arXiv subjects

Zhengguang Liu

Publications and source records attributed to Zhengguang Liu.

At least 19 recordsLinked to original sources

arXiv2025

A new class of one-step A-stable and L-stable schemes of high-order accuracy for parabolic type equations

Recently, a new class of BDF schemes proposed in [F. Huang and J. Shen, SIAM J Numer. Anal., 62.4, 1609--1637] for the parabolic type equations are studied in this paper. The basic idea is based on the Taylor expansions at time $t^{n+β}$ with $β>1$ being a tunable parameter. These new BDF schemes allow larger time steps at higher order r for stiff problems than that which allowed with a usual higher-order scheme. However, multi-step methods like BDF exhibit inherent disadvantages relative to one-step methods in practical implementations. In this paper, inspired by their excellent work, we construct a new class of high-order one-step schemes for linear parabolic-type equations. These new schemes, with several suitable $β_i$, can achieve A-stable, or even L-stable. Specially, the new scheme with special parameters $β_i$ can be regarded as the classical one-step Runge-Kutta scheme with a stabilized term. Besides, we provide two different techniques to construct the one-step high-order schemes: the first one is by choosing different parameters $β_i$, and the second one is by increasing the number of intermediate layers. Both methods have been proven to be highly effective and even exhibit superconvergence property. Finally, we also conducted several numerical experiments to support our conclusions.

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arXiv2025

Linear, decoupled and positivity-preserving staggered mesh schemes for general dissipative systems with arbitrary energy distributions

In this paper, we develop a novel staggered mesh (SM) approach for general nonlinear dissipative systems with arbitrary energy distributions (including cases with known or unknown energy lower bounds). Based on this framework, we propose several second-order semi-discrete schemes that maintain linearity, computational decoupling, and unconditional energy stability. Firstly, for dissipative systems with known energy lower bounds, we introduce a positive auxiliary variable $V(t)$ to substitute the total energy functional, subsequently discretizing it on staggered temporal meshes to ensure that the energy remains non-increasing regardless of the size of time step. The newly developed schemes achieve full computational decoupling, maintaining essentially the same computational expense as conventional implicit-explicit methods while demonstrating significantly improved accuracy. Furthermore, we rigorously establish the positivity preservation of the discrete variable $V^{n+1/2}$ which is a crucial property ensuring numerical stability and accuracy. Theoretical analysis confirms second-order temporal convergence for the proposed SM schemes. Secondly, for dissipative systems lacking well-defined energy lower bounds, we devise an alternative auxiliary variable formulation and extend the SM framework to maintain unconditional energy stability while preserving numerical effectiveness and accuracy. Finally, comprehensive numerical experiments, including benchmark problem simulations, validate the proposed schemes' efficacy and demonstrate their superior performance characteristics.

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arXiv2024

Improving the accuracy and consistency of the energy quadratization method with an energy-optimized technique

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.

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arXiv2023

Global error estimates of high-order fully decoupled schemes for the Cahn-Hilliard-Navier-Stokes model of Two-Phase Incompressible Flows

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.

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arXiv2023

An enhanced and highly efficient semi-implicit combined Lagrange multiplier approach with preserving original energy law for dissipative systems

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.

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arXiv2023

High-efficiency and positivity-preserving stabilized SAV methods for gradient flows

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(Ω) ) $ 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.

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arXiv2023

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

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.

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arXiv2023

The stabilized exponential-SAV approach preserving maximum bound principle for nonlocal Allen-Cahn equation

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.

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arXiv2023

A novel energy-optimal scalar auxiliary variable (EOP-SAV) approach for gradient flows

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.

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arXiv2022

A novel Lagrange Multiplier approach with relaxation for gradient flows

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.

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arXiv2022

Highly efficient exponential scalar auxiliary variable approaches with relaxation (RE-SAV) for gradient flows

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.

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arXiv2020

A highly efficient and accurate exponential semi-implicit scalar auxiliary variable (ESI-SAV) approach for dissipative system

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 $θ$. 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.

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arXiv2020

New SAV-pressure correction methods for the Navier-Stokes equations: stability and error analysis

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.

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arXiv2019

Step-by-step solving schemes based on scalar auxiliary variable and invariant energy quadratization approaches for gradient flows

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 $ϕ$ 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.

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arXiv2019

The exponential scalar auxiliary variable (E-SAV) approach for phase field models and its explicit computing

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.

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arXiv2019

Efficient invariant energy quadratization and scalar auxiliary variable approaches without bounded below restriction for phase field models

Recently introduced invariant energy quadratization (IEQ) and scalar auxiliary variable (SAV) approaches have proven to be very powerful ways to construct energy stable schemes for phase field models. Both methods require an assumption which the energy potential or energy is bounded from below. Under this assumption, a positive constant $C$ need to be added to keep the square root reasonable. However, it seems that it maybe more reasonable in both physics and mathematics if the energy potential or energy is arbitrary. Furthermore, if the bounded value from below is not obvious exactly, we are not easy to give a proper constant $C$ before calculation. In this paper, by adding a positive preserving function ($M(ϕ)$ in IEQ and $E_0(ϕ)$ in SAV ) instead of the positive constant in square root, we can improve this situation. The main contribution of this article is we proved that the positive preserving function can always be found for polynomial type $F(ϕ)$. We also proved the unconditional energy stability for all semi-discrete schemes carefully and rigorously. A comparative study of classical IEQ, SAV, MIEQ and MSAV approaches is considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and

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arXiv2019

Energy stable schemes for gradient flows based on novel auxiliary variable with energy bounded above

In this paper, we consider a novel auxiliary variable method to obtain energy stable schemes for gradient flows. The auxiliary variable based on energy bounded above does not limited to the hypothetical conditions adopted in previous approaches. We proved the unconditional energy stability for all the semi-discrete schemes carefully and rigorously. The novelty of the proposed schemes is that the computed values for the functional in square root are guaranteed to be positive. This method, termed novel auxiliary energy variable (NAEV) method does not consider any bounded below restrictions any longer. However, these restrictions are necessary in invariant energy quadratization (IEQ) and scalar auxiliary variable (SAV) approaches which are very popular methods recently. This property of guaranteed positivity is not available in previous approaches. A comparative study of classical SAV and NAEV approaches is considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy.

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arXiv2018

Fast procedures for Caputo fractional derivative and its applications to ordinary and partial differential equations

In this paper, we develop fast procedures for solving linear systems arising from discretization of ordinary and partial differential equations with Caputo fractional derivative w.r.t time variable. First, we consider a finite difference scheme to solve a two-sided fractional ordinary equation. Furthermore, we present a fast solution technique to accelerate Toeplitz matrix-vector multiplications arising from finite difference discretization. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the computational work from $O(N^{3})$ required by traditional methods to $O(Nlog^{2}N)$ and the memory requirement from $O(N^{2})$ to $O(N)$ without using any lossy compression, where $N$ is the number of unknowns. Two finite difference schemes to solve time fractional hyperbolic equations with different fractional order $γ$ are considered. We present a fast solution technique depending on the special structure of coefficient matrices by rearranging the order of unknowns. It helps to reduce the computational work from $O(N^2M)$ required by traditional methods to $O(N$log$^{2}N)$ and the memory requirement from $O(NM)$ to $O(N)$ without using any lossy compression, where $N=τ^{-1}$ and $τ$ is the size of time step, $M=h^{-1}$ and $h$ is the size of space step. Importantly, a fast method is employed to solve the classical time fractional diffusion equation with a lower coast at $O(MN$log$^2N)$, where the direct method requires an overall computational complexity of $O(N^2M)$. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

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