A NOVEL LAGRANGE MULTIPLIER APPROACH WITH RELAXATION FOR GRADIENT FLOWS. ZHENGGUANG LIUyAND XIAOLI LIz

2025-04-27 0 0 2.09MB 23 页 10玖币

A NOVEL LAGRANGE MULTIPLIER APPROACH WITH RELAXATION FOR

GRADIENT FLOWS. ∗

ZHENGGUANG LIU †AND XIAOLI LI*‡

Abstract. 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.

Key words. Lagrange Multiplier approach, Zero-factor approach, Gradient ﬂows, Relaxation, Energy stable, Numerical

examples.

AMS subject classiﬁcations. 65M12; 35K20; 35K35; 35K55; 65Z05

1. Introduction. Gradient ﬂows are a kind of important models to simulate many physical problems

such as the interface behavior of multi-phase materials, the interface problems of ﬂuid mechanics, environ-

mental science and material mechanics. In general, as the highly complex high-order nonlinear dissipative

systems, it is a great challenge to construct eﬀective and accurate numerical schemes with physical con-

straints such as energy dissipation and mass conservation. Many experts and scholars considered some

unconditionally energy stable schemes. These numerical schemes preserve the energy dissipation law which

does not depend on the time step. Some popular and widely used methods include convex splitting approach

[6, 13, 17], linear stabilized approach [16, 22], exponential time diﬀerencing (ETD) approach [4, 5, 18],

invariant energy quadratization (IEQ) approach [7, 19, 21, 25], scalar auxiliary variable (SAV) approach

[10, 14, 15], Lagrange multiplier approach [1] and so on.

Gradient ﬂow models are generally derived from the functional variation of free energy. In general, the

free energy E(φ) contains the sum of an integral phase of a nonlinear functional and a quadratic term:

(1.1) E(φ) = 1

2(φ, Lφ) + E1(φ) = 1

2(φ, Lφ) + ZΩ

F(φ)dx,

where Lis a symmetric non-negative linear operator, and E1(φ) = RΩF(φ)dxis nonlinear free energy. F(x)

is the energy density function. The gradient ﬂow from the energetic variation of the above energy functional

E(φ) in (1.1) can be obtained as follows:

(1.2) ∂φ

∂t =−Gµ, µ =Lφ+F0(φ),

where µ=δE

δφ is the chemical potential. Gis a positive operator. For example, G=Ifor the L2gradient

ﬂow and G=−∆ for the H−1gradient ﬂow.

It is not diﬃcult to ﬁnd that the above phase ﬁeld system satisﬁes the following energy dissipation law:

dtE= (δE

δφ ,∂φ

∂t ) = −(Gµ, µ)≤0,

∗We would like to acknowledge the assistance of volunteers in putting together this example manuscript and supplement.

This work is supported by National Natural Science Foundation of China (Grant Nos: 12001336, 11901489, 12131014).

†School of Mathematics and Statistics, Shandong Normal University, Jinan, China. Email: liuzhg@sdnu.edu.cn.

‡Shandong University, Jinan, Shandong, 250100, China. Email: xiaomath@sdu.edu.cn.

arXiv:2210.02723v1 [math.NA] 6 Oct 2022

2ZHENGGUANG LIU AND XIAOLI LI

which is a very important property for gradient ﬂows in physics and mathematics.

Recently, many SAV-type methods are developed to optimize the traditional SAV method. For example,

in [23], the authors introduced the generalized auxiliary variable method for devising energy stable schemes

for general dissipative systems. An exponential SAV approach in [12] is developed to modify the traditional

method to construct energy stable schemes by introducing an exponential SAV. In [8], the authors consider

a new SAV approach to construct high-order energy stable schemes. In [1], the authors introduce a new

Lagrange multiplier approach which is unconditionally energy stable with the original energy. However, the

new approach requires solving a nonlinear algebraic equation for the Lagrange multiplier which brings some

additional costs and theoretical diﬃculties for its analysis. Recently, Jiang et al. [9] present a relaxation

technique to construct a relaxed SAV (RSAV) approach to improve the accuracy and consistency noticeably.

