Construction of solutions for two-dimensional Riemann problems

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Abstract

Solutions to the scalar quasilinear equation σu(t, x) σt + ∑ i=1 2 σf1(u(t, x)) σxi = 0 for fi ε{lunate} C2:R → R with initial data given by a two-dimensional Riemann problem, are piecewise smooth if f1 f2 f, and f has at most one inflection point. We show that the "pieces" of this solution can be classified and are expressible in terms of two-dimensional nonlinear waves in analogy with the nonlinear rarefaction and shock waves of the Riemann problem in one spatial dimension. The two-dimensional waves can be expressed in almost-closed form. Explicit solutions are constructable from these waves. An application is illustrated by calculation of the interaction of water/oil banks in two-phase incompressible flow in reservoirs.

Original languageEnglish
Pages (from-to)615-630
Number of pages16
JournalComputers and Mathematics with Applications
Volume12
Issue number4-5 PART A
DOIs
StatePublished - 1986

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