Convex Splitting Method Research Articles

The Second-Order Numerical Approximation for a Modified Ericksen–Leslie Model

In this study, two numerical schemes with second-order accuracy in time for a modified Ericksen–Leslie model are constructed. The highlight is based on a novel convex splitting method for dealing with the nonlinear potentials, which is integrated with the second-order backward differentiation formula (BDF2) and leap frog method for temporal discretization and the finite element method for spatial discretization. The unconditional energy stability of both schemes is further demonstrated. Finally, several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed schemes.

Unconditionally energy-stable linear convex splitting algorithm for the L2 quasicrystals

Quasicrystals have extensive applications in material sciences. In this article, we develop an unconditional energy-dissipation-preserving, temporally second-order accurate, and linear method for solving a L2-gradient flow-based quasicrystal model. A truncated technique is introduced to regularize the nonlinear terms so that their second derivative is upper bounded. Based on the idea of the convex splitting method, we split free energy into the difference between two convex functionals. The convexity provides an appropriate choice of linear stabilization parameter. A linear and temporally second-order accurate algorithm is constructed utilizing the implicit-explicit Runge–Kutta method. In each time step, we can analytically prove that the proposed scheme satisfies the energy dissipation law. The Fourier spectral method is used to perform spatial discretization. Through numerical experiments, the accuracy, energy stability, and capability of the proposed method for quasicrystals are verified.

A convex splitting method for the time-dependent Ginzburg-Landau equation

A convex splitting method for the time-dependent Ginzburg-Landau equation

Efficient and energy stable numerical schemes for the two-mode phase field crystal equation

In this paper, we propose four time marching schemes for the two-mode phase field crystal equation with the periodic boundary condition. The first- and second-order schemes are based on the stabilized semi-implicit approach with a combination of the backward Euler method, Crank–Nicolson method, and second-order backward differentiation formula, respectively. Based on the convex splitting method and third-order backward differentiation formula, a third-order scheme is developed. Different stabilized terms are added so that the energy stability of the proposed schemes can be guaranteed. Theoretically, the unique solvability and energy stability of the proposed schemes are rigorously proved. The error estimate of the first-order scheme is also given. Various numerical examples and simulations in the 2D and 3D cases are presented to demonstrate the accuracy, stability, and efficiency of the proposed schemes.

The error analysis for the Cahn-Hilliard phase field model of two-phase incompressible flows with variable density

<abstract><p>In this paper, we consider the numerical approximations of the Cahn-Hilliard phase field model for two-phase incompressible flows with variable density. First, a temporal semi-discrete numerical scheme is proposed by combining the fractional step method (for the momentum equation) and the convex-splitting method (for the free energy). Second, we prove that the scheme is unconditionally stable in energy. Then, the $ L^2 $ convergence rates for all variables are demonstrated through a series of rigorous error estimations. Finally, by applying the finite element method for spatial discretization, some numerical simulations were performed to demonstrate the convergence rates and energy dissipations.</p></abstract>

A simple and explicit numerical method for the phase-field model for diblock copolymer melts

In this paper, we present a simple and explicit finite difference method for the phase-field model for diblock copolymer melts. A diblock copolymer is a polymer consisting of two types of different monomers bonded covalently to each other to form a single copolymer chain. When the temperature is below the critical temperature, the copolymer melt exhibits microphase separation. The mathematical model is derived from a total free energy functional which contains kinetic, gradient, double well, and long-range nonlocal potentials. The Saul’yev-type scheme based on a linearly stabilized convex splitting method is used for the discretizations. The proposed method is simple and computationally efficient because the scheme is explicit and it does not require any iterative procedures. The proposed scheme not only overcomes the severe time step restriction for the explicit scheme but also works well for the simulations of lamellar and hex-cylinder structures which are characteristic morphologies for diblock copolymer melts after phase separation. Furthermore, the proposed method can be easily applied to the simulations in complex computational domains. We present various numerical tests to demonstrate the performance of the proposed scheme.