In this paper, inspired by the new Lagrange multiplier approach and RSAV approach, we propose a

novel technique to construct the unconditional energy stable schemes for gradient ﬂows by introducing a

zero factor. Compared with the recently proposed SAV-type approach, the numerical schemes based on the

new zero-factor (ZF) method dissipate the original energy and do not require the explicitly treated part

of the free energy to be bounded from below. The core idea of the zero-factor approach is to introduce a

zero factor to modify the solution φn+1 of the baseline semi-implicit method at each time step. The value

of the introduced zero factor P(η) is controlled by energy stability. 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 approach almost preserves all the advantages of the new zero-factor approach. It is unconditionally

energy stable with respect to a modiﬁed energy that is closer to the original energy, and provides a very

simple calculation process. Our main contributions of this paper are:

(i). The new introduced RZF method can keep the original energy in most cases and provides a very

simple calculation process;

(ii). We prove that the zero factor schemes with speciﬁc P(η) lead to the popular SAV-type and Lagrange

multiplier methods;

(iii). The variation of the introduced zero factor is highly consistent with the nonlinear free energy which

implies that the introduced zero factor is very eﬃcient to capture the sharp dissipation of the nonlinear free

energy.

The paper is organized as follows. In Sect.2, we introduce a zero factor to construct a new zero-factor

approach to simulate a series of gradient ﬂows. In Sect.3, by using a relaxation technique, we propose a

relaxed ZF approach. Then the second-order Crank-Nicloson and BDF2 schemes based on RZF method

are constructed. In Sect.4, we brieﬂy illustrate that the RZF approach can be easily applied to simulate

the gradient ﬂow with several disparate nonlinear terms. Finally, in Sect.5, various 2D and 3D numerical

simulations are demonstrated to verify the accuracy and eﬃciency of our proposed schemes.

2. The Zero-Factor Approach. Introduce a scalar auxiliary function η(t) to construct a linear func-

tion P(η), and rewrite the gradient ﬂow (1.2) with a zero factor P(η) as follows:

(2.1)

∂φ

∂t =−Gµ,

µ=Lφ+F0(φ) + P(η)F0(φ),

dt ZΩ

F(φ)dx=ZΩ

F0(φ)φtdx+P(η)ZΩ

F0(φ)φtdx.

Here the zero factor P(η) is a linear zero function which can be chosen ﬂexibly, such as the following P1and

(2.2) P1(η) = k1η, P2(η) = k2ηt,

where k1and k2are any non-zero constants.

RELAXED ZERO-FACTOR APPROACH FOR GRADIENT FLOWS 3

Set the initial condition for η(t) to be η(0) = 0 for P1(η) or η(0) = c0for P2(η) where c0is an arbitrary

constant, then it is easy to see that the new system (2.1) is equivalent to the original system (1.2), i.e.,

P(η) = 0 in (2.1).

Taking the inner products of the ﬁrst two equations in the above equivalent system (2.1) with µand −φt,

respectively, then summing up the results together with the third equation, we obtain the original energy

dissipative law:

dtE=−(Gµ, µ)≤0,

It means that the linear functional P(η) here is to serve as a zero factor to enforce dissipation of the original

energy.

2.1. A second-order Crank-Nicloson ZF scheme. Before giving a detailed introduction, we let

N > 0 be a positive integer and set

∆t=T/N, tn=n∆t, for n≤N.

In the following, we will consider a second-order Crank-Nicolson scheme for the system (2.1). Discretize

the nonlinear functional F0(φ) explicitly and the other items implicitly in (2.1), and give the initial values

φ0=φ0(x), η(0) = c0, then couple with Crank-Nicolson formula, a second-order energy stable schemes can

be constructed as follows:

(2.3)

φn+1 −φn

∆t=−Gµn+1

µn+1

2=1

2Lφn+1 +1

2Lφn+F0(b

φn+1

2) + P(ηn+1

2)F0(b

φn+1

2),

(F(φn+1),1) −(F(φn),1) = F0(b

φn+1

2), φn+1 −φn+P(ηn+1

2)F0(b

φn+1

2), φn+1 −φn,

where b

φn+1

2=3

2φn−1

2φn−1.

