Hyperbolic Systems of Conservation Laws

3-point scheme a/at a/ax approximation Assume au/at au/ax av/at AVj+ BV(R Cauchy problem Chapter classical solution conservation form conservation laws constant continuous function converges convex function defined difference scheme 1.3 discontinuity dx dt E-scheme entropy condition entropy flux equation EXAMPLE Figure finite difference flux limiter function f Godunov's Godunov's scheme grad Harten Hence hyperbolic incremental coefficients L-stable L²(R Lagrangian coordinates Lax-Friedrichs scheme Lax-Wendroff scheme Leer's Lemma limit Lipschitz continuous monotone scheme Moreover nonlinear numerical flux numerical viscosity obtain Osher Proof Proposition prove Rankine-Hugoniot rarefaction result Riemann problem Riemann solver scalar conservation laws scheme is T.V.D. second order accurate sequence smooth smooth function solution of 1.1 strictly convex systems of conservation T.V.D. scheme Theorem 3.1 TV(u TV(v u₁ u₂ V₁ viscosity viscosity coefficient Vj+1 weak solution