Hyperbolic Systems of Conservation Laws

3-point apply approximation associated Assume belongs bounded Chapter 11 coefficients computation condition conservation form conservation laws consider consistent constant construct continuous converges convex corresponding defined definition denote depends derive difference scheme differential discontinuity dx dt E-scheme easy entropy condition entropy solution equation equivalent estimate EXAMPLE exists fact Figure formula function given gives Godunov grad hand Hence holds implies incremental inequality initial instance integral interval introduce jump Lemma limit linear method modified monotone Moreover Note numerical flux obtain order accurate piecewise Proof Proposition prove REMARK requirement result Riemann problem satisfies second order sequence shock shows simple smooth solution of 1.1 space strictly suppose term Theorem unique values variables viscosity wave weak solution write written yields