Unconditional Energy Stability Research Articles

A unified structure-preserving parametric finite element method for anisotropic surface diffusion

We propose and analyze a unified structure-preserving parametric finite element method (SP-PFEM) for the anisotropic surface diffusion of curves in two dimensions ( d = 2 ) (d=2) and surfaces in three dimensions ( d = 3 ) (d=3) with an arbitrary anisotropic surface energy density γ ( n ) \gamma (\boldsymbol {n}) , where n ∈ S d − 1 \boldsymbol {n}\in \mathbb {S}^{d-1} represents the outward unit vector. By introducing a novel unified surface energy matrix G k ( n ) \boldsymbol {G}_k(\boldsymbol {n}) depending on γ ( n ) \gamma (\boldsymbol {n}) , the Cahn-Hoffman ξ \boldsymbol {\xi } -vector and a stabilizing function k ( n ) : S d − 1 → R k(\boldsymbol {n}):\ \mathbb {S}^{d-1}\to {\mathbb R} , we obtain a unified and conservative variational formulation for the anisotropic surface diffusion via different surface differential operators, including the surface gradient operator, the surface divergence operator and the surface Laplace-Beltrami operator. A SP-PFEM discretization is presented for the variational problem. In order to establish the unconditional energy stability of the proposed SP-PFEM under a very mild condition on γ ( n ) \gamma (\boldsymbol {n}) , we propose a new framework via local energy estimate for proving energy stability/structure-preserving properties of the parametric finite element method for the anisotropic surface diffusion. This framework sheds light on how to prove unconditional energy stability of other numerical methods for geometric partial differential equations. Extensive numerical results are reported to demonstrate the efficiency and accuracy as well as structure-preserving properties of the proposed SP-PFEM for the anisotropic surface diffusion with arbitrary anisotropic surface energy density γ ( n ) \gamma (\boldsymbol {n}) arising from different applications.

Two linear energy stable lumped mass finite element schemes for the viscous Cahn–Hilliard equation on curved surfaces in 3D

The evolution of a dynamic system on complex curved 3D surfaces is essential for the understanding of natural phenomena, the development of new materials, and engineering design optimization. In this work, we study the viscous Cahn–Hilliard equation on curved surfaces and develop two linear energy stable finite element schemes based on the lumped mass method. Two stabilizing terms are added to ensure both the unique solvability and unconditional energy stability. We prove rigorously that two schemes are unconditionally energy stable . Numerical experiments are presented to verify theoretical results and to show the robustness and accuracy of the proposed method.

Efficient numerical scheme with full decoupling, second-order temporal accuracy and unconditional energy stability for a flow-coupled melt-convective dendritic solidification phase-field model

Efficient numerical scheme with full decoupling, second-order temporal accuracy and unconditional energy stability for a flow-coupled melt-convective dendritic solidification phase-field model

On the phase-field algorithm for distinguishing connected regions in digital model

In this paper, we propose a novel model for the discrimination of complex three-dimensional connected regions. The modified model is grounded on the Allen–Cahn equation. The modified equation not only maintains the original interface dynamics, but also avoids the unbounded diffusion behavior of the original Allen–Cahn equation. This advantage enables us to accurately populate and extract the complex connectivity region of the target part. The model is discretized employing a semi-implicit Crank–Nicolson scheme, ensuring second-order accuracy in both time and space. This paper provides a rigorous proof of the unconditional energy stability of our method, thereby affirming the numerical stability and the physical rationality of the solution. We validate the discriminative ability of the proposed model for 3D complex connected regions.

Unconditionally energy stable high-order BDF schemes for the molecular beam epitaxial model without slope selection

In this paper, we consider a class of k-order (3≤k≤5) backward differentiation formulas (BDF-k) for the molecular beam epitaxial (MBE) model without slope selection. Convex splitting technique along with k-th order Douglas-Dupont regularization term τnk(−Δ)kD_kϕn (D_k represents a truncated BDF-k formula) is added to the numerical schemes to ensure unconditional energy stability. The stabilized convex splitting BDF-k (3≤k≤5) methods are unique solvable unconditionally. Then the modified discrete energy dissipation laws are established by using the discrete gradient structures of BDF-k (3≤k≤5) formulas and processing k-th order explicit extrapolations of the concave term. In addition, based on the discrete energy technique, the L2 norm stability and convergence of the stabilized BDF-k (3≤k≤5) schemes are obtained by means of the discrete orthogonal convolution kernels and the convolution type Young inequalities. Numerical results are carried out to verify our theory and illustrate the validity of the proposed schemes.

