Hyperbolic systems of conservation laws

0sher 3-point scheme a/at a/axj approximation Assume au/at au/ax C.F.L. condition Cauchy problem Chapter classical solution compact support conservation form conservation laws consider constant continuous function converges convex function defined DEFINITI0N difference scheme 1.3 discontinuity dx dt E-scheme entropy condition entropy flux entropy inequality equation EXAMPLE Figure finite difference fj(u flux limiter function f given Godunov's Godunov's scheme grad Harten Hence hyperbolic incremental coefficients integral jump condition Lagrangian coordinates Lax-Wendroff scheme Leer's Lemma limit Lipschitz continuous LOO(Rd monotone scheme Moreover nonlinear numerical flux obtain PR0P0SITI0N Proof prove put in conservation rarefaction REMARK result Riemann problem scalar conservation laws scheme is T.V.D. second order accurate Section sequence smooth function solution of 1.1 strictly convex systems of conservation T.V.D. scheme Theorem 3.1 u0 belongs upwind scheme viscosity vj+1 weak solution