Strong bounded variation estimates for the multi-dimensional finite volume approximation of scalar conservation laws
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We prove a uniform bounded variation estimate for finite volume approximations of the nonlinear scalar conservation law ∂_t α + div(uf(α)) = 0 in two dimensions with an initial data of bounded variation. We assume that the divergence of the velocity div(u) is of bounded variation instead of the classical assumption that div(u) is zero. A uniform bounded variation estimate provides compactness for finite volume approximations in L^p spaces, which is essential to prove the existence of a solution for a partial differential equation (p.d.e.) with nonlinear terms in α, when uniqueness of the solution is not available. The finite volume schemes analysed in this article are set on nonuniform Cartesian grids. A uniform bounded variation estimate for finite volume solutions of the conservation law ∂_t α + div(F(t,x,α)) = 0, where div_xF≠0 on nonuniform Cartesian grids is also proved. Results from numerical tests are provided, and they complement theoretical findings.
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