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：对演化型偏微分方程进行数值积分是研究其解动力学行为的核心手段。尽管数值模拟在科学计算中占据基础地位，但其数学可靠性往往难以量化，特别是当关注的是给定模拟的特定输出结果，而非离散参数趋近于零的渐近状态时。本文提出一种基于计算机辅助证明的方法，用于对具有周期性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。我们在傅里叶空间中基于常数变易公式构造等价零点求解问题。通过切比雪夫插值与区域分解技术，最终采用牛顿-康托罗维奇型论证完成证明。该过程的最终输出包括：轨道存在性的数学证明，以及该轨道与数值计算近似值之间的保证误差界。我们通过Fisher方程、Swift-Hohenberg方程、Ohta-Kawasaki方程及Kuramoto-Sivashinsky方程的数值结果展示了该方法的普适性。预期该严格积分器将为研究偏微分方程中连接轨道的边值问题提供基础工具。