Energy Stable Schemes Research Articles

Generalized SAV approaches for gradient systems

We propose in this paper three generalized auxiliary scalar variable (G-SAV) approaches for developing, efficient energy stable numerical schemes for gradient systems. The first two G-SAV approaches allow a range of functions in the definition of the SAV variable, furthermore, the second G-SAV approach only requires the total free energy to be bounded from below as opposed to the requirement that the nonlinear part of the free energy to be bounded from below. On the other hand, the third G-SAV approach is unconditionally energy stable with respect to the original free energy as opposed to a modified energy. Ample numerical results for various gradient systems are presented to validate the effectiveness and accuracy of the proposed G-SAV approaches.

A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance

In this paper, we propose and analyze a positivity-preserving, energy stable numerical scheme for a certain type of reaction-diffusion systems involving the Law of Mass Action with the detailed balance condition. The numerical scheme is constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of reaction trajectories. The fact that both the reaction and diffusion parts dissipate the same free energy opens a path of designing an energy stable, operator splitting scheme for these systems. At the reaction stage, we solve equations of reaction trajectories by treating all the logarithmic terms in the reformulated form implicitly due to their convex nature. The positivity-preserving property and unique solvability can be theoretically proved, based on the singular behavior of the logarithmic function around the limiting value. Moreover, the energy stability of this scheme at the reaction stage can be proved by a careful convexity analysis. Similar techniques are used to establish the positivity-preserving property and energy stability for the standard semi-implicit solver at the diffusion stage. As a result, a combination of these two stages leads to a positivity-preserving and energy stable numerical scheme for the original reaction-diffusion system. Several numerical examples are presented to demonstrate the robustness of the proposed operator splitting scheme.

An efficient second-order energy stable BDF scheme for the space fractional Cahn–Hilliard equation

The space fractional Cahn-Hilliard phase-field model is more adequate and accurate in the description of the formation and phase change mechanism than the classical Cahn-Hilliard model. In this article, we propose a temporal second-order energy stable scheme for the space fractional Cahn-Hilliard model. The scheme is based on the second-order backward differentiation formula in time and a finite difference method in space. Energy stability and convergence of the scheme are analyzed, and the optimal convergence orders in time and space are illustrated numerically. Note that the coefficient matrix of the scheme is a $2 \times 2$ block matrix with a Toeplitz-like structure in each block. Combining the advantages of this special structure with a Krylov subspace method, a preconditioning technique is designed to solve the system efficiently. Numerical examples are reported to illustrate the performance of the preconditioned iteration.

Highly accurate, linear, and unconditionally energy stable large time-stepping schemes for the Functionalized Cahn–Hilliard gradient flow equation

This paper aims to develop a class of large time-stepping schemes for solving the Functionalized Cahn–Hilliard (FCH) gradient flow equation. By using the Invariant Energy Quadratization (IEQ) approach, we successfully construct several highly efficient linear schemes that allow large time step sizes to be used in computations. All the developed schemes with a fixed time step size are proved to be unconditionally uniquely solvable and unconditionally energy stable in theory. Moreover, we show that the obtained semi-discrete systems can be solved efficiently by using the preconditioned Krylov subspace methods, since all the linear operators of these systems are proved to be symmetric and positive definite. Finally, several numerical experiments are carried out to verify the accuracy, efficiency and energy stability of the developed IEQ schemes. We observe that our IEQ schemes are more accurate than the Scalar Auxiliary Variable (SAV) schemes. This can be attributed to the fact that the IEQ method can preserve the operator structures of the original equation as much as possible.

