Collaborative Research: Numerical Methods for Shallow Water Equations and Related Models

The project is aimed at developing accurate, efficient, and robust numerical methods for shallow water equations and related models, with particular reference to problems that admit non-smooth (discontinuous) solutions and to problems that involve highly disparate scales and therefore are difficult to solve numerically. Shallow water and related models are widely used as a mathematical framework to study water flows in rivers and coastal areas as well as to investigate a variety of phenomena in atmospheric sciences and oceanography. These models are systems of time-dependent partial differential equations (PDEs) that are derived using physical properties such as conservation of mass and momentum, and hydrostatic or barotropic approximations. The principal part of the proposed research will be focused on the development of new methods for solving problems involving complicated nonlinear wave phenomena, problems with complex computational domains and moving interfaces. The resulting methods, while based on high-order shock-capturing finite-volume schemes and non-dissipative mesh-free particle methods, will incorporate special numerical techniques such as numerical balancing between the terms that are balanced in the original system of PDEs (development of well-balanced schemes), ensuring positivity of all fluid layers (this is absolutely necessary for both accurate description of dry and near dry states and enforcement of nonlinear stability), accurate and efficient operator splitting, accurate and efficient interface tracking, and others that will be in the focus of the proposed research project.

The proposed project will contribute significantly toward development of computational methods for shallow water and related models. Special attention will be paid to applications arising in oceanography and atmospheric sciences, in which the Coriolis forces (due to the Earth's rotation), thermodynamics, and turbulent effects have to be taken into account. The problems under study include, among others, formation and propagation of atmospheric fronts and ocean currents, propagation of tsunami waves and their on-shore arrival, as well as propagation of pollutants in various environments. The numerical methods under design will provide considerably more powerful tools for studying a variety of internal and surface water waves, including tsunami and rogue waves. These extreme waves, which arise both in deep and shallow water, have a significant impact on the safety of people and infrastructure, and are responsible for damage to ships, oil platforms, coastlines, and sea bottoms and for changes to the biological environment. Thus, understanding the physics of these extreme waves is an important task that may even contribute to saving lives.