Search

We propose an energy-optimized invariant energy quadratization method to solve the gradient flow models in this paper, which requires only one linear energy-optimized step to correct the auxiliary variables on each time step. In addition to inheriting the benefits of the baseline and relaxed invariant energy quadratization method, our approach has several other advantages. Firstly, in the process of correcting auxiliary variables, we can directly solve linear programming problem by the energy-optimized technique, which greatly simplifies the nonlinear optimization problem in the previous relaxed invariant energy quadratization method. Secondly, we construct new linear unconditionally energy stable schemes by applying backward Euler formulas and Crank-Nicolson formula, so that the accuracy in time can reach the first- and second-order. Thirdly, comparing with relaxation technique, the modified energy obtained by energy-optimized technique is closer to the original energy, meanwhile the accuracy and consistency of the numerical solutions can be improved. Ample numerical examples have been presented to demonstrate the accuracy, efficiency and energy stability of the proposed schemes., Comment: 35 pages, 39 figures

In this paper we construct new fully decoupled and high-order implicit-explicit (IMEX) schemes for the two-phase incompressible flows based on the new generalized scalar auxiliary variable approach with optimal energy approximation (EOP-GSAV) for Cahn-Hilliard equation and consistent splitting method for Navier-Stokes equation. These schemes are linear, fully decoupled, unconditionally energy stable, only require solving a sequence of elliptic equations with constant coefficients at each time step, and provide a new technique to preserve the consistency between original energy and modified energy. We derive that numerical solutions of these schemes are uniformly bounded without any restriction on time step size. Furthermore, we carry out a rigorous error analysis for the first-order scheme and establish optimal global error estimates for the phase function, velocity and pressure in two and three-dimensional cases. Numerical examples are presented to validate the proposed schemes., Comment: arXiv admin note: text overlap with arXiv:2009.09353

The scalar auxiliary variable (SAV)-type methods are very popular techniques for solving various nonlinear dissipative systems. Compared to the semi-implicit method, the baseline SAV method can keep a modified energy dissipation law but doubles the computational cost. The general SAV approach does not add additional computation but needs to solve a semi-implicit solution in advance, which may potentially compromise the accuracy and stability. In this paper, we construct a novel first- and second-order unconditional energy stable and positivity-preserving stabilized SAV (PS-SAV) schemes for $L^2$ and $H^{-1}$ gradient flows. The constructed schemes can reduce nearly half computational cost of the baseline SAV method and preserve its accuracy and stability simultaneously. Meanwhile, the introduced auxiliary variable is always positive while the baseline SAV cannot guarantee this positivity-preserving property. Unconditionally energy dissipation laws are derived for the proposed numerical schemes. We also establish a rigorous error analysis of the first-order scheme for the Allen-Cahn type equation in $l^{\infty}(0,T; H^1(\Omega) ) $ norm. In addition we propose an energy optimization technique to optimize the modified energy close to the original energy. Several interesting numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed methods.

Recently, a new Lagrange multiplier approach was introduced by Cheng, Liu and Shen in \cite{cheng2020new}, which has been broadly used to solve various challenging phase field problems. To design original energy stable schemes, they have to solve a nonlinear algebraic equation to determine the introduced Lagrange multiplier, which can be computationally expensive, especially for large-scale and long-time simulations involving complex nonlinear terms. This paper presents an essential improved technique to modify this issue, which can be seen as a semi-implicit combined Lagrange multiplier approach. In general, the new constructed schemes keep all the advantages of the Lagrange multiplier method and significantly reduce the computation costs. Besides, the new proposed BDF2 scheme dissipates the original energy, as opposed to a modified energy for the classical Lagrange multiplier approach in \cite{cheng2020new}. We further construct high-order BDF$k$ schemes based on the new proposed approach. In addition, we establish a general framework for extending our constructed method to dissipative systems. Finally several examples have been presented to demonstrate the effectiveness of the proposed approach.

