https://github.com/mrp01/heatfun

Exploring the heat equation as part of a Special Topic for Approximation of Functions.
https://github.com/mrp01/heatfun

chebyshev-interpolation differential-equations heat-equation spectral-methods

Last synced: about 2 months ago
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Exploring the heat equation as part of a Special Topic for Approximation of Functions.

• Host: GitHub
• URL: https://github.com/mrp01/heatfun
• Owner: MrP01
• Created: 2022-12-19T19:10:22.000Z (about 2 years ago)
• Default Branch: master
• Last Pushed: 2023-05-05T21:09:29.000Z (over 1 year ago)
• Last Synced: 2024-04-16T03:20:36.043Z (9 months ago)
• Topics: chebyshev-interpolation, differential-equations, heat-equation, spectral-methods
• Language: TeX
• Homepage:
• Size: 37.2 MB
• Stars: 1
• Watchers: 1
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md

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README

        
# HeatFun - a spectral solver

Exploring the :fire: heat equation :fire: as part of a Special Topic for Approximation of Functions.

:rocket: **Solving PDEs using Spectral Methods in the Chebyshev basis by example of the Heat Equation.** :rocket:

A Special Topic on Approximation of Functions.

![Screenshot of the spectral heat equation solver](report/figures/screenshot.png)

_Screenshot of the graphical user interface. After entering an initial expression

u0(x), depicted in grey, the simulation will run upon pressing ’Start’. The solution at

time t, depicted in blue, is represented as a Chebyshev series of degree 29._

## Usage

HeatFun depends on Qt6, uses conan as a package manager and does analysis in Python and MatLab.

The project uses an out-of-source build:

```bash

mkdir -p build/

cd build

conan install ..

cmake -DCMAKE_BUILD_TYPE=Release ..

make -j 4

```

**To run the graphical user interface**:

```bash

./bin/main

```

**For usage as a Python module**:

```python3

import pathlib

import sys

BASE = pathlib.Path(__file__).parent.parent

sys.path.append(str(BASE / "build" / "lib"))

import heatfun

heatfun.solve(...)

```

## Motivation and Background

Partial differential equations are notoriously hard to solve. One more possible approach

to make way in this important class of problems is by the technique of spectral methods,

incidentally closely related to finite element methods. The key idea is to perform the

problem solution by representation of the occurring functions in a certain basis. For

non-periodic problem settings, Chebyshev series are a fantastic choice.

This work shall attempt to numerically solve the heat equation ut = αuxx

with Dirichlet boundary conditions over the domain [−1, 1] × [0, T] by representing the spatial component as a Chebfun (Chebyshev series) and moving on

in time by the Forward Euler numerical scheme.

The implementation, centered around what we will refer to as TschebFun, including three major algorithms TschebFun::interpolantThrough(),

TschebFun::evaluateOn() and TschebFun::derivative(), is done manually

in C++, extended to work as a Python module and for demonstration, even

features a high-level graphical interface to play with. Finally, we will compare

the numerical results with the output of Chebfun’s high-level pde15s().

Download the report [here](https://raw.githubusercontent.com/MrP01/HeatFun/master/report/report.pdf).