Learning SciML PDE Discretization

This repository documents my learning journey and technical preparation for contributing to the SciML ecosystem, specifically towards PDE discretization and numerical methods projects.

My goal is to prepare for a Google Summer of Code (GSoC) project focused on:

Discretizations of Partial Differential Equations (PDEs)

within the SciML organization.

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Background

I am a Computer Engineering student with strong foundations in:

• Differential Equations (AA)
• Linear Algebra
• Calculus and numerical reasoning

While exploring the SciML ecosystem, I encountered and resolved an installation-related issue and successfully contributed a merged pull request to the project. SciML/SciMLBase.jl#1258 This experience motivated me to deeply understand the numerical and architectural foundations behind SciML's PDE tooling.

This repository serves as a public learning log showing my progress toward becoming an effective open-source contributor.

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Learning Objectives

The main goals of this repository are:

• Understand numerical discretization of PDEs
• Learn Finite Difference and Method of Lines approaches
• Study internal design of SciML PDE tools
• Explore how mathematical models become executable solvers
• Prepare meaningful contributions to SciML libraries

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Topics Covered

Numerical PDE Foundations

• Finite Difference Method (FDM)
• Method of Lines (MOL)
• Stability intuition
• Grid discretization

SciML Ecosystem

• DifferentialEquations.jl
• ModelingToolkit.jl
• MethodOfLines.jl

Software Understanding

• Reading scientific computing source code
• Understanding solver abstractions
• Experiment-driven learning

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Repository Structure

learning-sciML-pde/
│
├── mini-projects/        # Small PDE implementations
├── WENO-NonUniform-Math  # WENO prototype functions for uniform and non-uniform grids
├── benchmarks/           # Performance and allocation tests
├── myNotes
└── source-study/         # Internal library exploration

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Progress

This repository is updated continuously as I:

• implement numerical methods from scratch,
• reproduce SciML examples,
• analyze internal implementations,
• and document insights gained during learning.

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Motivation

Rather than only applying to projects, I believe the best way to contribute to open source scientific computing is to learn in public and progressively integrate into the ecosystem.

This repository represents that process.

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Long-Term Goal

To contribute production-quality improvements to PDE discretization tooling in SciML and help make advanced numerical computing more accessible to users.

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Status

Active learning and preparation in progress.

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Mini Project 1

Finite Difference implementation of the 1D Heat Equation to understand PDE discretization fundamentals.

Mini Project 2

Heat Equation solved using SciML's MethodOfLines.jl discretization pipeline. Comparison between manual finite differences and automated discretization.

Mini Project 3: Non-Uniform 1D Heat Equation (Proof of Concept)

A custom implementation of the 1D Heat Equation using an Explicit Finite Difference Method on a non-uniform grid.

• Mapped a uniform spatial array into a clustered grid using a cubic function to focus on high-gradient areas.
• Dynamically calculated local spatial steps (h_left and h_right) for the second derivative stencil.
• Demonstrated the severe time-step penalty (microscopic dt requirement) caused by the explicit stability criterion on dense grids, highlighting the necessity of implicit solvers provided by MethodOfLines.jl.

Mini Project 4: Non-Uniform Grid Advection-Diffusion Model

This project demonstrates the implementation of a microscopic gradient dynamics model (Advection-Diffusion PDE) utilizing a non-uniform vector grid.

• Key Feature: Discretization is performed using MOLFiniteDifference with an explicit UpwindScheme on an AbstractVector grid.
• Open Source Contribution: During the development of this model, a BoundsError bug in the core MethodOfLines.jl library regarding vector grid offsets was identified and successfully patched. You can review the fix here: SciML/MethodOfLines.jl#533.

Mini Project 5: Fornberg Algorithm Benchmarking

Performance testing of the Fornberg algorithm for dynamic finite difference weight generation on non-uniform grids.

• Utilized BenchmarkTools.jl to profile execution time and memory allocations.
• Result: Proved that node-by-node dynamic weight calculation has zero performance overhead and zero extra memory allocation compared to uniform grids, clearing the path for MethodOfLines.jl integration.

Mini Project 6: WENO Scheme Math on Non-Uniform Grids

Theoretical analysis and code prototyping of WENO smoothness indicators ($\beta_k$) for clustered grids.

• Demonstrated why classic hardcoded fractional weights (like 13/12 or 1/4) cause bounds errors and fail on non-uniform domains.
• Prototyped the mathematical necessity of dynamically calculating indicators based on local geometric distances ($h_k$) to prepare for full Lagrange interpolation in the GSoC project.

Mini Project 7: 2D Non-Uniform Heat Equation

Extended the non-uniform grid logic to multi-dimensional PDEs to explore spatial stability.

• Implemented the 2D Heat Equation from scratch using a clustered cubic grid on both the X and Y axes.
• Dynamically calculated 2D Laplacian stencils ($u_{xx} + u_{yy}$) for variable geometric distances.
• Demonstrated the severe time-step constraints (microscopic dt) in explicit 2D schemes on clustered grids, further emphasizing the need for robust implicit solvers in MethodOfLines.jl.

Mini Project 8: Finite Volume Method (FVM) Geometry on Non-Uniform Grids

Prototyped the geometric foundations required for Finite Volume Method (FVM) mass conservation on clustered grids. This is a direct prerequisite for solving complex, multi-domain industrial problems like the DFN Lithium-Ion Battery Model.

• Generated a staggered grid architecture, dynamically calculating Cell Centers (Nodes) and Cell Faces (Interfaces) for a cubic clustered vector.
• Computed local Cell Volumes ($\Delta V$) and verified strict mass conservation (the sum of variable cell volumes perfectly equals the total domain length).
• Result: Successfully laid the geometric groundwork needed to extend MethodOfLines.jl's codegen to support FVM on variable spacing.