A novel fully-decoupled, second-order time-accurate, unconditionally energy stable scheme for a flow-coupled volume-conserved phase-field elastic bending energy model

Different from the classical phase-field elastic bending model of lipid vesicles that uses a penalty term to conserve volume approximately, in this paper, a new model with accurate volume conservation is first established. Then, for its coupling system with the incompressible flow, we design a highly efficient scheme which is linear and energy stable. More importantly, this scheme is second-order time-accurate and fully-decoupled and it only needs to solve several independent linear equations with constant coefficients at each time step to obtain a numerical solution with second-order time accuracy. The key idea is to introduce two types of nonlocal auxiliary variables, one of which is linearize the nonlinear potential, and the other is used to introduce an ordinary differential equation to deal with the nonlinear coupling terms that satisfy the “zero-energy-contribution” feature. We strictly prove the solvability and unconditional energy stability and conduct numerical simulations in 2D and 3D to demonstrate the accuracy and stability of the scheme numerically. To the best of the author's knowledge, the decoupling method developed in this paper is the first second-order fully-decoupled scheme for the flow-coupled phase-field model.

Novel linear decoupled and unconditionally energy stable numerical methods for the modified phase field crystal model

In this paper, we propose a novel numerical approach to construct unconditionally energy stable schemes for the modified phase field crystal (MPFC) model. The new technique is based on the invariant energy quadratization (IEQ) method. The numerical schemes based on IEQ approach lead to time-dependent dense matrices, thus the fast Fourier transform (FFT) is difficult to be applied to solve the systems directly. By introducing a new auxiliary variable to replace the original one, we derive a novel equivalent MPFC system. A step-by-step solving approach, termed 3S-IEQ method is considered to solve this equivalent system. Some totally decoupled, linear and unconditional energy stable semi-implicit schemes based on 3S-IEQ method are very easy to construct. More importantly, the phase function ϕ and auxiliary variable η can be calculated step-by-step. Meanwhile, the proposed approach only needs to solve linear equation with constant coefficients which is easy to use FFT directly in calculation. Some numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.

A linear, high-order, and unconditionally energy stable scheme for the epitaxial thin film growth model without slope selection

The epitaxial thin film growth model without slope selection is the L2-gradient flow of energy with a logarithmic potential in terms of the gradient of a height function. A challenge to numerically solving the model is how to treat the nonlinear term to preserve energy stability without compromising accuracy and efficiency. To resolve this problem, we present a high-order energy stable scheme by placing the linear and nonlinear terms in the convex and concave parts, respectively, and employing the specially designed implicit–explicit Runge–Kutta method. As a result, our scheme is linear, high-order accurate in time, and unconditionally energy stable. We show analytically that the scheme is unconditionally uniquely solvable and energy stable. Numerical experiments are presented to demonstrate the accuracy, efficiency, and energy stability of the proposed scheme.

Unconditionally energy stable discontinuous Galerkin schemes for the Cahn–Hilliard equation

In this paper, we introduce novel discontinuous Galerkin (DG) schemes for the Cahn–Hilliard equation, which arises in many applications. The method is designed by integrating the mixed DG method for the spatial discretization with the Invariant Energy Quadratization (IEQ) approach for the time discretization. Coupled with a spatial projection, the resulting IEQ–DG schemes are shown to be unconditionally energy dissipative, and can be efficiently solved without resorting to any iteration method. Both one and two dimensional numerical examples are provided to verify the theoretical results, and demonstrate the good performance of IEQ–DG in terms of efficiency, accuracy, and preservation of the desired solution 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 accurate backward differentiation formula (BDF) numerical scheme for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization 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-Dupont regularization term, in the form of $-A \Delta t^2 \Delta_N^2 ( u^{n+1} - u^n)$, is added in the numerical scheme. 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, the optimal rate convergence analysis and error estimate are derived in details, in the $\ell^\infty (0,T; \ell^2) \cap \ell^2 (0,T; H_h^2)$ norm, with the help of a linearized estimate for the nonlinear error terms. %This convergence estimate is the first such result for a third order accurate scheme for a gradient flow. Some numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence. The long time simulation results for $\varepsilon=0.02$ (up to $T=3 \times 10^5$) have indicated a logarithm law for the energy decay, as well as the power laws for growth of the surface roughness and the mound width. In particular, the power index for the surface roughness and the mound width growth, created by the third order numerical scheme, is more accurate than those produced by certain second order energy stable schemes in the existing literature.

