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1. DECOUPLED, LINEAR, AND ENERGY STABLE FINITE ELEMENT METHOD FOR THE CAHN–HILLIARD–NAVIER–STOKES–DARCY PHASE FIELD MODEL

   Gao, Y; He, X; Mei, L; Yang, X. (January 2018, SIAM journal on scientific computing)

   In this paper, we consider the numerical approximation for a phase field model of the coupled two-phase free flow and two-phase porous media flow. This model consists of Cahn– Hilliard–Navier–Stokes equations in the free flow region and Cahn–Hilliard–Darcy equations in the porous media region that are coupled by seven interface conditions. The coupled system is decoupled based on the interface conditions and the solution values on the interface from the previous time step. A fully discretized scheme with finite elements for the spatial discretization is developed to solve the decoupled system. In order to deal with the difficulties arising from the interface conditions, the decoupled scheme needs to be constructed appropriately for the interface terms, and a modified discrete energy is introduced with an interface component. Furthermore, the scheme is linearized and energy stable. Hence, at each time step one need only solve a linear elliptic system for each of the two decoupled equations. Stability of the model and the proposed method is rigorously proved. Numerical experiments are presented to illustrate the features of the proposed numerical method and verify the theoretical conclusions. 

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2. Fully decoupled energy-stable numerical schemes for two-phase coupled porous media and free flow with different densities and viscosities

   https://doi.org/10.1051/m2an/2023012  

   Gao, Yali; He, Xiaoming; Lin, Tao; Lin, Yanping (May 2023, ESAIM: Mathematical Modelling and Numerical Analysis)

   In this article, we consider a phase field model with different densities and viscosities for the coupled two-phase porous media flow and two-phase free flow, as well as the corresponding numerical simulation. This model consists of three parts: a Cahn–Hilliard–Darcy system with different densities/viscosities describing the porous media flow in matrix, a Cahn–Hilliard–Navier–Stokes system with different densities/viscosities describing the free fluid in conduit, and seven interface conditions coupling the flows in the matrix and the conduit. Based on the separate Cahn–Hilliard equations in the porous media region and the free flow region, a weak formulation is proposed to incorporate the two-phase systems of the two regions and the seven interface conditions between them, and the corresponding energy law is proved for the model. A fully decoupled numerical scheme, including the novel decoupling of the Cahn–Hilliard equations through the four phase interface conditions, is developed to solve this coupled nonlinear phase field model. An energy-law preservation is analyzed for the temporal semi-discretization scheme. Furthermore, a fully discretized Galerkin finite element method is proposed. Six numerical examples are provided to demonstrate the accuracy, discrete energy law, and applicability of the proposed fully decoupled scheme. 

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3. Improved Interval Reachability Bounds for Nonlinear Discrete-Time Systems using an Efficient One-Dimensional Partitioning Method

   https://doi.org/10.23919/ACC50511.2021.9483003  

   Mu, Bowen; Yang, Xuejiao; Shen, Kai; Scott, Joseph K. (May 2021, 2021 American Control Conference (ACC))

   This article presents a new method for accurately enclosing the reachable sets of nonlinear discrete-time systems with unknown but bounded disturbances. This method is motivated by the discrete-time differential inequalities method (DTDI) proposed by Yang and Scott, which exhibits state-of-the-art accuracy at low cost for many problems, but suffers from theoretical limitations that significantly restrict its applicability. The proposed method uses an efficient one-dimensional partitioning scheme to approximate DTDI while avoiding the key technical assumptions that limit it. Numerical result shows that this approach matches the accuracy of DTDI when DTDI is applicable, but, unlike DTDI, is valid for arbitrary systems. 

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4. Fully discrete error analysis of first‐order low regularity integrators for the Allen‐Cahn equation

   https://doi.org/10.1002/num.23017  

   Doan, Cao‐Kha; Hoang, Thi‐Thao‐Phuong; Ju, Lili (September 2023, Numerical Methods for Partial Differential Equations)

   Abstract The Allen‐Cahn equation satisfies the maximum bound principle, that is, its solution is uniformly bounded for all time by a positive constant under appropriate initial and/or boundary conditions. It has been shown recently that the time‐discrete solutions produced by low regularity integrators (LRIs) are likewise bounded in the infinity norm; however, the corresponding fully discrete error analysis is still lacking. This work is concerned with convergence analysis of the fully discrete numerical solutions to the Allen‐Cahn equation obtained based on two first‐order LRIs in time and the central finite difference method in space. By utilizing some fundamental properties of the fully discrete system and the Duhamel's principle, we prove optimal error estimates of the numerical solutions in time and space while the exact solution is only assumed to be continuous in time. Numerical results are presented to confirm such error estimates and show that the solution obtained by the proposed LRI schemes is more accurate than the classical exponential time differencing (ETD) scheme of the same order. 

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5. A linear second-order maximum bound principle-preserving BDF scheme for the Allen-Cahn equation with a general mobility

   https://doi.org/10.1090/mcom/3843  

   Hou, Dianming; Ju, Lili; Qiao, Zhonghua (November 2023, Mathematics of Computation)

   In this paper, we propose and analyze a linear second-order numerical method for solving the Allen-Cahn equation with a general mobility. The proposed fully-discrete scheme is carefully constructed based on the combination of first and second-order backward differentiation formulas with nonuniform time steps for temporal approximation and the central finite difference for spatial discretization. The discrete maximum bound principle is proved of the proposed scheme by using the kernel recombination technique under certain mild constraints on the time steps and the ratios of adjacent time step sizes. Furthermore, we rigorously derive the discrete $H^1$error estimate and energy stability for the classic constant mobility case and the $L^{\mathrm{\infty <\#comment/>}}$error estimate for the general mobility case. Various numerical experiments are also presented to validate the theoretical results and demonstrate the performance of the proposed method with a time adaptive strategy. 

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