The project focuses on developing advanced numerical methods to solve systems governed by evolutionary partial differential equations (PDEs) that depend on various parameters. These equations are fundamental in diverse fields such as fluid dynamics, environmental modeling, epidemiology, finance, and social sciences. The aim is to create efficient tools capable of addressing the challenges posed by hyperbolic systems of balance laws, kinetic equations, and delay-PDEs.

A significant focus is on designing well-balanced (WB) numerical methods that maintain equilibrium states with high precision. This is crucial for accurately detecting small signals in systems with perturbations. Additionally, implicit-explicit (IMEX) methods will be developed to efficiently handle multiscale systems, where stiff terms are treated implicitly to avoid restrictive time steps, and non-stiff terms are handled explicitly to optimize computational effort. Combining WB and IMEX methods is one of this project's goals, enabling the effective solution of balance laws and accurate equilibrium detection.

The techniques will be applied to linear hyperbolic and kinetic equations with dissipative terms, with further exploration of their extension to mildly nonlinear equations. The project also seeks to develop efficient numerical approaches for delay-PDEs, which are widely used in the mathematical modeling of systems such as epidemics. Overall, this work aims to produce robust and impactful methods for solving complex PDE systems across various applications, including uncertainty quantification.