C1-PPU Schemes for Efficient Simulation of Coupled Flow and Transport with Gravity

It is revealed by recent studies that in the presence of counter-current flow due to buoyancy, nonlinear convergence problems may be pronounced when the popular PPU scheme is used to approximate the numerical flux. The PPU numerical flux is non-differentiable across the co-current/counter-current flow regimes and thus may lead to oscillations or even divergence in the Newton iterations. Recently proposed methods address improved smoothness of the numerical flux. In this paper we devise and analyze an alternative numerical flux scheme called C1-PPU that allows a smooth variation between the co-current/counter-current flow regimes as well as an optimal balance between the scalar nonlinearity and accuracy of the flux function. The C1-PPU scheme involves a novel use of the flux limiter concept from the context of high-resolution methods. Numerical examples including 1D scalar transport problem and 2D heterogeneous problem with fully-coupled flow and transport are presented. The results indicate that in addition to smoothness, nonlinearity may also be critical for convergence behavior and thus needs to be considered in the design of an efficient numerical flux scheme. Moreover, the results show that our C1-PPU scheme exhibits superior convergence properties for large time steps compared to the other alternatives.