## Abstract

Solutions to the scalar quasilinear equation σu(t, x) σt + ∑ i=1 2 σf_{1}(u(t, x)) σx_{i} = 0 for f_{i} ε{lunate} C^{2}:R → R with initial data given by a two-dimensional Riemann problem, are piecewise smooth if f_{1} f_{2} 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 language | English |
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Pages (from-to) | 615-630 |

Number of pages | 16 |

Journal | Computers and Mathematics with Applications |

Volume | 12 |

Issue number | 4-5 PART A |

DOIs | |

State | Published - 1986 |