Kinetic Approaches for Multi-Scale Problems in Quantum Chemistry and Seismology

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

This research project concerns an interdisciplinary topic that is of general interest to both computational mathematicians and scientists from other areas. This project will lead to quantitative understanding of the optical properties of crystals and in imaging Earth rock structures clearly, simply, and quickly. More importantly, these new methods will provide important advances in computational mathematics, leading to practical benefit in the fields of materials science and seismology. This work is motivated by recent work on Frozen Gaussian Approximation (FGA) and its unique interpretation of high-frequency waves that suggests many new numerical techniques embedded. State-of-the-art numerical methods will be developed and used in various applications. The work will have impact in a wide range of problems of significant interest in physical sciences and industry. Specifically, the project investigates the following topics: 1. To theoretically study the FGA method in more depth. This includes analyzing the asymptotic accuracy of FGA, and integrating it with the Tailored Finite Point/Cell Method (TFPM/TFCM). 2. To study the dynamics of electrons in crystals by computing the Schrödinger equation with a lattice potential in both the Lagrangian and Eulerian approaches. 3. To compute seismic wave propagation in heterogeneous or non-lateral media based on FGA and its combination with TFPM/TFCM. 4. To survey the practical performance of developed numerical methods on, e.g., tsunami and earthquake models.