Paul Andries Zegeling

Hong Zhang, Paul Andries Zegeling

Motivated by observations of saturation overshoot, this paper investigates numerical modeling of two-phase flow incorporating dynamic capillary pressure. The effects of the dynamic capillary coefficient, the infiltrating flux rate and the initial and boundary values are systematically studied using a travelling wave ansatz and efficient numerical methods. The travelling wave solutions may exhibit monotonic, non-monotonic or plateau-shaped behaviour. Special attention is paid to the non-monotonic profiles. The travelling wave results are confirmed by numerically solving the partial differential equation using an accurate adaptive moving mesh solver. Comparisons between the computed solutions using the Brooks-Corey model and the laboratory measurements of saturation overshoot verify the effectiveness of our approach.

Hong Zhang, Yunrui Guo, Weibin Li et al.

We present a robust and accurate numerical method to solve the modified Buckley-Leverett equation in two-phase porous media flow with dynamic capillary pressure effect. A symmetric interior penalty discontinuous Galerkin method is used to discretize the equation in the space direction. For accuracy and stability issues, the third-order strong stability preserving implicit-explicit Runge-Kutta method is adopted to solve the nonlinear semi-discrete system: the linear diffusion term is discretized implicitly while the nonlinear flux term is discretized explicitly. The spatial accuracy of the discontinuous Galerkin method depends on the limiters applied to the solution: we test a minmod-TVB limiter, a simple WENO limiter and a high-order shock-capturing moment limiter to demonstrate that a suitable shock capturing moment limiter leads to more accurate approximation of solution. A set of representative numerical experiments are presented to show the accuracy and efficiency of the proposed approach. The results indicate that the moment limiter proposed by Moe et al. [Arxiv:1507.03024, 2015] is the most suitable one to be used in solving the modified Buckley- Leverett equation, and high order schemes perform much better than lower order schemes. Our simulation results are consistent with the previous results in Kao et al.[J. Sci. Comput., 64(3) (2015), 837-857], Zhang and Zegeling [J. Comput. Phys., 345 (2017), 510-527 and Commun. Comput. Phys., 22(4) (2017), 935-964].

Han Zhou, Paul Andries Zegeling

We propose and study a class of numerical schemes to approximate time fractional differential equations. The methods are based on the approximation of the Caputo fractional derivative by continuous piecewise polynomials, which is strongly related to the backward differentiation formulae for the integer-order case. We investigate their theoretical properties, such as the local truncation error and global error analyses with respect to a sufficiently smooth solution, and the numerical stability in terms of the stability region and $A(\fracπ{2})$-stability by refining the technique proposed in \cite{LubichC:1986b}. Numerical experiments are given to verify the theoretical investigations.