Compactness estimates for difference schemes for conservation laws with discontinuous flux

05/30/2023

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by   Kenneth H. Karlsen, et al.

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We establish quantitative compactness estimates for finite difference schemes used to solve nonlinear conservation laws. These equations involve a flux function f(k(x,t),u), where the coefficient k(x,t is BV-regular and may exhibit discontinuities along curves in the (x,t) plane. Our approach, which is technically elementary, relies on a discrete interaction estimate and the existence of one strictly convex entropy. While the details are specifically outlined for the Lax-Friedrichs scheme, the same framework can be applied to other difference schemes. Notably, our compactness estimates are new even in the homogeneous case (k≡ 1).