The nonlocal Allen-Cahn equation with nonlocal diffusion operator is a generalization of the classical Allen-Cahn equation. It satisfies the energy dissipation law and maximum bound principle (MBP), and is important for simulating a series of physical and biological phenomena involving long-distance interactions in space. In this paper, we construct first- and second-order (in time) accurate, unconditionally energy stable and MBP-preserving schemes for the nonlocal Allen-Cahn type model based on the stabilized exponential scalar auxiliary variable (sESAV) approach. On the one hand, we have proved the MBP and unconditional energy stability carefully and rigorously in the fully discrete levels. On the other hand, we adopt an efficient FFT-based fast solver to compute the nearly full coefficient matrix generated from the spatial discretization, which improves the computational efficiency. Finally, typical numerical experiments are presented to demonstrate the performance of our proposed schemes.

The nonlocal Cahn-Hilliard (NCH) equation with nonlocal diffusion operator is more suitable for the simulation of microstructure phase transition than the local Cahn-Hilliard (LCH) equation. In this paper, based on the exponential semi-implicit scalar auxiliary variable (ESI-SAV) method, the highly effcient and accurate schemes in time with unconditional energy stability for solving the NCH equation are proposed. On the one hand, we have demostrated the unconditional energy stability for the NCH equation with its high-order semi-discrete schemes 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 FFT and FCG 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.

In recent years, the scalar auxiliary variable (SAV) approach has become very popular and hot in the design of linear, high-order and unconditional energy stable schemes of gradient flow models. However, the nature of SAV-based numerical schemes preserving modified energy dissipation limits its wider application. A relaxation technique to correct the modified energy for the baseline SAV method (RSAV) was proposed by Zhao et al. and Shen et al.. The RSAV approach is unconditionally energy stable with respect to a modified energy that is closer to the original free energy, and provides a much improved accuracy when compared with the SAV approach. In this paper, inspired by the RSAV approach, we propose a novel technique to correct the modified energy of the SAV approach, which can be proved to be an optimal energy approximation. We construct new high-order implicit-explicit schemes based on the proposed energy-optimal SAV (EOP-SAV) approach. The constructed EOP-SAV schemes not only provide an improved accuracy but also simplify calculation, and can be viewed as the optimal relaxation. We also prove that the numerical schemes based on the EOP-SAV approach are unconditionally energy stable. Compared with the RSAV approach, the proposed EOP-SAV approach does not need introduce any relaxed factors and can share the similar procedure for error estimates. Several interesting numerical examples have been presented to demonstrate the accuracy and effectiveness of the proposed methods., Comment: Scalar auxiliary variable, Gradient flow, Relaxation, Optimal, Error analysis

In this paper, we propose a novel Lagrange Multiplier approach, named zero-factor (ZF) approach to solve a series of gradient flow problems. The numerical schemes based on the new algorithm are unconditionally energy stable with the original energy and do not require any extra assumption conditions. We also prove that the ZF schemes with specific zero factors lead to the popular SAV-type method. To reduce the computation cost and improve the accuracy and consistency, we propose a zero-factor approach with relaxation, which we named the relaxed zero-factor (RZF) method, to design unconditional energy stable schemes for gradient flows. The RZF schemes can be proved to be unconditionally energy stable with respect to a modified energy that is closer to the original energy, and provide a very simple calculation process. The variation of the introduced zero factor is highly consistent with the nonlinear free energy which implies that the introduced ZF method is a very efficient way to capture the sharp dissipation of nonlinear free energy. Several numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method.

