision ($\Delta E < 10^{-8}$).

→ View detailed results

Project 5: Oscillations on a String

String oscillations — power spectrum

Mean power spectrum of transverse string oscillations showing discrete normal-mode peaks. FFT spectral analysis reveals harmonic structure and damping characteristics from finite-difference wave equation solver.

→ View detailed results

Project 6: Diffusion

Diffusion MSD

Mean squared displacement from 3D random-walk Monte Carlo simulation, confirming Einstein’s diffusion law $\langle r^2 \rangle = 6Dt$. Ensemble of $10^6$ particle trajectories with reflective boundary conditions.

→ View detailed results

The Fulton Supercomputer at BYU

The Fulton Supercomputer (managed by the BYU Office of Research Computing) is a HPC cluster providing the processing backbone for my numerical modeling and simulation work. It manages over 35,000 CPU cores and 360+ GPUs, supported by a 6 PB parallel filesystem.

Architecture & Resources (2026 Specs):

Compute Nodes AMD EPYC 7763 (128 cores/node), Intel Xeon Platinum 8568Y+ (96 cores/node)
High-Memory Up to 2 TB of DDR5 RAM for memory-intensive simulations
GPU Acceleration NVIDIA H200 (141GB), L40S (48GB), A100 (80GB)
Interconnect 100 Gb/s InfiniBand and RoCE v2 low-latency networking
Storage 6 PB parallel filesystem via /fslhome and local NVMe scratch
Me at Fulton Supercomputer

Me at the BYU HPC cluster, February 2026

Workflow & Scheduling

Job Management

I utilize SLURM (Simple Linux Utility for Resource Management) to orchestrate simulations. This involves writing batch scripts that request specific hardware constraints to optimize performance, such as:

Environment & Compilation

Development is performed on RHEL 9.4 login nodes using Remote - SSH and the linux terminal to manage my development and computational resources. I manage software dependencies via the module load system, typically involving:

Physics Implementation

On this cluster, I implement numerical solvers for complex potentials where the computation scales as $O(N^{2+})$. For a system of $N$ particles, the potential $V$ is calculated as:

\[V = \sum_{i < j} \frac{q_i q_j}{4\pi\epsilon_0 |\mathbf{r}_i - \mathbf{r}_j|}\]

By utilizing OpenMP for multi-threading and MPI for node-to-node communication, I can distribute these calculations, significantly reducing the wall-time required for high-resolution datasets.

// Example: Basic OpenMP Parallelization for Force Calculation
#pragma omp parallel for reduction(+:total_energy)
for (int i = 0; i < N; ++i) {
    for (int j = i + 1; j < N; ++j) {
        total_energy += calculate_interaction(particles[i], particles[j]);
    }
}

Simulation Deployment (This Portfolio):

Project Configuration CPUs Wall Time Use Case
Project 4 (SOR) CPU-only, #threads=16 16 30–45 min Strong-scaling study of iterative PDE solver
Project 6 (Diffusion) CPU-only, #threads=32 32 2-3 min Parallelizing independent particle trajectories
Project 7 (Ising) CPU-only, #threads=128 128 10–45 min Large parameter-space sweep with checkerboard MCMC
Project 8 (Molecular Dynamics) CPU-only, #threads=8 8 10 min–2 hrs Thread-local force accumulation (race condition mitigation)

Key Advantages for This Work:

Quick Start

Every project follows the same workflow:

# 1. Navigate to a project
cd "Project 7: The Ising Model"

# 2. Build
make release

# 3. Run
./bin/main

# 4. Visualize
python3 plotting.py

Build Toolchain

Tool Version Notes
Compiler clang++ / g++ (C++17) Platform-detecting Makefiles
OpenMP Homebrew libomp (macOS) / native (Linux) Projects 4–7
zlib System For .npz output via cnpy
Python 3.x numpy, matplotlib, pandas
SLURM BYU Supercomputer Batch scheduling for HPC runs

Prerequisites

# macOS
brew install libomp
pip3 install numpy matplotlib pandas

# Linux (e.g., BYU Supercomputer)
module load gcc python   # or equivalent
pip3 install numpy matplotlib pandas

Repository Structure

CPP_Workspace/
|-- README.md                          # <- This file (Portfolio Hub)
|
|-- Project 1: realistic projectile motion/    # RK4 ballistics with drag & Magnus
|-- Project 2: driven damped oscillations/     # Nonlinear pendulum & chaos
|-- Project 3: Celestial Dynamics/             # N-body gravitational simulation
|-- Project 4: Overrelaxation/                 # 3D Laplace solver (SOR + OpenMP)
|-- Project 5: Occilations on a string/        # Wave equation FD + FFT spectral
|-- Project 6: Diffusion/                      # 3D random-walk Monte Carlo
|-- Project 7: The Ising Model/                # 3D Metropolis MCMC (HPC)
|
|-- Personal Project 1: 3n+1/                 # Collatz conjecture exploration
|-- Personal Project 2: Idelic Numbers/        # Euler's idoneal number sieve
|
|-- include/                                   # Shared libraries
|   |-- cnpy.h                                 # NPZ I/O library
|   `-- vector3d.h                             # 3D vector utilities
|-- src/
|   `-- cnpy.cpp                               # cnpy implementation
|
|-- reference/                                 # Study notes & cheatsheets
|   |-- CPP_BASICS.md
|   |-- FUNCTIONS_REFERENCE.md
|   |-- NUMERICAL_PHYSICS.md
|   |-- OPTIMIZATION_TIPS.md
|   `-- SLURM_FORMATING.md
|
`-- Template Makefile                          # Reusable build template

Nels Buhrley — BYU-Idaho Physics, 2025–2026