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Validated integration of semilinear parabolic PDEs

Jan Bouwe van den Berg, Maxime Breden, Ray Sheombarsing

TL;DR

A computer-assisted proof methodology to perform rigorous time integration for scalar semilinear parabolic PDEs with periodic boundary conditions and expects that this rigorous integrator can form the basis for studying boundary value problems for connecting orbits in partial differential equations.

Abstract

Integrating evolutionary partial differential equations (PDEs) is an essential ingredient for studying the dynamics of the solutions. Indeed, simulations are at the core of scientific computing, but their mathematical reliability is often difficult to quantify, especially when one is interested in the output of a given simulation, rather than in the asymptotic regime where the discretization parameter tends to zero. In this paper we present a computer-assisted proof methodology to perform rigorous time integration for scalar semilinear parabolic PDEs with periodic boundary conditions. We formulate an equivalent zero-finding problem based on a variations of constants formula in Fourier space. Using Chebyshev interpolation and domain decomposition, we then finish the proof with a Newton--Kantorovich type argument. The final output of this procedure is a proof of existence of an orbit, together with guaranteed error bounds between this orbit and a numerically computed approximation. We illustrate the versatility of the approach with results for the Fisher equation, the Swift--Hohenberg equation, the Ohta--Kawasaki equation and the Kuramoto--Sivashinsky equation. We expect that this rigorous integrator can form the basis for studying boundary value problems for connecting orbits in partial differential equations.

Validated integration of semilinear parabolic PDEs

TL;DR

A computer-assisted proof methodology to perform rigorous time integration for scalar semilinear parabolic PDEs with periodic boundary conditions and expects that this rigorous integrator can form the basis for studying boundary value problems for connecting orbits in partial differential equations.

Abstract

Integrating evolutionary partial differential equations (PDEs) is an essential ingredient for studying the dynamics of the solutions. Indeed, simulations are at the core of scientific computing, but their mathematical reliability is often difficult to quantify, especially when one is interested in the output of a given simulation, rather than in the asymptotic regime where the discretization parameter tends to zero. In this paper we present a computer-assisted proof methodology to perform rigorous time integration for scalar semilinear parabolic PDEs with periodic boundary conditions. We formulate an equivalent zero-finding problem based on a variations of constants formula in Fourier space. Using Chebyshev interpolation and domain decomposition, we then finish the proof with a Newton--Kantorovich type argument. The final output of this procedure is a proof of existence of an orbit, together with guaranteed error bounds between this orbit and a numerically computed approximation. We illustrate the versatility of the approach with results for the Fisher equation, the Swift--Hohenberg equation, the Ohta--Kawasaki equation and the Kuramoto--Sivashinsky equation. We expect that this rigorous integrator can form the basis for studying boundary value problems for connecting orbits in partial differential equations.
Paper Structure (52 sections, 28 theorems, 238 equations, 6 figures, 1 table)

This paper contains 52 sections, 28 theorems, 238 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sequence spaces
  4. Chebyshev interpolation
  5. A fixed point Theorem
  6. Functional analytic setup
  7. An equivalent zero-finding problem
  8. Finite dimensional reduction
  9. A posteriori analysis
  10. Construction of $\mathcal{L}$
  11. General estimates
  12. An $\ell^1_\nu$ dual estimate
  13. Controlling convolution products
  14. Interpolation errors
  15. Truncation errors in Fourier space
  16. ...and 37 more sections

Key Result

Theorem 1.5

Consider the Swift-Hohenberg equation with $\alpha = 5$, $L=6\pi$, $t_{\text{end}}=3/2$, and $f(x) = 0.4\cos\left(\frac{2\pi x}{L}\right)-0.3\cos\left(\frac{4 \pi x}{L}\right)$. Let $\bar{u}=\bar{u}(t,x)$ be the function represented in Figure fig:SH, and whose precise description in terms of Fourier-Chebyshev coefficients can be downloa

Figures (6)

  • Figure 1: The approximate solution $\bar{u}$ of \ref{['eq:SH']}, which has been validated in Theorem \ref{['th:SH']}, depicted twice with different views.
  • Figure 2: The approximate solution $\bar{u}$ of \ref{['eq:Fisher']}, which has been validated in Theorem \ref{['th:Fisher']}.
  • Figure 3: The approximate solution $\bar{u}$ of \ref{['eq:SH']}, which has been validated in Theorem \ref{['th:SH_less_precise']}.
  • Figure 4: The approximate solution $\bar{u}$ of \ref{['eq:KS']}, which has been validated in Theorem \ref{['th:KS']}.
  • Figure 5: The approximate solution $\bar{u}$ of \ref{['eq:OK']}, which has been validated in Theorem \ref{['th:OK']}.
  • ...and 1 more figures

Theorems & Definitions (97)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 2.1
  • Lemma 2.2
  • Definition 2.3: Chebyshev points
  • Remark 2.4
  • ...and 87 more