For the past few years, scalar auxiliary variable (SAV) and SAV-type approaches became very hot and efficient methods to simulate various gradient flows. Inspired by the new SAV approach in \cite{huang2020highly}, we propose a novel technique to construct a new exponential scalar auxiliary variable (E-SAV) approach to construct high-order numerical energy stable schemes for gradient flows. To improve its accuracy and consistency noticeably, we propose an E-SAV approach with relaxation, which we named the relaxed E-SAV (RE-SAV) method for gradient flows. The RE-SAV approach preserves all the advantages of the traditional SAV approach. In addition, we do not need any the bounded-from-below assumptions for the free energy potential or nonlinear term. Besides, the first-order, second-order and higher-order unconditionally energy stable time-stepping schemes are easy to construct. Several numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method., Comment: arXiv admin note: substantial text overlap with arXiv:2010.11728, arXiv:2001.00812

The scalar auxiliary variable (SAV) approach is a very popular and efficient method to simulate various phase field models. To save the computational cost, a new SAV approach is given by introducing a new variable $\theta$. The new SAV approach can be proved to save nearly half CPU time of the original SAV approach while keeping all its other advantages. In this paper, we propose a novel technique to construct an exponential semi-implicit scalar auxiliary variable (ESI-SAV) approach without introducing any extra variables. The new proposed method also only needs to solve one linear equation with constant coefficients at each time step. Furthermore, the constructed ESI-SAV method does not need the bounded below restriction of nonlinear free energy potential which is more reasonable and effective for various phase field models. Meanwhile it is easy to construct first-order, second-order and higher-order unconditionally energy stable time-stepping schemes. Other than that, the ESI-SAV approach can be proved to be effective to solve the non-gradient but dissipative system such as Navier-Stokes equations. Several numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method., Comment: arXiv admin note: text overlap with arXiv:1912.09263, arXiv:2001.00812

We construct new first- and second-order pressure correction schemes using the scalar auxiliary variable (SAV) approach for the Navier-Stokes equations. These schemes are linear, decoupled and only require a sequence of solving Poisson type equations at each time step. Furthermore, they are unconditionally energy stable. We also establish rigorous error estimates for the velocity and pressure approximation of the first-order scheme without any condition on the time step.

In this paper, we propose several novel numerical techniques to deal with nonlinear terms in gradient flows. These step-by-step solving schemes, termed 3S-SAV and 3S-IEQ schemes, are based on recently popular scalar auxiliary variable (SAV) and invariant energy quadratization (IEQ) approaches. In these constructed numerical methods, the phase function $\phi$ and auxiliary variable can be calculated step-by-step. Compared with the traditional SAV/IEQ approaches, there are many advantages for the novel 3S-SAV/3S-IEQ schemes. Firstly, we do not need the restriction of the bounded from below of the nonlinear free energy potential/density function. Secondly, the auxiliary variable combined with nonlinear function can be treated totally explicitly in the 3S-SAV/3S-IEQ approaches. Specially, for solving the discrete scheme based on IEQ approach, the linear system usually involves variable coefficients which change at each time step. However, the discrete scheme based on 3S-IEQ approach leads to linear equation with constant coefficients. Two comparative studies of traditional SAV/IEQ and 3S-SAV/3S-IEQ approaches are considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy., Comment: arXiv admin note: substantial text overlap with arXiv:1912.09263

In this paper, we consider an exponential scalar auxiliary variable (E-SAV) approach to obtain energy stable schemes for a class of phase field models. This novel auxiliary variable method based on exponential form of nonlinear free energy potential is more effective and applicable than the traditional SAV method which is very popular to construct energy stable schemes. The first contribution is that the auxiliary variable without square root removes the bounded from below restriction of the nonlinear free energy potential. Then, we prove the unconditional energy stability for the semi-discrete schemes carefully and rigorously. Another contribution is that we can discrete the auxiliary variable combined with the nonlinear term totally explicitly. Such modification is very efficient for fast calculation. Furthermore, the positive property of $r$ can be guaranteed which is very important and reasonable for the models' equivalence. Besides, for complex phase field models with two or more unknown variables and nonlinear terms, we construct a multiple E-SAV (ME-SAV) approach to enhance the applicability of the proposed E-SAV approach. A comparative study of classical SAV and E-SAV approaches is considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy.