| We consider the initial value problem for a scalar conservation law {dollar}partialsb{lcub}t{rcub}u(x,t){dollar} + div{dollar}sb{lcub}x{rcub}f(u(x,t)){dollar} = 0, {dollar}u(x,0) = usb0(x),x in IRsp{lcub}d{rcub}{dollar}, {dollar}tge{dollar} 0, (CL) where f, the flux function, is known and {dollar}usb0{dollar}, the initial data is given.; It is known that even for smooth initial data, discontinuities can form in the solution. This means we need to solve the conservation law in a class of discontinuous functions and we have to consider the weak formulation of the problem. This introduces an additional difficulty, since weak solutions are not uniquely determined.; Several authors have introduced uniqueness criteria, now called entropy conditions, for selecting a unique weak solution. Lax conjectured that there is a canonical way of determining a "reasonable" unique weak solution. We prove that a set of natural properties of the solution operator is sufficient to ensure the uniqueness and that it completely determines the solution operator. Since all known entropy conditions imply these properties of the solution operator, we prove in some sense their equivalence.; A survey of the known entropy conditions and the proof of their equivalence is contained in Chapter 3.; The rest of the thesis is devoted to rates of convergence of approximate solutions to the entropy solution.; Chapter 2 contains estimates of convergence rates of the viscosity solution for the linear advection equation and for Burgers's equation. We derive uniform estimates for the pointwise convergence of viscosity solutions to the entropy solution of Burgers's equation.; Chapter 4 contains the main result of the thesis. Monotone finite difference schemes, like the Lax-Friedrichs scheme, the Godunov scheme, and others, are known to be formally first order accurate. However, general error estimates, show convergence of order only one half in {dollar}Lsp1{dollar}. For linear equations it is known that these estimates are sharp. We prove, by constructing appropriate initial data, that the estimate of order one half is sharp for nonlinear problems as well. This proves the inefficiency of numerical methods based on linear approximation. |
