An Error Estimate for the Approximate Solution of a Porous Media Diphasic Flow Equation

In this paper we present an error estimate for the approximate solution of the nonlinear hyperbolic equation u _t + div (f(u(x, t))v(x)) = 0 by an implicit finite volume scheme, using an Engquist-Osher numerical flux. We show that the error is of order % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGcaa % qaaiaadUgacqGHRaWkdaGcaaqaaiaadIgaaSqabaaabeaaaaa!3A93!$$\sqrt {k + \sqrt h } $$ , where h and k are respectively the space and the time steps size parameters. The error estimate shows that the convergence of this scheme is possible with an unbounded CFL condition. This result is extended to other numerical fluxes and explicit scheme in [3].