# 2023-08-28 First Day#

## CSCI-5636: Numerical solution of partial differential equations (PDE)#

## Instructor: Jed Brown, jed.brown@colorado.edu, ECOT 824#

## Meeting: Mon, Wed, Fri at 1:25-2:15p in ECCS 1B14 (and virtually)#

# What are PDE?#

## partial derivatives#

Gradient \(\mathbf g(\mathbf x) = \nabla u(\mathbf x)\)

scalar field to vector field

Curl \(\mathbf c(\mathbf x) = \nabla\times \mathbf g(\mathbf x)\)

vector field to vector field

Divergence \(d(\mathbf x) = \nabla\cdot \mathbf c(\mathbf x)\)

vector field to scalar field

time derivatives: \(\frac{\partial u(t, \mathbf x)}{\partial t}\)

## what do they describe?#

heat transfer

electrostatics, electromagnetics

solid mechanics

fluid mechanics

quantum mechanics

population dynamics

Find \(u\) such that

# Structural mechanics#

# What does it mean to discretize a PDE?#

Find \(u\) such that

## How is it satisfied?#

At grid points with rules to approximate derivatives

Finite Difference / collocation

A weak (integral) over elements with rules to reconstruct from element averages and define fluxes

Finite Volume

A weak form over elements with solution and “test functions” in the same space

Finite Element / Galerkin

## How expensive is it?#

Suppose our domain is \(\Omega = (0, 1)^3\)

```
using Plots
# a common "good" discretiation
error(h) = min(1, 10*h^4)
function cost(h)
n = 1/h # number of points per dimension
N = n^3 # total number of grid points in 3D
N
end
function cost2(h)
N = cost(h)
N^2
end
```

```
cost2 (generic function with 1 method)
```

```
plot([cost, cost2], error, .01, 1, xscale=:log10, yscale=:log10,
xlabel="Cost", ylabel="Error")
```

# What goes in? What comes out?#

Equations

Conservation

Variational principles

Materials

Geometry

meshing needed?

Boundary conditions

essential/Dirichlet

natural/Neumann

mixed

Initial conditions

Discrete solutions

Sampled on a grid

Time series

Quantities of Interest (QoI)

Heat flux

Maximum temperature

Maximum stress

Lift, drag

Eigensolutions

Resonant frequencies/modes

Orbitals

# Who works with PDE solvers?#

Numerical analysts

Domain scientists and engineers

Materials scientists

HPC specialists

Optimizers

Statisticians

# What skills should they have?#

Breakout groups: pick one role and make a list of essential and nice-to-have skills, plus at least one question. Pick one member to report out.

…

…

# Computer Science = Runnable Abstraction Science#

## In the “grinder”#

Mathematical language and principles for each major class of method

What is covered by theory

When you’re venturing off-trail

Analytic tools to predict and debug

Realistic cost and exploitable structure

Performance on modern hardware

Algebraic solvers and time integrators

Write/modify stand-alone code

Select and use (parallel) libraries

## How can you trust the solution?#

Verification: solving the problem right

Validation: solving the right problem

## Abstractions/collaboration#

Domain scientists, engineers

Optimizers, statisticians

Abstractions that reduce cognitive load

Metrics/visualization for decisions

# On programming languages#

## C#

Reliable and popular for libraries (PETSc, etc.). “Simple”, unsafe, capable of encapsulation.

## C++#

Popular with applications and some libraries. Powerful, but complicated and unsafe.

## Fortran#

The OG of numerical computing. Good for “array programming”, but encapsulation is hard. Unsafe depending on dialect.

## Julia#

For ground-up examples in class and in activities. Capable of high performance, expressive multiple dispatch, works well in a notebook. Library ecosystem for PDEs is limited, but rapidly improving.

## Python#

Good access to JIT, libraries like FEniCS. Poor native performance, but good libraries to compiled code.

## Rust#

New compiled language. Good performance, encapsulation, safety, static analysis. Poor libraries (so far).

You don’t need to know any of these well and you can choose the language for your project.

# Intrinsic motivation#

# I won’t grade and I won’t waste your time#

I will provide feedback

I will be a mentor and guide navigating this field

I will help you get what you want out of the class (with a dash of wholesome context)

My mom said, “basically, your professor is asking you to be an adult”. That was too flexible.

In other classes, I didn’t feel like I could bounce back, but in this one I did.

# What about the letter grade?#

Collect a portfolio of the work and insights you’re most proud of

Individual meetings during the last week of the semester (mid-term preview)

We’ll have a conversation and you’ll propose a grade based on your portfolio

I’ll trust you.

*I can nudge upward when students are too modest – pretty common. In rare cases, I may adjust down.*

# What I need from you: Growth Mindset#

# How will the semester look?#

## Lecture periods#

Refresh, introduce, activity/group discussion, reflect

Pointers to further resources/activities

## In-class and homework activities#

(Short) coding, experiments, presentation of results

Use the math and programming tools of the field

Open-ended, go further occasionally

## Learning plans#

Write, track, and revise a personal learning plan.

Meet/chat approx weekly to check in with partner

## Projects (second half of semester)#

Dig into community software for solving PDEs

Short presentation on how the community works

Key stakeholders

Strengths and weaknesses

Discuss and critique

## Community contribution#

Tutorial, documentation, performance study, comparison, new features, new application

Reflect in a short presentation

# Tools#

## Git and GitHub#

GitHub CLassroom to manage repositories

Write using notebooks and markdown

Review using GitHub tools and nbgrader

## Jupyter#

RISE slides available on website

Activities with scaffolding

Nbgrader for writing feedback

Works with many languages; we’ll mostly use Julia

Collaborative mode (new)

## JupyterHub: coding.csel.io#

Nothing to install, persistent storage

# A note about illnesses#

There is a new covid variant circulating.

Updated vaccines aren’t available yet.

High-quality masks are effective.

You’re welcome to attend virtually (and please do if you have symptoms or have had exposures).

## Resources#

CDPHE Wastewater Data by county

I have a 3yo in childcare and a 6yo in first grade.

Illnesses circulate often and sometimes they need to stay home.

I’ll likely need to teach remotely at some times, or otherwise make it up to you

# Choices in scoping the class#

## Theory#

Analysis first, confirm using numerics. Limited to simpler models.

## Applied#

Numerics first, pointers to useful theory.

## Build from scratch#

Limited to simpler models, but you’ll understand everything under the hood.

## Build using libraries#

More installation and software layers, but can solve more interesting problems.