Provably bounds-preserving high-order numerical methods in liquid column chromatography

• Applicants: Professor Dr.-Ing. Gregor Gassner; Professor Dr.-Ing. Eric von Lieres und Wilkau
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 548805630

The idea of this research project is to bring recent developments in the numerical analysis of structure preserving high-order Discontinuous Galerkin (DG) methods to the field of chromatography process engineering. Packed bed liquid chromatography plays a pivotal role in the downstream processing of biotechnological manufacturing processes. It enables the separation and purification of product molecules derived from fermentation broth or natural sources. Mathematical modeling and numerical simulation have been integral to chromatography research for decades, aiding in understanding underlying mechanisms and identifying critical separation factors. Many investigations such as model calibration/parameter estimation, process screening, design and optimization, uncertainty quantification, or robustness analysis, require (hundreds of) thousands of simulations. Moreover, model predicitve control requires computer simulations to perform substantially faster than the modeled physical process. Thus, the numerical efficiency of individual forward simulations is most relevant. A major advantage of the considered nodal collocation spectral element DG method DGSEM is their high order accuracy per grid node with compact differentiation and surface integral operators that allow for smart data layouts and enable efficient computational kernels. The high-order method is further improved by split-formulations that enable mathematically provable energy (and entropy) estimates. In this project, we aim to perform a full continuous and discrete energy analysis of the two-dimensional general rate model (GRM). A disadvantage of DG (and high-order methods in general) is their tendency for spurious solution oscillations at sharp gradients (and discontinuities) that might even violate physical solution bounds (like positivity). To fix this issue and bring the full DG potential to the applications, we aim to introduce provably bounds preserving limiting techniques based on algebraic flux corrections, namely the invariant domain preserving or monolithic convex blending techniques. Here, the idea is to use, in a controlled mathematically guided way, a convex combination of the high order DG scheme with a provably bounds preserving low order discretization, such that the resulting high-resolution scheme is provably bounds preserving. Finally, our aim is to extend the open-source simulation software CADET with these newest mathematical and numerical developments to provide it to the community.

• DFG Programme: Research Grants