Colloquium

This lecture is concerned with numerical methods for hyperbolic equations which play an important role in mathematical modeling and numerical simulation of fluid flows. A brief introduction to the theory of scalar conservation laws (linear advection, Burgers' equation, Buckley-Leverett equation for flows in porous media) is given. In the one-dimensional case, the discretization in space is performed using conservative finite difference schemes. Numerical difficulties related to the existence of discontinuous and, possibly, nonunique solutions are discussed. The upwind difference scheme as well as the Lax-Wendroff and Beam-Warming methods are introduced in the context of linear advection and extended to nonlinear conservation laws. A family of total variation diminishing (TVD) discretizations is constructed by blending numerical fluxes of first and second order so as to satisfy the conditions of Harten's TVD theorem. The properties of standard flux limiters are illustrated by one-dimensional numerical examples. The state of the art in the development of high-resolution schemes for hyperbolic systems, multidimensional problems, and unstructured meshes is reviewed. Simulation results are presented for the Euler equations of gas dynamics in 1D and 2D.