An Energy Stable Second-Order Accurate Scheme for Microphase Separation of Periodic Diblock Copolymers

An Energy Stable Second-Order Accurate Scheme for Microphase Separation of Periodic Diblock Copolymers

An Efficient and Convergent Finite Element Scheme for Cahn--Hilliard Equations with Dynamic Boundary Conditions

The Cahn--Hilliard equation is a widely used model that describes amongst others phase separation processes of binary mixtures or two-phase flows. In the recent years, different types of boundary conditions for the Cahn--Hilliard equation were proposed and analyzed. In this publication, we are concerned with the numerical treatment of a recent model which introduces an additional Cahn--Hilliard type equation on the boundary as closure for the Cahn--Hilliard equation in the domain [C. Liu, H. Wu, Arch. Ration. Mech. An., 2019]. By identifying a mapping between the phase-field parameter and the chemical potential inside of the domain, we are able to postulate an efficient, unconditionally energy stable finite element scheme. Furthermore, we establish the convergence of discrete solutions towards suitable weak solutions of the original model. This serves also as an additional pathway to establish existence of weak solutions. Furthermore, we present simulations underlining the practicality of the proposed scheme and investigate its experimental order of convergence.

A Second-Order Energy Stable BDF Numerical Scheme for the Viscous Cahn-Hilliard Equation with Logarithmic Flory-Huggins Potential

A Second-Order Energy Stable BDF Numerical Scheme for the Viscous Cahn-Hilliard Equation with Logarithmic Flory-Huggins Potential

An unconditionally stable semi-implicit CutFEM for an interaction problem between an elastic membrane and an incompressible fluid

In this paper we introduce a finite element method for the Stokes equations with a massless immersed membrane. This membrane applies normal and tangential forces affecting the velocity and pressure of the fluid. Additionally, the points representing this membrane move with the local fluid velocity. We design and implement a high-accuracy cut finite element method (CutFEM) which enables the use of a structured mesh that is not aligned with the immersed membrane, and we formulate a time discretization that yields an unconditionally energy stable scheme. We prove that the stability is not restricted by the parameter choices that constrained previous finite element immersed boundary methods and illustrate the theoretical results with numerical simulations.

Bound/Positivity Preserving and Energy Stable Scalar auxiliary Variable Schemes for Dissipative Systems: Applications to Keller--Segel and Poisson--Nernst--Planck Equations

Bound/Positivity Preserving and Energy Stable Scalar auxiliary Variable Schemes for Dissipative Systems: Applications to Keller--Segel and Poisson--Nernst--Planck Equations

Unconditionally energy stable second-order numerical schemes for the Functionalized Cahn–Hilliard gradient flow equation based on the SAV approach

In this paper, we devise and analyse three highly efficient second-order accurate (in time) schemes for solving the Functionalized Cahn–Hilliard (FCH) gradient flow equation where an asymmetric double-well potential function is considered. Based on the Scalar Auxiliary Variable (SAV) approach, we construct these schemes by splitting the FCH free energy in a novel and ingenious way. Utilizing the Crank–Nicolson formula, we firstly construct two semi-discrete second-order numerical schemes, which we denote by CN-SAV and CN-SAV-A, respectively. To be more specific, the CN-SAV scheme is constructed based on the fixed time step, while the CN-SAV-A scheme is a variable time step scheme. The BDF2-SAV scheme is another second-order scheme in which the fixed time step should be used. It is designed by applying the second-order backward difference (BDF2) formula. All the constructed schemes are proved to be unconditionally energy stable and uniquely solvable in theory. To the best of our knowledge, the CN-SAV-A scheme is the first unconditionally energy stable, second-order scheme with variable time steps for the FCH gradient flow equation. In addition, an effective adaptive time selection strategy introduced in Christlieb et al., (2014) is slightly modified and then adopted to select the time step for the CN-SAV-A scheme. Finally, several numerical experiments based on the Fourier pseudo-spectral method are carried out in two and three dimensions, respectively, to confirm the numerical accuracy and efficiency of the constructed schemes.

