An Extension of Essentially Non-Oscillatory Shock-Capturing Schemes to Multi-Dimensional Systems of Conservation Laws

In recent years, a class of numerical schemes for solving hyperbolic partial differential equations has been developed which generalizes the first-order method of Godunov to arbitrary order of accuracy. High-order accuracy is obtained, wherever the solution is smooth, by an essentially non-oscillatory (ENO) piecewise polynomial reconstruction procedure, which yields high-order pointwise information from the cell averages of the solution at a given point in time. When applied to piecewise smooth initial data, this reconstruction enables a flux computation that provides a time update of the solution which is of high-order accuracy, wherever the function is smooth, and avoids a Gibbs phenomenon at discontinuities. The promising results of Harten et al., in the use of ENO schemes in solving the one-dimensional Euler equations of gas dynamics, have aroused considerable interest in the aerodynamic community. However, the application of these schemes to areas of scientific and industrial interest, such as aircraft configuration, obviously requires compressible flow solutions in more than one spatial dimension. It is this extension of ENO schemes to multi-dimensional application to which this study is dedicated. In particular, a two-dimensional extension is proposed for the Euler equations of gas dynamics. Among the issues to be considered in this extension are achieving formal high-order two-dimensional spatial accuracy, the implementation of boundary conditions and applications to general curvilinear coordinates