The high-order exponential semi-implicit scalar auxiliary variable approach for the general nonlocal Cahn-Hilliard equation

The nonlocal Cahn-Hilliard equation with nonlocal diffusion operator is more suitable for the simulation of microstructure phase transition than the local Cahn-Hilliard equation. In this paper, based on the exponential semi-implicit scalar auxiliary variable method, the highly efficient and accurate schemes (in time) with unconditional energy stability for solving the nonlocal Cahn-Hilliard equation are proposed. On the one hand, we have demonstrated the unconditional energy stability and mass conservation for the nonlocal Cahn-Hilliard equation with its high-order numerical schemes in the continuous and discrete level carefully and rigorously. On the other hand, in order to reduce the calculation and storage cost in numerical simulation, we use the fast solver based on fast Fourier transform and conjugate gradient approach for spatial discretization. Some numerical simulations involving the Gaussian kernel are presented and show the stability, accuracy, efficiency and unconditional energy stability of the proposed schemes.

Efficiently linear and unconditionally energy-stable time-marching schemes with energy relaxation for the phase-field surfactant model

In this article, we consider consistent, efficient, and accurate numerical approximations for a recently developed Cahn–Hilliard type phase-field surfactant model enjoying the bounded-from-below condition of total energy. The phase-field surfactant model naturally satisfies the energy dissipation property in time. Based on this feature, we construct linearly decoupled and unconditionally energy-stable time-marching schemes based on an efficient variant of scalar auxiliary variable approach. To enhance the consistency of discrete modified energy and discrete original energy, a practical energy relaxation technique is adopted. We analytically proved the unique solvability and unconditional energy stability of the proposed schemes. In each time step, only two linear elliptic type equations with constants coefficients need to be solved and several algebraic operations need to be performed. Therefore, the proposed schemes are highly efficient and easy to implement. The space is discretized by the finite difference method and the linear multigrid algorithm is used to accelerate the convergence. The numerical results indicate that the proposed method not only has desired accuracy in time but also satisfies the energy dissipation law even if larger time steps are used. Furthermore, the surfactant laden spinodal decompositions and surfactant absorptions can be well simulated.

Constraint-preserved numerical schemes with decoupling structure for the Ericksen–Leslie model with variable density

In this paper, first- and second-order numerical schemes are developed for the Ericksen–Leslie model with variable density. The influence of variable density leads to some conflicts between decoupling and unconditional energy stability. We overcome it by special discretization and extra-fractional-step method. The fully decoupled structures are obtained by using the scalar auxiliary variable (SAV) method with only one SAV. This allows only one additional ODE to be solved so that the schemes are computationally cheaper. The non-convex constraint |b|=1 is preserved by rewriting the orientation field with polar coordinates. It also improves computational efficiency because the vector function of the orientation field is replaced by scalar function. We first reformulate the Ericksen–Leslie model as an equivalent new form by introducing a SAV and polar coordinate form of the orientation field. Secondly, two linear schemes and the corresponding unconditional energy stabilities are established. Then, we show the detailed implementations for proving that the schemes are fully decoupled and uniquely solvable. Finally, the convergence rates and energy dissipations are tested by performing some numerical experiments. The evolutionary simulations are also given.

A linear second-order convex splitting scheme for the modified phase-field crystal equation with a strong nonlinear vacancy potential

The modified phase-field crystal (MPFC) equation was extended to the MPFC equation with a strong nonlinear vacancy potential (VMPFC) to include vacancies. In this paper, we develop a linear, second-order, and unconditionally energy stable scheme for solving the VMPFC equation by proposing a new linear convex splitting, using the Crank–Nicolson formula to the contractive part and the second-order Adams–Bashforth formula to the expansive part, and adding a second-order Douglas–Dupont regularization. The unconditional energy stability of the developed scheme is proved. By solving the VMPFC equation, vacancies which exist in real materials can be observed in numerical simulations.

Efficient second-order accurate scheme for fluid–surfactant systems on curved surfaces with unconditional energy stability

Accurately simulating the interplay between fluids and surfactants is a challenge, especially when ensuring both mass conservation and guaranteed energy stability. This study proposes a highly accurate numerical scheme for the water–oil–surfactant system coupled with the Navier–Stokes equation. We use the second-order accurate discrete operators on triangular grids representing these surfaces. We use the Crank–Nicolson method to achieve the second-order temporal accuracy. To handle high-order nonlinearities, we employ the consistency-enhanced Scalar Auxiliary Variable method. Additionally, the projection method tackles the Navier–Stokes equations. These techniques guarantee the unconditional stability. Consequently, larger time steps are permissible. Our method achieves the high accuracy with lower computational cost. This efficiency stems from solely utilizing surface information, eliminating the need for calculations across the entire 3D space. The proposed scheme’s accuracy, stability, and robustness can be confirmed through various numerical experiments.

