Hyperbolic Systems of Conservation Laws

3-point scheme a/at a/ax approximation Assume au/at au/ax BV(R C.F.L. condition Cauchy problem Chapter classical solution conservation form conservation laws constant continuous function converges convex function d/dx defined difference scheme 1.3 discontinuity du/at du/dx dx dt E-scheme entropy condition entropy flux equation EXAMPLE Figure flux limiter function f given Godunov's Godunov's scheme grad Hence incremental coefficients integral L-stable L²(Rd Lagrangian coordinates Lax-Friedrichs scheme Lax-Wendroff scheme Leer's Lemma limit Lipschitz continuous monotone scheme Moreover nonlinear numerical flux obtain problem 1.1 Proof Proposition prove put in conservation Rankine-Hugoniot rarefaction result Riemann problem Riemann solver 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(uo TV(v u₁ upwind scheme v₁ viscosity Vj+1 weak solution ΔΧ ζε