Support: National Science Foundation (NSF) 

Period of Performance:  September 1st, 2022 – August 31st, 2025

Budget: $369,000

Summary: The project is aimed at developing accurate, efficient, and robust numerical methods for nonlinear hyperbolic systems of PDEs and and related models. In particular, we are interested in the derivation of of structure preserving and asymptotic preserving numerical methods, which goes beyond basic properties of consistency, stability and convergence and requires a revision of standard strategies.

In this proposal, we design such methods by “mimicking” the theoretical properties and ensuring that relevant structures of the underlying PDE system and the asymptotic behavior of its solutions are maintained at the discrete numerical level. Specifically, we will be working on further development of several classes of numerical methods:

(i) well-balanced and positivity preserving numerical schemes, i.e., the methods which are capable of exactly preserving physically relevant steady-state solutions as well as maintaining the positivity of the computed quantity of interest;
(ii) asymptotic preserving schemes, which provide accurate and efficient numerical solutions in certain stiff and/or asymptotic regimes of physical interest;
(iii) well-balanced and asymptotic preserving methods for problems with uncertain data.