A Novel Fully Decoupled Scheme for the MHD System with Variable Density

Abstract In this paper, we first establish a novel first-order, fully decoupled, unconditionally stable time discretization scheme for the MHD system with variable density. This scheme successfully decouples all the coupling terms by combining the gauge-Uzawa method and the scalar auxiliary variable (SAV) method. And we prove its unconditional energy stability. Then we give the first-order finite element scheme and its implementation. Furthermore, we perform a rigorous error analysis of the proposed numerical scheme. Finally, we perform some numerical experiments to demonstrate the effectiveness of the decoupling scheme.

Highly efficient, robust and unconditionally energy stable second order schemes for approximating the Cahn-Hilliard-Brinkman system

In this paper, we present an efficient, robust, and unconditionally energy stable second-order scheme for solving the Cahn-Hilliard-Brinkman (CHB) model, which mathematically describes multiphase flow in porous media. Solving the CHB model is significantly challenging due to its high coupling and nonlinearity. Here, we utilize the scalar auxiliary variable (SAV) method to handle the nonlinear term in the Cahn-Hilliard system, followed by decoupling the model through the implicit-explicit (IMEX) Crank-Nicolson approach. Our constructed scheme employs an adaptive time step, enhancing the robustness of the scheme compared to a fixed time step. The unconditional energy stability of our scheme that integrates artificial stabilized term, is assured easily owing to the SAV method and is theoretically proven in the paper. Finally, numerical experiments employing the Fourier pseudo-spectral method to discrete the spatial variables were conducted in two-dimensional space to validate the numerical accuracy and efficiency of the proposed approach.

The subdivision-based IGA-EIEQ numerical scheme for the Navier–Stokes equations coupled with Cahn–Hilliard phase-field model of two-phase incompressible flow on complex curved surfaces

We develop an accurate and robust numerical scheme for solving the incompressible hydrodynamically coupled Cahn–Hilliard system of the two-phase fluid flow system on complex surfaces. Our algorithm leverages a number of efficient techniques, including the subdivision-based isogeometric analysis (IGA) method for spatial discretization, the explicit Invariant Energy Quadratization (EIEQ) method for linearizing nonlinear potentials, the Zero-Energy-Contribution (ZEC) method for decoupling, and the projection method for the Navier–Stokes equation to facilitate fully decoupled type implementations. The integration of these methodologies results in a fully discrete scheme with desired properties such as linearity, second-order temporal accuracy, full decoupling, and unconditional energy stability. The implementation of the scheme is straightforward, requiring the solution of a few elliptic equations with constant coefficients at each time step. The rigorous stability proof of unconditional energy stability and the implementation procedure are given in detail. Numerous numerical simulations on complex curved surfaces are carried out to verify the effectiveness of the proposed numerical scheme.

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.

A linear, second-order accurate, positivity-preserving and unconditionally energy stable scheme for the Navier–Stokes–Poisson–Nernst–Planck system

Electrokinetic phenomena involving the transport of ions in fluids generally appears in biological system, hydrodynamics and geodynamics. The underlining mathematical models involve the nonlinear coupling between the fluid motion and ion transportation. Performing efficient, accurate and structure preserving computation for these electrohydrodynamic models is challenging due to their multiphysics and complex nature. The Navier–Stokes–Poisson–Nernst–Planck (NSPNP) system is a widely used electrokinetic model in the existing literature. In this paper, a linear, second-order accurate, positivity-preserving and unconditionally energy stable scheme is developed for the NSPNP system. First, a square root transformation is employed to ensure the positivity of ion concentrations. Next, a specific ordinary differential equation is introduced to deal with the nonlinear coupling terms satisfying the “zero-energy-contribution” feature, which is combined with the scalar auxiliary variable method for nonlinear potentials to achieve a decoupled energy stable scheme. Therefore, only several linearly independent equations need to be solved at each time step. The staggered grid finite difference method is used for the spatial discretizations, and the unconditional energy stability of the fully discrete scheme is established. Finally, the unique solvability of the scheme is proved. Numerical experiments are carried out to demonstrate the accuracy and performance of the proposed scheme, and the effect of electric Coulomb force on the flow and motion of ions are also illustrated.

