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.

翻译：积分演化偏微分方程是研究解动力学的基本要素。事实上，数值模拟是科学计算的核心，但其数学可靠性往往难以量化，尤其是在关注特定模拟输出而非离散化参数趋于零的渐近状态时。本文提出一种计算机辅助证明方法，用于对具有周期边界条件的标量半线性抛物型偏微分方程进行严格时间积分。我们基于傅里叶空间中的常数变易公式构建等价零点寻找问题。通过采用切比雪夫插值与区域分解技术，最终借助牛顿-康托罗维奇型论证完成证明。该过程的最终输出是轨道存在性的证明，以及该轨道与数值计算近似解之间具有保证误差界的严格结果。我们通过Fisher方程、Swift-Hohenberg方程、Ohta-Kawasaki方程和Kuramoto-Sivashinsky方程的计算结果展示了该方法的普适性。我们预期该严格积分器能为研究偏微分方程连接轨道的边值问题奠定基础。