Construction of solutions for two-dimensional Riemann problems

Solutions to the scalar quasilinear equation σu(t, x) σt + ∑ i=1 2 σf₁(u(t, x)) σx_i = 0 for f_i 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.