A fully-decoupled energy stable scheme for the phase-field model of non-Newtonian two-phase flows

<abstract><p>In this paper, we first propose a novel fully-decoupled, linear and second-order time accurate scheme to solve the phase-field model of non-Newtonian two-phase flows; the developed scheme is based on a stabilized Scalar Auxiliary Variable (SAV) approach. We strictly prove the unconditional energy stability of the scheme and conduct a numerical simulation to show the accuracy and stability of the proposed scheme. Moreover, we can observe that the parameter $ r $ in non-Newtonian fluids can affect spatial patterns during phase transitions, which directly enables us to design and perform optimal control experiments in engineering processes.</p></abstract>

The subdivision-based IGA-EIEQ numerical scheme for the Cahn–Hilliard–Darcy system of two-phase Hele-Shaw flow on complex curved surfaces

This paper investigates the numerical approximation of a two-phase Hele-Shaw fluid flow system, which consists of a phase-field Cahn–Hilliard model coupled with the Darcy flow, confined on a complex curved surface. To avoid geometric errors caused by the surface approximation and additional approximation errors introduced by numerical methods, for spatial discretization, we consider the recently developed strategy of subdivision-based isogeometric analysis (IGA), where we can accurately represent complex surfaces of arbitrary topology due to its high smoothness and hierarchical refinement properties. By incorporating multiple methods of temporal discretizations, including the Explicit-Invariant-Energy-Quadratization (EIEQ) approach for linearizing nonlinear potentials, the Zero-Energy-Contribution (ZEC) method for decoupling of nonlinear couplings, and the projection method for decoupling of linear couplings between the fluid velocity and pressure, and the finite-element method based on IGA for spatial discretizations, we arrive at a fully discrete scheme that possesses desirable properties such as decoupling, linearity, second-order accuracy in time, and unconditional energy stability. At each time step, only several elliptic equations with constant coefficients are needed to be solved. We also present the rigorous proof of unconditional energy stability, accompanied by numerous numerical examples to verify the stability and accuracy of the numerical scheme, such as the Saffman–Taylor fingering instability occurring on various complex curved surfaces induced by rotational forces.

A fully decoupled linearized and second‐order accurate numerical scheme for two‐phase magnetohydrodynamic flows

SummaryIn this paper, we analyze the numerical approximation of two‐phase magnetohydrodynamic flows. Firstly, an equivalent new system is designed by introducing two scalar auxiliary variables. One of variables is used to linearize the phase field function and the other is used to deal with the highly coupled and nonlinear terms. Secondly, by combining with a novel decoupling technique based on the “zero‐energy‐contribution” feature and the pressure correction method, the linearized second order BDF numerical scheme, which has the advantage of fully decoupled structure, is constructed. Furthermore, we strictly prove the unconditional energy stability and error analysis of the scheme, and give a detailed implementation procedure that only requires to calculate several linear elliptic equations with constant coefficients. Finally, the results of numerical simulations are presented to validate the rates of convergence and energy stability.

A splitting discontinuous Galerkin projection method for the magneto-hydrodynamic equations

This paper is focused on the numerical approximations of the incompressible magneto-hydrodynamic equations. Our interest is to develop a novel decoupled, linear and unconditional energy stable fully-discrete scheme, which is achieved by the interior penalty discontinuous Galerkin (DG) method for spatial discretization, the stabilizing strategy and implicit-explicit (IMEX) scheme used to handle the nonlinear coupling terms, and a rotational pressure-correction method for the Navier-Stokes equations. We prove the unique solvability, unconditional energy stability and optimal error estimates of the proposed scheme rigorously. We further present several numerical examples to demonstrate the accuracy, stability, and efficiency of the proposed scheme numerically.

An unconditionally energy-stable and orthonormality-preserving iterative scheme for the Kohn-Sham gradient flow based model

We propose an unconditionally energy-stable, orthonormality-preserving, component-wise splitting iterative scheme for the Kohn-Sham gradient flow based model in the electronic structure calculation. We first study the scheme discretized in time but still continuous in space. The component-wise splitting iterative scheme changes one wave function at a time, similar to the Gauss-Seidel iteration for solving a linear equation system. At the time step n, the orthogonality of the wave function being updated to other wave functions is preserved by projecting the gradient of the Kohn-Sham energy onto the subspace orthogonal to all other wave functions known at the current time, while the normalization of this wave function is preserved by projecting the gradient of the Kohn-Sham energy onto the subspace orthogonal to this wave function at tn+1/2. The unconditional energy stability is nontrivial, and it comes from a subtle treatment of the two-electron integral as well as a consistent treatment of the two projections. Rigorous mathematical derivations are presented to show our proposed scheme indeed satisfies the desired properties. We then study the fully-discretized scheme, where the space is further approximated by a conforming finite element subspace. For the fully-discretized scheme, not only the preservation of orthogonality and normalization (together we called orthonormalization) can be quickly shown using the same idea as for the semi-discretized scheme, but also the highlight property of the scheme, i.e., the unconditional energy stability can be rigorously proven. The scheme allows us to use large time step sizes and deal with small systems involving only a single wave function during each iteration step. Several numerical experiments are performed to verify the theoretical analysis, where the number of iterations is indeed greatly reduced as compared to similar examples solved by the Kohn-Sham gradient flow based model in the literature.