Nonlinear Multigrid Implementation for the Two-Dimensional Cahn–Hilliard Equation

We present a nonlinear multigrid implementation for the two-dimensional Cahn–Hilliard (CH) equation and conduct detailed numerical tests to explore the performance of the multigrid method for the CH equation. The CH equation was originally developed by Cahn and Hilliard to model phase separation phenomena. The CH equation has been used to model many interface-related problems, such as the spinodal decomposition of a binary alloy mixture, inpainting of binary images, microphase separation of diblock copolymers, microstructures with elastic inhomogeneity, two-phase binary fluids, in silico tumor growth simulation and structural topology optimization. The CH equation is discretized by using Eyre’s unconditionally gradient stable scheme. The system of discrete equations is solved using an iterative method such as a nonlinear multigrid approach, which is one of the most efficient iterative methods for solving partial differential equations. Characteristic numerical experiments are conducted to demonstrate the efficiency and accuracy of the multigrid method for the CH equation. In the Appendix, we provide C code for implementing the nonlinear multigrid method for the two-dimensional CH equation.

Estimation of a Permeability Field within the Two-Phase Porous Media Flow Using Nonlinear Multigrid Method

Estimation of spatially varying permeability within the two-phase porous media flow plays an important role in reservoir simulation. Usually, one needs to estimate a large number of permeability values from a limited number of observations, so the computational cost is very high even for a single field-model. This paper applies a nonlinear multigrid method to estimate the permeability field within the two-phase porous media flow. Numerical examples are provided to illustrate the feasibility and effectiveness of the proposed estimation method. In comparison with other existing methods, the most outstanding advantage of this method is the computational efficiency, computational accuracy, and antinoise ability. The proposed method has a potential applicability to a variety of parameter estimation problems.

Nonlinear Multigrid for Reservoir Simulation

Summary A feasibility study is presented on the effectiveness of applying nonlinear multigrid methods for efficient reservoir simulation of subsurface flow in porous media. A conventional strategy modeled after global linearization by means of Newton’s method is compared with an alternative strategy modeled after local linearization, leading to a nonlinear multigrid method in the form of the full-approximation scheme (FAS). It is demonstrated through numerical experiments that, without loss of robustness, the FAS method can outperform the conventional techniques in terms of algorithmic and numerical efficiency for a black-oil model. Furthermore, the use of the FAS method enables a significant reduction in memory usage compared with conventional techniques, which suggests new possibilities for improved large-scale reservoir simulation and numerical efficiency. Last, nonlinear multilevel preconditioning in the form of a hybrid-FAS/Newton strategy is demonstrated to increase robustness and efficiency.