This is a summary of five lectures delivered at the CIME course on "Advanced Numerical Approximation of Nonlinear Hyperbolic Equations" held in Cetraro, Italy, on June 1997.

Following the introductory lecture I -- which provides a general overview of approximate solution to nonlinear conservation laws, the remaining lectures deal with the specifics of four complementing topics:

• Lecture II. Finite-difference methods - non-oscillatory central schemes;
• Lecture III. Spectral approximations - the Spectral Viscosity method;
• Lecture IV. Convergence rate estimates - a Lip' convergence theory;
• Lecture V. Kinetic approximations - regularity of kinetic formulations.

4in

Acknowledgment.I thank Alfio Quarteroni for the invitation, B. Cockburn, C. Johnson & C.-W Shu for the team discussions, the Italian participants for their attention, and the hosting Grand Hotel at San Michele for its remarkably unique atmosphere. Research was supported in part by ONR grant #N00014-91-J-1076 and NSF grant #97-06827.

1in

AMS Subject Classification. Primary 35L65, 35L60. Secondary 65M06, 65M12, 65M15, 65M60, 65M70

 
• Contents
• Non-oscillatory central schemes
  • Introduction
  • A Short Guide to Godunov-Type schemes
    • Upwind schemes
    • Central schemes
  • Central schemes in one-space dimension
    • The second-order Nessyahu-Tadmor scheme
    • The third-order central scheme
  • Central schemes in two space dimensions
    • Boundary conditions
  • Incompressible Euler equations
    • The vorticity formulation
    • The velocity formulation
• References
• Kinetic Formulations and Regularity
  • Regularizing effect in one-space dimension
  • Velocity averaging lemmas ()
  • Regularizing effect revisited ()
    • Kinetic and other approximations
  • Degenerate parabolic equations
  • The isentropic equations
• References
• About this document ...