New High-Resolution Semi-Discrete Central Schemes: Derivation, Applications and Local Error Analysis

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Central schemes may serve as universal finite-difference methodsfor numerically solving hyperbolic conservation laws, Hamilton-Jacobiequations and closely related convection-diffusion equations. Suchschemes are not tied to the specific eigen-structure of the problem,and hence can be implemented in a straightforward manner as black-boxsolvers for a wide variety of nonlinear equations governing thespontaneous evolution of large gradient phenomena. The first-order Lax-Friedrichs scheme is the forerunner for suchcentral schemes. The second-order Nessyahu-Tadmor scheme offers high resolution while retaining the simplicity of Riemann-solver-free approach. In the convective regime the improved resolution of the Nessyahu-Tadmor scheme and its generalizations is achieved by using high-order piecewise polynomial reconstructions and high-order quadrature formulas for computing the flux integrals. At the same time, this family of staggered central schemes suffers from excessivenumerical viscosity when a sufficiently small time step is enforced,e.g., due to the presence of (degenerate) diffusive term.Recently Kurganov and Tadmor introduced a new family of centralschemes, which retain the simplicity of staggered central schemes,yet they enjoy a smaller numerical viscosity. In particular, theseschemes admit a simple semi-discrete formulation. This project aims to develop new, minimally dissipative fully- and semi-discrete central schemes for conservation laws. The main ideas behind the construction of these new schemes is the use of more precise information of the localpropagation speed, and realizing the (non-smooth part of the) approximatesolution in terms of its cell averages integrated over the nonsymmetric Riemann fans of varying size.Hyperbolic conservation laws, Hamilton-Jacobi equations and convection-diffusion equations are of great practical importance. They govern avariety of physical phenomena that appear in fluid mechanics, gasdynamics, magnetohydrodynamics, astrophysics, groundwater flow,meteorology, semiconductors, reactive flows, two-phase flow in oilreservoirs, non-Newtonian flows, front propagation and several otherareas. Financial modeling, traffic flow, differential games, optimalcontrol and image enhancement are among the most recent applicationsof the above models.Genuinely multidimensional high-resolution semi-discrete central schemes provide a rather simple and universal method for solving these problems. At the same time, the computationalefficiency of central schemes is extremely high. For example, recent numerical experiments in three-dimensional magnetohydrodynamics demonstrate that using central schemes allows to achieve the desired resolution about 25 times faster in comparison with other methods. In general, the advantage of the new semi-discrete central schemes over alternative upwind methods is particularly amplified when they are used to solve complicated multidimensional systems arising in practice.The proposed schemes will be also applied to such important problemsas compressible and incompressible Euler and Navier-Stokes equations,multi-phase model of geometric optics, multicomponent flow andcompressible bubbles models, moving boundaries problems, shockreflection problem for the unsteady transonic small disturbanceequation and others.