Analysis of numerical methods for nonlinear hyperbolic conservation laws.

摘要

This thesis is devoted to the study of some a priori and a posteriori error estimates of numerical methods of nonlinear hyperbolic conservation laws.; The first result is a new priori error estimate. For multidimensional flux-splitting monotone schemes defined on non-Cartesian grids, we identify a class of consistent schemes and prove that the L ∞ (0, T; L1 ($Rd$))-norm of the error goes to zero as (Δx) 1/2 when the discretization parameter Δx goes to zero. Moreover, we show that non-consistent schemes can converge at optimal rates of (Δx)1/2 because the conservation form of the schemes and the consistency of the numerical fluxes allow the regularity properties of the approximate solution to compensate for their lack of consistency.; The second result is a new posteriori error estimate for nonlinear convection-diffusion equation which physically models nonlinear hyperbolic conservation laws when the diffusion effects become negligible. First, we prove a continuous dependency result in an L1-like norm, from which a simple new, posteriori error estimate is obtained. Then we show how this new posteriori error estimate helps us to understand the nonlinear hyperbolic case.