Transformed Model Reduction for Partial Differential Equations with Sharp Inner Layers

Abstract

SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2178-A2201, August 2024.
Abstract. Small parameters in partial differential equations can give rise to solutions with sharp inner layers that evolve over time. However, the standard model reduction method becomes inefficient when applied to these problems due to the slow decaying Kolmogorov [math]-width of the solution manifold. To address this issue, a natural approach is to transform the equation in such a way that the transformed solution manifold exhibits a fast decaying Kolmogorov [math]-width. In this paper, we focus on the Allen–Cahn equation as a model problem. We employ asymptotic analysis to identify slow variables and perform a transformation of the partial differential equations accordingly. Subsequently, we apply the proper orthogonal decomposition method and a QR discrete empirical interpolation method (qDEIM) technique to the transformed equation with the slow variables. Numerical experiments demonstrate that the new model reduction method yields significantly improved results compared to direct model reduction applied to the original equation. Furthermore, this approach can be extended to other equations, such as the convection equation and the Burgers equation. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/toreanony/TransformedModelReduction and in the supplementary materials (TransformedModelReduction-master.zip [19.1KB]).