A Positivity-Preserving Second-Order BDF Scheme for the Cahn-Hilliard Equation with Variable Interfacial Parameters

We present and analyze a new second-order finite difference scheme for the Macromolecular Microsphere Composite hydrogel, Time-Dependent Ginzburg-Landau (MMC-TDGL) equation, a Cahn-Hilliard equation with Flory-Huggins-deGennes energy potential. This numerical scheme with unconditional energy stability is based on the Backward Differentiation Formula (BDF) method in time derivation combining with Douglas-Dupont regularization term. In addition, we present a pointwise bound of the numerical solution for the proposed scheme in the theoretical level. For the convergent analysis, we treat three nonlinear logarithmic terms as a whole and deal with all logarithmic terms directly by using the property that the nonlinear error inner product is always non-negative. Moreover, we present the detailed convergent analysis in $ℓ^∞$(0,$T$;$H_h^{-1}$)∩$ℓ^2$(0,$T$;$H_h^1$) norm. At last, we use the local Newton approximation and multigrid method to solve the nonlinear numerical scheme, and various numerical results are presented, including the numerical convergence test, positivity-preserving property test, spinodal decomposition, energy dissipation and mass conservation properties.     

A Third Order BDF Energy Stable Linear Scheme for the No-Slope-Selection Thin Film Model

In this paper we propose and analyze a (temporally) third order accuratebackward differentiation formula (BDF) numerical scheme for the no-slope-selection(NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectraldiscretization in space. The surface diffusion term is treated implicitly, while the nonlinear chemical potential is approximated by a third order explicit extrapolation formula for the sake of solvability. In addition, a third order accurate Douglas-Dupontregularization term, in the form of $−A∆t^2∆^2_N(u^{n+1}−u^n)$, is added in the numericalscheme. A careful energy stability estimate, combined with Fourier eigenvalue analysis, results in the energy stability in a modified version, and a theoretical justification of the coefficient $A$ becomes available. As a result of this energy stability analysis, a uniform in time bound of the numerical energy is obtained. And also, theoptimal rate convergence analysis and error estimate are derived in details, in the $ℓ^∞(0,T;ℓ^2)∩ℓ^2(0,T;H^2_h)$ norm, with the help of a linearized estimate for the nonlinear error terms. Some numerical simulation results are presented to demonstrate theefficiency of the numerical scheme and the third order convergence. The long timesimulation results for $ε = 0.02$ (up to $T = 3×10^5$) have indicated a logarithm law forthe energy decay, as well as the power laws for growth of the surface roughness andthe mound width. In particular, the power index for the surface roughness and themound width growth, created by the third order numerical scheme, is more accuratethan those produced by certain second order energy stable schemes in the existingliterature.     

Decoupled,Energy Stable Numerical Scheme for the Cahn-Hilliard-Hele-Shaw System with Logarithmic Flory-Huggins Potential

In this paper, a decoupling numerical method for solving Cahn-Hilliard-Hele-Shaw system with logarithmic potential is proposed. Combing with a convex-splitting of the energy functional, the discretization of the Cahn-Hilliard equation intime is presented. The nonlinear term in Cahn-Hilliard equation is decoupled fromthe pressure gradient by using a fractional step method. Therefore, to update the pressure, we just need to solve a Possion equation at each time step by using an incrementalpressure-correction technique for the pressure gradient in Darcy equation. For logarithmic potential, we use the regularization procedure, which make the domain forthe regularized functional $F$($ф$) is extended from (−1,1) to (−∞,∞). Further, the stability and the error estimate of the proposed method are proved. Finally, a series ofnumerical experiments are implemented to illustrate the theoretical analysis.     

A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations

In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [11] for convection-diffusion equations, which relies on a special kernel-based formulation of the solutions and successive convolution. However, disadvantages appear when we extend the previous method to our equations, such as inefficient choice of parameters and unprovable stability for high-dimensional problems. To overcome these difficulties, a new kernel-based formulation is designed to approach the spatial derivatives. It maintains the good properties of the original one, including the high order accuracy and unconditionally stable for one-dimensional problems, hence allowing much larger time step evolution compared with other explicit schemes. In addition, without extra computational cost, the proposed scheme can enlarge the available interval of the special parameter in the formulation, leading to less errors and higher efficiency. Moreover, theoretical investigations indicate that it is unconditionally stable for multi-dimensional problems as well. We present numerical tests for one- and two-dimensional scalar and system, demonstrating the designed high order accuracy and unconditionally stable property of the scheme.     

Construction of a Minimum Energy Path for the VT Flash Model by the String Method Coupled with the Exponential Time Differencing Scheme

Flash calculation plays significant roles in petroleum and chemical industries. Since Michelsen proposed his milestone studies in 1982, through several decadesof development, the current research interests on flash calculation have been shiftedfrom accuracy to efficiency, but the ultimate goal remains the same; that is accuratedetermination of equilibrium phase amounts and compositions at a given condition.On the other hand, finding the transition route and its related saddle point is oftencrucial to understand the whole energy landscape of flash models, which would provide new insights for designing numerical algorithms or optimizing existing ones. Inthis study, an efficient numerical approach is developed by coupling the string methodwith the exponential time differencing (ETD) scheme to investigate the minimum energy paths and first-order saddle points of VT flash models with Peng-Robinson equation of state. As a promising alternative to the conventional approach, VT flash calculates phase equilibria under a new variable specification of volume and temperature. The Rosenbrock-type ETD scheme is used to reduce the computational difficulty caused by the high stiffness of the model systems. The proposed ETD-Stringmethod successfully calculates the minimum energy paths of single-component andtwo-component VT flash models with strong stiffness. Numerical results also showgood feasibility and accuracy in calculation of equilibrium phase amounts and compositions.