Boundary value problems for the N-wave interaction equations

We consider boundary value problems for the N-wave interaction equations in one and two space dimensions, posed for x [greater-or-equal, slanted] 0 and x,y [greater-or-equal, slanted] 0, respectively. Following the recent work of Fokas, we develop an inverse scattering formalism to solve these problems by considering the simultaneous spectral analysis of the two ordinary differential equations in the associated Lax pair. The solution of the boundary value problems is obtained through the solution of a local Riemann–Hilbert problem in the one-dimensional case, and a nonlocal Riemann–Hilbert problem in the two-dimensional case.

Conservative upwind difference schemes for the Euler equations for real gas flows in a duct

In a recent paper [P. Glaister, Conservative upwind difference schemes for compressible flows in a Duct, Comput. Math. Appl. 56 (2008) 1787–1796] numerical schemes based on a conservative linearisation are presented for the Euler equations governing compressible flows of an ideal gas in a duct of variable cross-section, and in [P. Glaister, Conservative upwind difference schemes for compressible flows of a real gas, Comput. Math. Appl. 48 (2004) 469–480] schemes based on this philosophy are presented for real gas flows with slab symmetry. In this paper we seek to extend these ideas to encompass compressible flows of real gases in a duct. This will incorporate the handling of additional terms arising out of the variable geometry and the non-ideal nature of the gas.