FAST AND ACCURATE PHASE EQUILIBRIUM CALCULATIONS FOR CONDENSATE SHALE GAS RESERVOIRS

Phase equilibrium calculations of complex fluids in shale gas reservoirs require the establishment of advanced numerical models that consider capillary effects, and the design of fast and reliable algorithms to handle the various components in the reservoir fluids in practical working conditions. In this study, we develop a thermodynamically consistent VT-type pore-scale flash calculation scheme based on realistic equations of state suitable for oil/gas reservoirs, e.g. the Peng-Robinson equation of state. The effect of capillarity has been incorporated in the scheme for a more accurate description of the thermodynamic properties of shale gas, and the diffuse interface model is applied to establish a dynamic evolution scheme in the phase equilibrium process, and a convex splitting method is used to model the evolution of compositional moles and volume. In order to accelerate the iterative flash calculations for realistic reservoir fluids containing a large number of components, a self-adaptive deep learning algorithm is developed in this paper with a novel structure to achieve wider applicability to various components in different fluids. The input and output features of the neural network are selected as the key thermodynamic features on the basis of thermodynamic analysis, and the network hyper-parameters have been carefully tuned to achieve a better performance on both accuracy and efficiency. Advanced deep learning technics resolving overfitting problems have been applied in our algorithm. The trained model significantly accelerates the conventional flash calculation based on iterative methods, while a good prediction accuracy has been preserved. Phase stability test and phase splitting calculations are automatically incorporated in our prediction, and we can significantly capture the effect of capillarity on phase equilibrium behaviors. Such a fast, accurate and reliable shale gas phase equilibrium calculation scheme using deep learning algorithms can provide an initial phase distribution field with physical meanings for subsequent multiphase flow simulations, while the number of phases can be also determined. The thermodynamic information and analysis can also be used as a thermodynamic basis for a multiphase numerical model with built-in physical conservation.

A non-iterative and unconditionally energy stable method for the Swift–Hohenberg equation with quadratic–cubic nonlinearity

Most implicit methods for the Swift–Hohenberg (SH) equation with quadratic–cubic nonlinearity require costly iterative solvers at each time step. In this paper, a non-iterative method for obtaining approximate solutions of the SH equation which is based on the convex splitting idea is presented. By regularizing the cubic–quartic function in the energy for the SH equation and adding an extra linear stabilizing term, we arrive at a non-iterative convex splitting method, where the operator involved is linear and positive and has constant coefficients. We further prove the unconditional energy stability of the method. Numerical examples illustrating the accuracy, efficiency, and energy stability of the proposed method are provided.

A novel Cahn–Hilliard–Navier–Stokes model with a nonstandard variable mobility for two-phase incompressible fluid flow

In this study, we present a novel Cahn–Hilliard–Navier–Stokes (CHNS) system with a nonstandard variable mobility for two-phase incompressible fluid flow. Unlike the classical constant mobility, the developed variable mobility has decreasing values nearby the interface and increasing values away from the interface, which minimizes the dynamics of the Cahn–Hilliard (CH) model nearby the interface. An unconditionally stable convex splitting method is used to solve the CH equation and the projection method is used to solve the NS equation. As benchmark tests, the Rayleigh–Taylor instability, drop deformation, and rising bubble are performed to show the accuracy and practicability of the proposed model. The computational results indicate that the proposed model accurately captures the interfacial position and keeps the interface region from being too much distorted.

A phase-field model and its efficient numerical method for two-phase flows on arbitrarily curved surfaces in 3D space

Herein, we present a phase-field model and its efficient numerical method for incompressible single and binary fluid flows on arbitrarily curved surfaces in a three-dimensional (3D) space. An incompressible single fluid flow is governed by the Navier–Stokes (NS) equation and the binary fluid flow is governed by the two-phase Navier–Stokes–Cahn–Hilliard (NSCH) system. In the proposed method, we use a narrow band domain to embed the arbitrarily curved surface and extend the NSCH system and apply a pseudo-Neumann boundary condition that enforces constancy of the dependent variables along the normal direction of the points on the surface. Therefore, we can use the standard discrete Laplace operator instead of the discrete Laplace–Beltrami operator. Within the narrow band domain, the Chorin’s projection method is applied to solve the NS equation, and a convex splitting method is employed to solve the Cahn–Hilliard equation with an advection term. To keep the velocity field tangential to the surface, a velocity correction procedure is applied. An effective mass correction step is adopted to preserve the phase concentration. Computational results such as convergence test, Kevin–Helmholtz instability, and Rayleigh–Taylor instability on curved surfaces demonstrate the accuracy and efficiency of the proposed method.