Taking the inner products of ﬁrst two equation in (2.3) with µn+1

2and −φn+1−φn

∆trespectively, and

multiplying the third equation with ∆t, then combining these equations, we obtain the above Crank-Nicolson

scheme satisﬁes the following original energy dissipative law:

(2.4) E(φn+1)− E(φn) = −∆t(Gµn+1

2, µn+1

2)≤0,

where E(φn) = 1

2(Lφn, φn)+(F(φn),1).

The Crank-Nicolson scheme (2.3) is nonlinear for the variables φn+1 and ηn+1. We now show how to

solve it eﬃciently. Combining the ﬁrst two equations in (2.3), we can obtain the following linear matrix

equation

(I+1

2∆tGL)φn+1 = (I−1

2∆tGL)φn−∆tGF0(b

φn+1

2)− P(ηn+1

2)∆tGF0(b

φn+1

2).

Noting that the coeﬃcient matrix A= (I+1

2∆tGL) is a symmetric positive matrix, then we obtain

(2.5) φn+1 =A−1(I−1

2∆tGL)φn−∆tGF0(b

φn+1

2)− P(ηn+1

2)∆tA−1GF0(b

φn+1

=φn+1 +P(ηn+1

2)qn+1,

Here φn+1 and qn+1 can be solved directly by φnand b

φn+1

2as follows:

(2.6) φn+1 =A−1(I−1

2∆tGL)φn−∆tGF0(b

φn+1

2), qn+1 =−∆tA−1GF0(b

φn+1

2).

4ZHENGGUANG LIU AND XIAOLI LI

Combining the equation (2.5) with the third equation in (2.3), we have

(2.7) F(φn+1 +P(ηn+1

2)qn+1),1−(F(φn),1)

=h1 + P(ηn+1

2)iF0(b

φn+1

2), pn+1 +P(ηn+1

2)qn+1 −φn.

One can see that to solve above nonlinear numerical scheme (2.7), we need to solve ηn+1 by the Newton

iteration as the initial condition. The computational complexity depends on F(φ). The computational cost

is equal to the Lagrange Multiplier approach which was proposed by Shen et al. [1].

Remark 2.1. In principle one can choose any linear function to be zero factor P(η)in equation (2.1).

A special case is P(η) = η(t)−1, then the zero factor method leads to the new Lagrange multiplier approach

in [1].

Remark 2.2. From the equation (2.5), we notice that φn+1 is the solution of the baseline semi-implicit

Crank-Nicolson scheme. Hence the core idea of the zero factor approach is to introduce a zero factor to

modify the solution φn+1 at each time step. The value of the zero factor is controlled by energy stability.

2.2. A revisit of the SAV-type approach. In this subsection, we will review the SAV-type approach

and prove that the introduced scalar auxiliary variables can be seen as the speciﬁc zero factors. Furthermore,

we can modify the SAV-type methods to construct new schemes which dissipate the original energy.

The key for the SAV approach is to introduce a scalar variable r(t) = pE(φ) + Cwhere E1(φ) =

(F(φ),1) is the nonlinear free energy and rewrite the gradient ﬂows (1.2) as the following equivalent system:

(2.8)

∂φ

∂t =−Gµ,

µ=Lφ+r(t)

pE1(φ) + CF0(φ),

dt =1

2pE1(φ) + C(F0(φ), φt).

A second-order Crank-Nicloson SAV scheme for above equivalent system is as follows:

(2.9)

φn+1 −φn

∆t=−Gµn+1

µn+1

2=1

2Lφn+1 +1

2Lφn+rn+1

qE1(b

φn+1

2) + C

F0(b

φn+1

2),

rn+1 −rn

∆t=1

2qE1(b

φn+1

2) + C

(F0(b

φn+1

2),φn+1 −φn

∆t).

Here rn+1

2= (rn+1 +rn)/2.

