Portal de Periódicos da CAPES

On Nonlocal Monotone Difference Schemes for Scalar Conservation Laws

1986; American Mathematical Society; Volume: 47; Issue: 175 Linguagem: Inglês

10.2307/2008080

1088-6842

Bradley J. Lucier,

Numerical methods for differential equations

We provide error analyses for explicit, implicit, and semi-implicit monotone finitedifference schemes on uniform meshes with nonlocal numerical fluxes.We are motivated by finite-difference discretizations of certain long-wave (Sobolev) regularizations of the conservation laws that explicitly add a dispersive term as well as a nonlinear dissipative term.We also develop certain relationships between dispersion and stability in finite-difference schemes.Specifically, we find that discretization and explicit dispersion have identical effects on the amount of artificial dissipation necessary for stability.1. Introduction.We analyze a class of monotone numerical methods for the approximate solution of the hyperbolic conservation laws «, + /(«), = 0, xeR,ie(0,r],We give convergence results with error estimates for explicit, implicit, and semi-implicit finite-difference schemes on uniform meshes with nonlocal numerical fluxes.The motivating examples for these methods are finite-difference discretizations of a Sobolev-type regularization of (C), (S) u,+f(u)x-vg{u)xx-a2uxxl = 0. Equation (S), which regularizes (C) by adding a term simulating dispersive effects (~a2uxxl) as well as dissipation (-vg(u)xx), has been studied in [20] as a singularperturbation of (C); one can find other references there.For example, the implicit difference scheme that we consider is (1.1) d,U," + dxf{U"+l)i -ud2g(Un + 1)i -a2d2d,U," = 0, i e Z, n > 0, where dxW,n = (W''+l -W?_x)/2h, d2xW?= (Wi"+l -2Wi" + W^J/h2, and dtW?for any mesh function W. The positive parameters A and Ai are the mesh size and the time step, respectively.Such methods are similar to finite-difference and finite-element schemes introduced by Douglas et al. [8], and to artificial time methods, introduced by Jameson and Baker [14], for finding steadystate solutions of the Euler equations.