A box scheme for transcritical flow

The accurate computer simulation of river and pipe ow is of great importance in the design of urban drainage networks. The use of implicit numerical schemes allows the time step to be chosen on the basis of accuracy rather than stability, o ering a potential computational saving over explicit methods. The highly successful Box Scheme is an implicit method which can be used to model a wide range of subcritical and supercritical ows. However, care must be taken over the modelling of transcritical ows since, unless the correct internal boundary conditions are imposed, the scheme becomes unstable. The necessity of accurately tracking all the critical interfaces and treating them accordingly can be algorithmically complex and in practice the underlying mathematical model is often modi ed to ensure that the ow remains essentially subcritical. Such a modi cation however inevitably leads to additional errors and incorrect qualitative behaviour can be observed. In this paper we show how the technique of ‘residual distribution’ can be successfully implemented in order to accurately model unsteady transcritical ow without the need to know a priori which regions of the computational domain correspond to subcritical and supercritical ow. When used in conjunction with a form of arti cial smoothing, the resulting method generates very high resolution results even for transcritical problems involving shocks, as can be seen in the numerical results