Math @ Duke

Publications [#246927] of JianGuo Liu
Papers Published
 Lefloch, PG; Liu, JG, Generalized monotone schemes, discrete paths of extrema, and discrete entropy conditions,
Mathematics of Computation, vol. 68 no. 227
(1999),
pp. 10251055
(last updated on 2019/02/21)
Abstract: Solutions of conservation laws satisfy the monotonicity property: the number of local extrema is a nonincreasing function of time, and local maximum/minimum values decrease/increase monotonically in time. This paper investigates this property from a numerical standpoint. We introduce a class of fully discrete in space and time, high order accurate, difference schemes, called generalized monotone schemes. Convergence toward the entropy solution is proven via a new technique of proof, assuming that the initial data has a finite number of extremum values only, and the fluxfunction is strictly convex. We define discrete paths of extrema by tracking local extremum values in the approximate solution. In the course of the analysis we establish the pointwise convergence of the trace of the solution along a path of extremum. As a corollary, we obtain a proof of convergence for a MUSCLtype scheme that is second order accurate away from sonic points and extrema.


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