Porous Three-Dimensional Scaffold Generation for 3D Printing

In this paper, we present an efficient numerical method for arbitrary shaped porous structure generation for 3D printing. A phase-field model is employed for modeling phase separation phenomena of diblock copolymers based on the three-dimensional nonlocal Cahn–Hilliard (CH) equation. The nonlocal CH equation is a fourth-order nonlinear partial differential equation. To efficiently solve the governing equation, an unconditionally gradient stable convex splitting method for temporal discretization with a Fourier spectral method for the spatial discretization is adopted. The standard fast Fourier transform is used to speed up the computation. A new local average concentration function is introduced to the original nonlocal CH equation so that we can locally control the morphology of the structure. The proposed algorithm is simple to implement and complex shaped structures can also be implemented with corresponding signed distance fields. Various numerical tests are performed on simple and complex structures. The computational results demonstrate that the proposed method is efficient to generate irregular porous structures for 3D printing.

Numerical analysis for the Cahn-Hilliard-Hele-Shaw system with variable mobility and logarithmic Flory-Huggins potential

In this paper, a fully discrete scheme is proposed for solving the Cahn-Hilliard-Hele-Shaw system with a logarithmic potential and concentration dependent mobility. Using the regularization procedure, the domain for the logarithmic free energy density function F(ϕ) is extended from (−1,1) to (−∞,∞). A convex-splitting method of the energy is adopted in time and the mixed finite element method is used in space. Furthermore, we prove the stability of the proposed numerical method and establish the optimal error estimate in the energy norm. Finally, numerical results are presented to support our theoretical analysis.

A diffuse interface model and semi-implicit energy stable finite element method for two-phase magnetohydrodynamic flows

In this paper, we propose a diffuse interface model and finite element approximation for two-phase magnetohydrodynamic (MHD) flows with different viscosities and electric conductivities. An energy stable scheme, which is based on the finite element method for the spatial discretization and first order semi-implicit scheme combined with convex splitting method for the temporal discretization, is proposed to solve this new model. The numerical scheme is proved to be mass-conservative and energy law preserving. By Leray–Schauder fixed point theorem, the existence of solutions to the numerical scheme is shown. The uniqueness of the numerical solutions is obtained. Utilizing the stability of the numerical scheme and the compactness method, the existence of the weak solutions to the two-phase MHD model is established as well. Furthermore, given more regularity on the weak solution, the convergence of the numerical scheme is derived. Finally, numerical experiments are provided to verify the theoretical results and validate the proposed model.

A Second-order Energy Stable Fourier Spectral Method for the Fractional Cahn-Hilliard Equation

In this paper, we presented a second order accurate (in time) energy stable numerical scheme for the Fractional Cahn-Hilliard (CH) equation. Combining the convex splitting method, we applied the implicit backward differentiation formula (BDF2) to derive second order temporal accuracy, and we used the Fourier spectral method for space discrete to obtain the fully discretization scheme. Then we discussed the unique solvability and energy stability. A few numerical experiments were presented to conclude the article.

Convex splitting method for the calculation of transition states of energy functional

Among numerical methods for partial differential equations arising from steepest descent dynamics of energy functionals (e.g., Allen–Cahn and Cahn–Hilliard equations), the convex splitting method is well-known to maintain unconditional energy stability for a large time step size. In this work, we show how to use the convex splitting idea to find transition states, i.e., index-1 saddle points of the same energy functionals. Based on the iterative minimization formulation (IMF) for saddle points [14], we introduce the convex splitting method to minimize the auxiliary functional at each cycle of the IMF. We present a general principle of constructing convex splitting forms for these auxiliary functionals and show how to avoid solving nonlinear equations. The new numerical scheme based on the convex splitting method allows for large time step sizes. The new methods are tested for the one dimensional Ginzburg–Landau energy functional in the search of the Allen–Cahn or Cahn–Hilliard types of transition states. We provide the numerical results of transition states for the two dimensional Landau–Brazovskii energy functional for diblock copolymers.

Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations

We propose a semi-discrete scheme for 2D Keller-Segel equations based on a symmetrization reformation, which is equivalent to the convex splitting method and is free of any nonlinear solver. We show that, this new scheme is unconditionally stable as long as the initial condition does not exceed certain threshold, and it asymptotically preserves the quasi-static limit in the transient regime. Furthermore, we prove that the fully discrete scheme is conservative and positivity preserving, which makes it ideal for simulations. The analogical schemes for the radial symmetric cases and the subcritical degenerate cases are also presented and analyzed. With extensive numerical tests, we verify the claimed properties of the methods and demonstrate their superiority in various challenging applications.

Unconditionally stable methods for gradient flow using Convex Splitting Runge–Kutta scheme

We propose a Convex Splitting Runge–Kutta (CSRK) scheme which provides a simple unified framework to solve a gradient flow in an unconditionally gradient stable manner. The key feature of the scheme is a combination of a convex splitting method and a specially designed multi-stage two-additive Runge–Kutta method. Our methods are high order accurate in time and assure the gradient (energy) stability for any time step size. We provide detailed proof of the unconditional energy stability and present issues on the practical implementations. We demonstrate the accuracy and stability of the proposed methods using numerical experiments of the Cahn–Hilliard equation.

Numerical modeling of isothermal compositional grading by convex splitting methods

In this paper, an isothermal compositional grading process is simulated based on convex splitting methods with the Peng-Robinson equation of state. We first present a new form of gravity/chemical equilibrium condition by minimizing the total energy which consists of Helmholtz free energy and gravitational potential energy, and incorporating Lagrange multipliers for mass conservation. The time-independent equilibrium equations are transformed into a system of transient equations as our solution strategy. It is proved our time-marching scheme is unconditionally energy stable by the semi-implicit convex splitting method in which the convex part of Helmholtz free energy and its derivative are treated implicitly and the concave parts are treated explicitly. With relaxation factor controlling Newton iteration, our method is able to converge to a solution with satisfactory accuracy if a good initial estimate of mole compositions is provided. More importantly, it helps us automatically split the unstable single phase into two phases, determine the existence of gas-oil contact (GOC) and locate its position if GOC does exist. A number of numerical examples are presented to show the performance of our method.

First and second order numerical methods based on a new convex splitting for phase-field crystal equation

The phase-field crystal equation derived from the Swift–Hohenberg energy functional is a sixth order nonlinear equation. We propose numerical methods based on a new convex splitting for the phase-field crystal equation. The first order convex splitting method based on the proposed splitting is unconditionally gradient stable, which means that the discrete energy is non-increasing for any time step. The second order scheme is unconditionally weakly energy stable, which means that the discrete energy is bounded by its initial value for any time step. We prove mass conservation, unique solvability, energy stability, and the order of truncation error for the proposed methods. Numerical experiments are presented to show the accuracy and stability of the proposed splitting methods compared to the existing other splitting methods. Numerical tests indicate that the proposed convex splitting is a good choice for numerical methods of the phase-field crystal equation.

Local Discontinuous Galerkin Method and High Order Semi-Implicit Scheme for the Phase Field Crystal Equation

In this paper, we present a local discontinuous Galerkin (LDG) method and two unconditionally energy stable schemes for the phase field crystal (PFC) equation. The semidiscrete energy stability of the LDG method is proved first. The PFC equation is a sixth order nonlinear partial differential equation (PDE), which leads to the severe time step restriction ($\Delta t=\mathcal{O}(\Delta x^6)$) of explicit time discretization methods to maintain stability. Due to this, we introduce semi-implicit first order and second order time discretization methods which are based on the convex splitting principle of a discrete energy and prove the corresponding unconditional energy stabilities. To improve the temporal accuracy, the spectral deferred correction (SDC) method and a high order semi-implicit Runge--Kutta method combining with the first--order convex splitting method are adopted for the PFC equation with constant and degenerate mobility, respectively. The equations at the implicit time level are nonlinear and ...