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

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Central schemes may serve as universal finite-difference methods fornumerically solving hyperbolic conservation and balance laws,Hamilton-Jacobi equations, and related problems. Such schemes are nottied to the specific eigen-structure of the problem, and hence can beimplemented in a straightforward manner for a wide variety of nonlinearequations governing the spontaneous evolution of large gradientphenomena. This project aims to further improve the family of Godunov-type centralschemes, recently developed by Kurganov et al. The main ideas behind theconstruction of the new, less dissipative central schemes are to usemore precise estimate of the smooth and nonsmooth parts of the solutionby considering non-rectangular control volumes; to use a more accurateprojection of the evolved data onto the original, non-staggered grid;and to avoid the loss of information when very accurate fully-discreteschemes are reduced to a much simpler semi-discrete form.The second main goal of the project is the application of central schemesto various multi-phase and multi-fluid flow models, the Saint-Venantsystems of shallow water equations (which describe flows in rivers andcoastal areas), multi-layer shallow water systems, models of transport ofpollutant in shallow water, the Euler equations of gas dynamics subjectto a static gravitational field, chemotaxis models, reactive flows (inparticular, the models describing stiff detonation waves), extendedthermodynamics, shallow water equations on a rotating sphere, acousticwave propagation, heterogeneous elasticity, granular material flows.Naturally, these applications involve multiple space dimensions, complexgeometries and moving boundaries/interfaces, This would require furtherdevelopment of the theory and implementation of central schemes. Inparticular, semi-discrete central schemes on unstructured and triangularmeshes will be derived, and different adaptive techniques will beincorporated 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 hyperbolicconservation and balance laws, Hamilton-Jacobi equations, and relatedproblems. In the past decade, a family of simple, universal,Riemann-solver-free finite volume central schemes has proven to bean appealing alternative to the more complicated and problem orientedupwind schemes. The advantages of central schemes are particularlyprominent when they are used to solve complicated multidimensionalsystems of PDEs arising in such important fields including fluidmechanics, 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 ofcentral schemes, and on their practical applications. The new centralschemes will be incorporated into a general-purpose adaptive meshrefinement (AMR) and adaptive moving mesh (AMM) codes, which will befreely accessible for the scientific and industrial communities. Thesecodes will serve as a reliable and robust "black-box-solver" for arather comprehensive class of time-dependent PDEs.