Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A simple rigorous integrator for semilinear parabolic PDEs

Jan Bouwe van den Berg, Maxime Breden

TL;DR

The paper develops a simple yet robust computer-assisted framework for rigorously integrating semilinear parabolic PDEs over long times by a fixed-point reformulation with a piecewise-constant approximation of the Jacobian, yielding explicit a posteriori error bounds. It introduces a time-discretization strategy based on Chebyshev polynomials and a Fourier-space representation in ll^1_ u, together with a systematic construction of a time-independent linear operator L that enables tractable semigroup estimates. By extending the setup with time-domain decomposition and carefully derived Y,Z,W bounds, the method achieves long-time, and even infinity-time, integration, and can certify the existence and stability of steady states. The authors demonstrate significant computational gains and rigorous validation on the Swift-Hohenberg, Ohta-Kawasaki, and Kuramoto-Sivashinsky equations, highlighting the approach’s practicality and potential for broader parabolic PDEs and boundary-value problems.

Abstract

Simulations of the dynamics generated by partial differential equations (PDEs) provide approximate, numerical solutions to initial value problems. Such simulations are ubiquitous in scientific computing, but the correctness of the results is usually not guaranteed. We propose a new method for the rigorous integration of parabolic PDEs, i.e., the derivation of rigorous and explicit error bounds between the numerically obtained approximate solution and the exact one, which is then proven to exist over the entire time interval considered. These guaranteed error bounds are obtained a posteriori, using a fixed point reformulation based on a piece-wise in time constant approximation of the linearization around the numerical solution. Our setup leads to relatively simple-to-understand estimates, which has several advantages. Most critically, it allows us to optimize various aspects of the proof, and in particular to provide an adaptive time-stepping strategy. In case the solution converges to a stable hyperbolic equilibrium, we are also able to prove this convergence, applying our rigorous integrator with a final, infinitely long timestep. We showcase the ability of our method to rigorously integrate over relatively long time intervals, and to capture non-trivial dynamics, via examples on the Swift--Hohenberg equation, the Ohta--Kawasaki equation and the Kuramoto--Sivashinsky equation. We expect that the simplicity and efficiency of the approach will enable generalization to a wide variety of other parabolic PDEs, as well as applications to boundary value problems.

A simple rigorous integrator for semilinear parabolic PDEs

TL;DR

The paper develops a simple yet robust computer-assisted framework for rigorously integrating semilinear parabolic PDEs over long times by a fixed-point reformulation with a piecewise-constant approximation of the Jacobian, yielding explicit a posteriori error bounds. It introduces a time-discretization strategy based on Chebyshev polynomials and a Fourier-space representation in ll^1_ u, together with a systematic construction of a time-independent linear operator L that enables tractable semigroup estimates. By extending the setup with time-domain decomposition and carefully derived Y,Z,W bounds, the method achieves long-time, and even infinity-time, integration, and can certify the existence and stability of steady states. The authors demonstrate significant computational gains and rigorous validation on the Swift-Hohenberg, Ohta-Kawasaki, and Kuramoto-Sivashinsky equations, highlighting the approach’s practicality and potential for broader parabolic PDEs and boundary-value problems.

Abstract

Simulations of the dynamics generated by partial differential equations (PDEs) provide approximate, numerical solutions to initial value problems. Such simulations are ubiquitous in scientific computing, but the correctness of the results is usually not guaranteed. We propose a new method for the rigorous integration of parabolic PDEs, i.e., the derivation of rigorous and explicit error bounds between the numerically obtained approximate solution and the exact one, which is then proven to exist over the entire time interval considered. These guaranteed error bounds are obtained a posteriori, using a fixed point reformulation based on a piece-wise in time constant approximation of the linearization around the numerical solution. Our setup leads to relatively simple-to-understand estimates, which has several advantages. Most critically, it allows us to optimize various aspects of the proof, and in particular to provide an adaptive time-stepping strategy. In case the solution converges to a stable hyperbolic equilibrium, we are also able to prove this convergence, applying our rigorous integrator with a final, infinitely long timestep. We showcase the ability of our method to rigorously integrate over relatively long time intervals, and to capture non-trivial dynamics, via examples on the Swift--Hohenberg equation, the Ohta--Kawasaki equation and the Kuramoto--Sivashinsky equation. We expect that the simplicity and efficiency of the approach will enable generalization to a wide variety of other parabolic PDEs, as well as applications to boundary value problems.
Paper Structure (29 sections, 15 theorems, 155 equations, 4 figures)

This paper contains 29 sections, 15 theorems, 155 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sequence space of Fourier coefficients
  4. Function space
  5. Approximation in time
  6. A fixed point Theorem
  7. Taking $\mathcal{L}$ constant over the entire integration time
  8. Construction of $\mathcal{L}$
  9. Y-bound
  10. Z-bounds
  11. W-bound
  12. Longer integration times
  13. How to adapt the setup
  14. $Y$-bound
  15. $Z$-bound
  16. ...and 14 more sections

Key Result

Theorem 1.2

Consider the Ohta--Kawasaki equation eq:OK with $\gamma = \sqrt{8}$, $\sigma=1/5$, $m=1/10$, $L=2\pi$, and ${u^{\mathrm{in}}}(x) = m+\frac{2}{10}\cos\left(\frac{2\pi x}{L}\right)+\frac{2}{10}\cos\left(\frac{4\pi x}{L}\right)$. The solution $u=u(t,x)$ satisfies where the precise description of the approximate solution $\bar{u}=\bar{u}(t,x)$ in terms of Chebyshev$\times$Fourier coefficients can be

Figures (4)

  • Figure 1: The global approximate solution $\bar{u}$ of the Ohta--Kawasaki equation \ref{['eq:OK']} which has been validated in Theorem \ref{['th:OK']}, represented on the time interval $[0,30]$.
  • Figure 2: The approximate solution $\bar{u}$ of the Swift--Hohenberg equation \ref{['eq:SH']}, which has been validated in Theorem \ref{['th:SH']}, depicted twice with different views.
  • Figure 3: Illustration of some of the important computational parameters used for the proof of Theorem \ref{['th:OK']}. The length $\tau_{m+1}-\tau_{m}$ of each subdomain is shown on the left (except for the last one which has infinite length), and the Chebyshev order $K^{(m)}$ used for the approximate solution on each subdomain is shown on the right.
  • Figure 4: The approximate solution $\bar{u}$ of the Kuramoto--Sivashinsky equation \ref{['eq:KS']}, which has been validated in Theorem \ref{['th:KS']}.

Theorems & Definitions (45)

  • Remark 1.1
  • Theorem 1.2
  • Remark 2.1: Notation
  • Lemma 2.2
  • Remark 2.3: Notation, continued
  • Lemma 2.4
  • Definition 2.5: Truncation of phase space
  • Definition 2.6: Spatial derivative
  • Definition 2.7
  • Remark 2.8: Notation
  • ...and 35 more