Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes

Abstract

This paper is devoted to the study of an error estimate of the nite volume approximation to the solution u 2 L 1 (IR N IR) of the equation ut + div(vf(u)) = 0, where v is a vector function depending on time and space. A \h 1=4 " error estimate for an initial value in BV (IR N) is shown for a large variety of nite volume monotoneous ux schemes, with an explicit or implicit time discretization. For this purpose, the error estimate is given for the general setting of approximate entropy solutions, where the error is expressed in terms of measures in IR N and IR N IR. The study of the implicit schemes involves the study of the existence and uniqueness of the approximate solution. The cases where an \h 1=2 " error estimate can be achieved are also discussed.

Mathematical study of a petroleum-engineering scheme

“…This technique was first used for the existence of a solution to a nonlinear discretization scheme in [21]. The idea of the proof is the following: if we can modify continuously the scheme to obtain a linear system and if the modification simultaneously preserves the estimates which were obtained in Propositions 4.1 and 4.2, then the scheme has at least one solution (since in the linear case, these estimates also prove that the linear system has a unique solution).…”

Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients

“…The case of finite difference schemes have been treated by Oleȋnik [40], Harten et al [24], Kuznetsov [36], Crandall and Majda [12], Sanders [44], Lucier [37], Osher and Tadmor [41], Cockburn and Gremaud [10], and many others. The study of finite volume methods is more recent and have been conducted by Champier et al [7], Vila [48], Cockburn et al [8,9], Kröner and Rokyta [32], Kröner et al [31], Noelle [38], Eymard et al [21], and Chainais-Hillairet [6], as well as many others. Among the cited papers, only [6,7,21,40] treat equations where the nonlinearity f depends on the spatial position x (and time t).…”

“…The study of finite volume methods is more recent and have been conducted by Champier et al [7], Vila [48], Cockburn et al [8,9], Kröner and Rokyta [32], Kröner et al [31], Noelle [38], Eymard et al [21], and Chainais-Hillairet [6], as well as many others. Among the cited papers, only [6,7,21,40] treat equations where the nonlinearity f depends on the spatial position x (and time t).…”

An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations

“…We now state the abstract theorem which will be used hereafter; this result follows from standard arguments of the topological degree theory (see [9] for an exposition of the theory and [13] for another utilisation for the same objective as here, namely the proof of existence of a solution to a numerical scheme). We are now in position to prove the existence of a solution to the discrete problem (2.3), for fairly general equations of states.…”