Applied Mathematics Colloquium – Alina Chertock (NC State)

Asymptotic Preserving Numerical Methods for Multiscale Problems

Abstract. Many phenomena in nature exhibit multiscale behaviors, which can be rather different in character. These phenomena can be categorized into two groups. On the one hand, there are problems featuring localized singularities, such as boundary or internal layers, shocks, and dislocations. On the other hand, there are problems, such as porous media flows, turbulent flows, and highly oscillating models, where microscopic and macroscopic scales coexist across the entire domain.

When several scales occur in a physical problem, using an approach that describes the phenomenon on a single scale is insufficient. Describing the problem at a microscopic level offers exceptional physical accuracy but is computationally impractical. Likewise, adopting a macroscopic description, where explicit equations for the macroscopic scale are used, effectively eliminating the other scales, is also unsuitable. As such, a multi-scale modeling strategy becomes essential. This involves employing different models to describe phenomena at various scales while balancing the trade-off between numerical accuracy and computational efficiency. The primary objective of multi-scale techniques is to develop numerical schemes that bridge the microscopic and macroscopic scales, outperforming the computational demands of solving the complete microscopic model while still delivering the desired level of accuracy.

Among many other approaches, a special class of numerical methods, known as Asymptotic Preserving (AP) schemes, was developed specifically for multiscale problems. The fundamental concept involves designing numerical techniques that maintain the asymptotic behavior across the transition from microscopic to macroscopic models within a discrete framework. Consequently, AP schemes seamlessly bridge the two scales: the transition between the two scales is implemented effortlessly in that a micro solver automatically becomes a macro solver if the numerical discretizations fail to resolve the physically small scales. As a result, the AP methodology offers straightforward, robust, and efficient computational tools for a wide array of multiscale problems, including kinetic, hyperbolic, and other physical problems. This talk provides an overview of the core concept, design principles, and several representable  AP schemes.

Applied Mathematics Seminar
There will be a post-colloquium tea from 4:30p-5:00p in PH 330.