Godunov-Type Central Schemes for Hyperbolic Problems: Further Development, Adaptation, and Applications

Tulane University, New Orleans LA

Investigators

Abstract

Central schemes may serve as universal finite-difference methods for numerically solving hyperbolic conservation and balance laws, Hamilton-Jacobi equations, and related problems. Such schemes are not tied to the specific eigen-structure of the problem, and hence can be implemented in a straightforward manner for a wide variety of nonlinear equations governing the spontaneous evolution of large gradient phenomena. This project aims to further improve the family of Godunov-type central schemes, recently developed by Kurganov et al. The main ideas behind the construction of the new, less dissipative central schemes are to use more precise estimate of the smooth and nonsmooth parts of the solution by considering non-rectangular control volumes; to use a more accurate projection of the evolved data onto the original, non-staggered grid; and to avoid the loss of information when very accurate fully-discrete schemes are reduced to a much simpler semi-discrete form. The second main goal of the project is the application of central schemes to various multi-phase and multi-fluid flow models, the Saint-Venant systems of shallow water equations (which describe flows in rivers and coastal areas), multi-layer shallow water systems, models of transport of pollutant in shallow water, the Euler equations of gas dynamics subject to a static gravitational field, chemotaxis models, reactive flows (in particular, the models describing stiff detonation waves), extended thermodynamics, shallow water equations on a rotating sphere, acoustic wave propagation, heterogeneous elasticity, granular material flows. Naturally, these applications involve multiple space dimensions, complex geometries and moving boundaries/interfaces, This would require further development of the theory and implementation of central schemes. In particular, semi-discrete central schemes on unstructured and triangular meshes will be derived, and different adaptive techniques will be incorporated into the central framework. Recent development of modern technology requires reliable, efficient, high-resolution methods for solving time-dependent partial differential equations (PDEs), including multidimensional systems of hyperbolic conservation and balance laws, Hamilton-Jacobi equations, and related problems. In the past decade, a family of simple, universal, Riemann-solver-free finite volume central schemes has proven to be an appealing alternative to the more complicated and problem oriented upwind schemes. The advantages of central schemes are particularly prominent when they are used to solve complicated multidimensional systems of PDEs arising in such important fields including fluid mechanics, gas dynamics, geophysics, meteorology, magnetohydrodynamics, astrophysics, multi-component flows, granular flows, reactive flows, semiconductors, non-Newtonian flows, geometric optics, traffic flow, image processing, financial, biological modeling, differential games, and optimal control. This project is focused on the further development and improvement of central schemes, and on their practical applications. The new central schemes will be incorporated into a general-purpose adaptive mesh refinement (AMR) and adaptive moving mesh (AMM) codes, which will be freely accessible for the scientific and industrial communities. These codes will serve as a reliable and robust "black-box-solver" for a rather comprehensive class of time-dependent PDEs.