Numerical approximations of the Navier–Stokes equation coupled with volume-conserved multi-phase-field vesicles system: Fully-decoupled, linear, unconditionally energy stable and second-order time-accurate numerical scheme

We consider the numerical approximation of the flow-coupled multi-phase-field elastic bending energy model of lipid vesicles. Based on the classical model with approximate volume conservation only, this paper first establishes a new model that can accurately conserve volume by adding some nonlocal terms to the model equation. Then, for the system coupled with the incompressible flow, we propose a novel numerical method to construct an effective scheme that is fully-decoupled, linear, unconditionally energy stable, and second-order time-accurate. The key idea to achieve the full decoupling nature is to introduce an ordinary differential equation to deal with the nonlinear coupling term that satisfies the so-called “zero-energy-contribution” property. Thus, in actual calculations, this scheme only needs to solve several independent linear equations with constant coefficients at each time step. We strictly prove the solvability and unconditional energy stability, and perform numerical simulations in 2D and 3D to verify the accuracy and stability of the scheme numerically.

Stability and convergence analysis for a new phase field crystal model with a nonlocal Lagrange multiplier

In this work, an energy stable numerical scheme is proposed to solve the PFC model with a nonlocal Lagrange multiplier. The construction of the numerical scheme is based on invariant energy quadratization (IEQ) technique to transform a nonlinear system into a linear system, while the time variables are discretized by second‐order scheme. The stability term in the new scheme can balance the influence of nonlinear term. Moreover, we obtain the results of unconditional energy stability, uniqueness, and uniform boundedness of numerical solution, and the numerical scheme is convergent with order of . Several numerical tests are conducted to confirm the theoretical results.

Arbitrarily high-order structure-preserving schemes for the Gross–Pitaevskii equation with angular momentum rotation

In this paper, we design a novel class of arbitrarily high-order structure-preserving numerical schemes for the time-dependent Gross–Pitaevskii equation with angular momentum rotation. Based on the idea of the scalar auxiliary variable approach which is proposed in the recent papers [J. Comput. Phys., 353 (2018) 407–416 and SIAM Rev., 61(2019) 474–506] for developing energy stable schemes for gradient flow systems, we firstly reformulate the Gross–Pitaevskii equation into an equivalent system with a modified energy conservation law. The reformulated system is then discretized by the Gauss collocation method in time and the standard Fourier pseudo-spectral method in space, respectively. We show that the proposed schemes can preserve the discrete mass and modified energy exactly. Numerical results are addressed to verify the efficiency and high-order accuracy of the proposed schemes.

Convergence of energy stable finite-difference schemes with interfaces

We extend the convergence results in Svärd and Nordström (2019) [7] for single-domain energy-stable high-order finite difference schemes, to include domains split into several grid blocks. The analysis also demonstrates that reflective boundary conditions enjoy the same convergence properties. Finally, we briefly indicate that these results (and the previous ones in [7]) also hold in multiple dimensions.

Positive-definiteness preserving and energy stable time-marching scheme for a diffusive Oldroyd-B electrohydrodynamic model

In this paper, we develop a new time-marching scheme for a viscoelastic electrohydrodynamic model that satisfies an energy dissipation law. The scheme is based on the logarithmic conformation formulation so that it naturally preserves the positive-definiteness of the conformation tensor. The backward Euler scheme with the semi-Lagrangian formulation is used to construct the time-stepping scheme, with finite difference method on staggered grid for spatial discretization. The scheme is proved to be unconditionally energy stable at the fully discrete level and a fixed-point iteration is designed for its efficient implementation. Numerical experiments are given to verify the convergence rates and the energy stability of the scheme. Finally, the flow structure within the electro-convection phenomena has been studied in detail, focusing on an understanding of the elastic effect of fluids.