Resumen de High-order hybridizable discontinuous galerkin formulation and implicit runge-kutta schemes for multiphase flow through porous media
Albert Costa Solé
This dissertation presents high-order hybridisable discontinuous Galerkin (HDG) formulations coupled with implicit Runge-Kutta (RK) methods for the simulation of one-phase flow and two-phase flow problems.
High-order-methods can reduce the computational cost while obtaining more accurate solutions with less dissipation and dispersion errors than low order methods. HDG is an unstructured, high-order accurate, and stable method. The stability is imposed using a single parameter. In addition, it is a conservative method at the element level, which is an important feature when solving PDEs in a conservative form. Moreover, a hybridization procedure can be applied to reduce the size of the global linear system. To keep the stability and accuracy advantages in transient problems, we couple the HDG method with high-order implicit RK schemes.
The first contribution is a stable high-order HDG formulation coupled with DIRK schemes for slightly compressible one-phase flow problem. We obtain an analytical expression for the stabilization parameter using the Engquist-Osher monotone flux scheme. The selection of the stabilization parameter is crucial to ensure the stability and to obtain the high-order properties of the method. We introduce the stabilization parameter in the Newton’s solver since we analytically compute its derivatives.
The second contribution is a high-order HDG formulation coupled with DIRK schemes for immiscible and incompressible two-phase flow problem. We set the water pressure and oil saturation as the main unknowns, which leads to a coupled system of two non-linear PDEs. To solve the resulting non-linear problem, we use a fix-point iterative method that alternatively solves the saturation and the pressure unknowns implicitly at each RK stage until convergence is achieved. The proposed fix-point method is memory-efficient because the saturation and the pressure are not solved at the same time.
The third contribution is a discretization scheme for the two-phase flow problem with the same spatial and temporal order of convergence. High-order spatial discretization combined with low-order temporal discretizations may lead to arbitrary small time steps to obtain a low enough temporal error. Moreover, high-order stable DIRK schemes need a high number of stages above fourth-order. Thus, the computational cost can be severely hampered because a non-linear problem has to be solved at each RK stage. Thus, we couple the HDG formulation with high-order fully implicit RK schemes. These schemes can be unconditionally stable and achieve high-order temporal accuracy with few stages. Therefore, arbitrary large time steps can be used without hampering the temporal accuracy. We rewrite the non-linear system to reduce the memory footprint. Thus, we achieve a better sparsity pattern of the Jacobian matrix and less coupling between stages. Furthermore, we have adapted the previous fix-point iterative method. We first compute the saturation at all the stages by solving a single non-linear system using the Newton-Raphson method. Next, we solve the pressure equation sequentially at each RK stage, since it does not couple the unknowns at different stages.
The last contribution is an efficient shock-capturing method for the immiscible and incompressible two-phase flow problem to reduce the spurious oscillations that may appear in the high-order approximations of the saturation. We introduce local artificial viscosity only in the saturation equation since only the saturation variable is non-smooth. To this end, we propose a shock sensor computed from the saturation and the post-processed saturation of the HDG method. This shock sensor is computationally efficient since the post-processed saturation is computed in an element-wise manner. Our methodology allows tracking the sharp fronts as they evolve since the shock sensor is computed at all RK stages.