Combining the ﬁrst two equations in above second-order scheme, we can obtain:

(2.10)

(I+1

2∆tGL)φn+1 = (I−1

2∆tGL)φn−∆trn+1

qE1(b

φn+1

2) + CGF0(b

φn+1

=(I−1

2∆tGL)−∆tGF0(b

φn+1

2)−

rn+1

qE1(b

φn+1

2) + C−1

∆tGF0(b

φn+1

2).

RELAXED ZERO-FACTOR APPROACH FOR GRADIENT FLOWS 5

Using the same deﬁnitions of φn+1 and qn+1 in (2.6), we can obtain φn+1 as follows:

(2.11)

φn+1 =A−1(I−1

2∆tGL)φn−∆tGF0(b

φn+1

2)−

rn+1

qE1(b

φn+1

2) + C−1

∆tA−1GF0(b

φn+1

=φn+1 +

rn+1

qE1(b

φn+1

2) + C−1

qn+1.

Compared above equation (2.11) with (2.6), we can obviously obtain that the key for the SAV approach is

to introduce a zero factor

(2.12) P(r) = r(t)

pE1(φ) + C−1.

It means that the core idea of the SAV scheme (2.9) is also to introduce a special zero factor to modify the

solution φn+1 which is the solution of the baseline semi-implicit Crank-Nicolson scheme at each time step.

The value of the zero factor P(r) is controlled by energy stability.

Inspired by the introduced ZF method, we can obtain a new SAV approach which is unconditionally

energy stable with the original energy by changing the third equation in the equivalent system (2.8):

(2.13)

∂φ

∂t =−Gµ,

µ=Lφ+r(t)

pE1(φ) + CF0(φ),

dt ZΩ

F(φ)dx=r(t)

pE1(φ) + CZΩ

F0(φ)φtdx.

A second-order Crank-Nicloson SAV scheme for above equivalent system (2.13) is as follows:

(2.14)

φn+1 −φn

∆t=−Gµn+1

µn+1

2=1

2Lφn+1 +1

2Lφn+rn+1

qE1(b

φn+1

2) + C

F0(b

φn+1

2),

(F(φn+1),1) −(F(φn),1) = rn+1

qE1(b

φn+1

2) + C

(F0(b

φn+1

2), φn+1 −φn).

Taking the inner products of ﬁrst two equation in (2.14) with µn+1

2and −φn+1−φn

∆trespectively, and mul-

tiplying the third equation with ∆t, then combining these equations, we obtain the above Crank-Nicolson

scheme satisﬁes the following original energy dissipative law:

(2.15) E(φn+1)− E(φn) = −∆t(Gµn+1

2, µn+1

2)≤0,

where E(φn) = 1

2(Lφn, φn)+(F(φn),1).

Remark 2.3. For other SAV-type approaches, the core idea is also to introduce a special zero factor to

modify the solution φn+1. For example, the zero factor P(r) = r(t)

exp(E1(φ)) −1for ESAV approach in [12].

3. The Relaxed Zero-Factor Approach. From above analysis, we notice that the scheme based on

the zero-factor approach dissipates the original energy but needs to solve a nonlinear algebraic equation

which brings some additional costs and theoretical diﬃculties for its analysis. In general, compared with the

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ANOVELLAGRANGEMULTIPLIERAPPROACHWITHRELAXATIONFORGRADIENTFLOWS.ZHENGGUANGLIUyANDXIAOLILI*zAbstract.Inthispaper,weproposeanovelLagrangeMultiplierapproach,namedzero-factor(ZF)approachtosolveaseriesofgradientowproblems.Thenumericalschemesbasedonthenewalgorithmareunconditionallyenergystablewiththeorigi...

展开>> 收起<<

A NOVEL LAGRANGE MULTIPLIER APPROACH WITH RELAXATION FOR GRADIENT FLOWS. ZHENGGUANG LIUyAND XIAOLI LIz.pdf

共23页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

A NOVEL LAGRANGE MULTIPLIER APPROACH WITH RELAXATION FOR GRADIENT FLOWS. ZHENGGUANG LIUyAND XIAOLI LIz

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: