HeatFun - a spectral solver

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

🚀 Solving PDEs using Spectral Methods in the Chebyshev basis by example of the Heat Equation. 🚀 A Special Topic on Approximation of Functions.

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:

mkdir -p build/
cd build
conan install ..
cmake -DCMAKE_BUILD_TYPE=Release ..
make -j 4

To run the graphical user interface:

./bin/main

For usage as a Python module:

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.