TY - JOUR

T1 - Construction of solutions for two-dimensional Riemann problems

AU - Lindquist, W. B.

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 1986

Y1 - 1986

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0000349914&partnerID=8YFLogxK

U2 - 10.1016/0898-1221(86)90185-9

DO - 10.1016/0898-1221(86)90185-9

M3 - Article

AN - SCOPUS:0000349914

VL - 12

SP - 615

EP - 630

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 4-5 PART A

ER -