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\documentclass[11pt]{article}

\usepackage[a5paper,margin=14mm]{geometry}

\usepackage{amsmath,amssymb,amsthm}

\usepackage{booktabs}

\newtheorem{theorem}{Theorem}

\title{Heat Flow on a Ring}

\author{Ada Kaya \and Kenji Sato \and Maya Ortiz}

\date{October 2026}

\begin{document}

\maketitle

\begin{abstract}

We solve the heat equation on a periodic domain with a Fourier

spectral method, and show that its error falls faster than any

power of the number of modes.

\end{abstract}

\section{The model}

Let $u(x,t)$ be the temperature on a ring of length $2\pi$. It obeys

\begin{equation}

\frac{\partial u}{\partial t} = \kappa \frac{\partial^2 u}{\partial x^2},

\qquad u(x,0) = f(x).

\label{eq:heat}

\end{equation}

Writing $u = \sum_k \hat u_k(t)\, e^{ikx}$ turns \eqref{eq:heat} into

$\hat u_k' = -\kappa k^2 \hat u_k$ for each mode, so

\begin{equation}

u(x,t) = \sum_{k \in \mathbb{Z}} \hat f_k \, e^{-\kappa k^2 t} e^{ikx}.

\end{equation}

\section{Convergence}

\begin{theorem}

If $f$ is smooth, the truncated sum $u_N$ satisfies

$\| u - u_N \|_2 \le C_m N^{-m}$ for every $m > 0$.

\end{theorem}

\begin{center}

\begin{tabular}{rcc}

\toprule

$N$ & Error & Time (ms) \\

\midrule

16 & $3.1 \times 10^{-4}$ & 0.2 \\

32 & $8.7 \times 10^{-9}$ & 0.4 \\

64 & $2.2 \times 10^{-15}$ & 0.9 \\

\bottomrule

\end{tabular}

\end{center}

\end{document}

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Heat flow›main›main.tex

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\title{Heat Flow on a Ring}

\author{Ada Kaya \and Kenji Sato \and Maya Ortiz}

\date{October 2026}

\begin{document}

\maketitle

\begin{abstract}

We solve the heat equation on a periodic domain with a Fourier

spectral method, and show that its error falls faster than any

power of the number of modes.

\end{abstract}

\section{The model}

Let $u(x,t)$ be the temperature on a ring of length $2\pi$. It obeys

\begin{equation}

\frac{\partial u}{\partial t} = \kappa \frac{\partial^2 u}{\partial x^2},

\qquad u(x,0) = f(x).

\label{eq:heat}

\end{equation}

Writing $u = \sum_k \hat u_k(t)\, e^{ikx}$ turns \eqref{eq:heat} into

$\hat u_k' = -\kappa k^2 \hat u_k$ for each mode, so

\begin{equation}

u(x,t) = \sum_{k \in \mathbb{Z}} \hat f_k \, e^{-\kappa k^2 t} e^{ikx}.

\end{equation}

\section{Convergence}

\begin{theorem}

If $f$ is smooth, the truncated sum $u_N$ satisfies

$\| u - u_N \|_2 \le C_m N^{-m}$ for every $m > 0$.

\end{theorem}

\begin{center}

\begin{tabular}{rcc}

\toprule

$N$ & Error & Time (ms) \\

\midrule

16 & $3.1 \times 10^{-4}$ & 0.2 \\

32 & $8.7 \times 10^{-9}$ & 0.4 \\

64 & $2.2 \times 10^{-15}$ & 0.9 \\

\bottomrule

\end{tabular}

\end{center}

\end{document}

CodeVisual

Heat Flow on a Ring

Ada KayaKenji SatoMaya Ortiz

October 2026

Abstract

We solve the heat equation on a periodic domain with a Fourier spectral method, and show that its error falls faster than any power of the number of modes.

1The model

Let $u(x,t)$ be the temperature on a ring of length $2\pi$ . It obeys

\frac{\partial u}{\partial t} = \kappa \frac{\partial^2 u}{\partial x^2}, \quad u(x,0) = f(x).

(1)

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Heat flow›main›main.tex

AKKSSLLive

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42⌄

\end{equation}

Writing $u = \sum_k \hat u_k(t)\, e^{ikx}$ turns \eqref{eq:heat} into

$\hat u_k' = -\kappa k^2 \hat u_k$ for each mode, so

\begin{equation}

u(x,t) = \sum_{k \in \mathbb{Z}} \hat f_k \, e^{-\kappa k^2 t} e^{ikx}.

\end{equation}

High modes decay fastest, so $u$ is smooth for every $t > 0$.Ada Kaya

\section{Convergence}

\begin{theorem}

If $f$ is smooth, the truncated sumKenji Sato $u_N$ satisfies

$\| u - u_N \|_2 \le C_m N^{-m}$ for every $m > 0$.

\end{theorem}

\begin{center}

\begin{tabular}{rcc}

\toprule

$N$ & Error & Time (ms) \\

\midrule

16 & $3.1 \times 10^{-4}$ & 0.2 \\

32 & $8.7 \times 10^{-9}$ & 0.4 \\

64 & $2.2 \times 10^{-15}$ & 0.9 \\

\bottomrule

\end{tabular}

\end{center}

\end{document}

NotificationsLive

Kenji Sato approved your pull request #4Address the referee’s comments1m ago
Ada Kaya commented on your pull request #4Address the referee’s comments14m ago
Ada Kaya merged your pull request #3 into mainFix typos in section 12h ago

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main the truncated sum
referee-fixes the Fourier sum
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Heat flow›History

main3 versions

Merge pull request #3 from typo-fixes

AKAda Kaya·2h agoWz5cH7yP1d

Tighten the abstract and Theorem 1

AKAda Kaya·6h agoHn4kT8sQ2v

Add the convergence table

KSKenji Sato·1d agok3Fq9Lm2Ta

Compared with the version before: Tighten the abstract and Theorem 1

main.texModified+3−3

@@ -14,8 +14,8 @@

1414

1515 \begin{abstract}

1616 We solve the heat equation on a periodic domain with a Fourier

17−spectral method, and show that its error is small

18−for smooth initial data.

17+spectral method, and show that its error falls faster than any

18+power of the number of modes.

1919 \end{abstract}

2020

2121 \section{The model}

@@ -33,7 +33,7 @@

3333

3434 \section{Convergence}

3535 \begin{theorem}

36−If $f$ is smooth, the partial sum $u_N$ satisfies

36+If $f$ is smooth, the truncated sum $u_N$ satisfies

3737 $\| u - u_N \|_2 \le C_m N^{-m}$ for every $m > 0$.

3838 \end{theorem}

3939

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