Structure-Preserving Discretization of Fractional Vector Calculus using Discrete Exterior Calculus

2025-05-02 0 0 1.54MB 25 页 10玖币

侵权投诉

Structure-Preserving Discretization of Fractional Vector Calculus

using Discrete Exterior Calculus

Alon Jacobsona, Xiaozhe Hua,∗

aDepartment of Mathematics, Tufts University, Medford, 02155, MA, USA

Abstract

Fractional vector calculus is the building block of the fractional partial diﬀerential equations

that model non-local or long-range phenomena, e.g., anomalous diﬀusion, fractional elec-

tromagnetism, and fractional advection-dispersion. In this work, we reformulate a type of

fractional vector calculus that uses Caputo fractional partial derivatives and discretize this

reformulation using discrete exterior calculus on a cubical complex in the structure-preserving

way, meaning that the continuous-level properties curlαgradα=0and divαcurlα= 0 hold

exactly on the discrete level. We discuss important properties of our fractional discrete ex-

terior derivatives and verify their second-order convergence in the root mean square error

numerically. Our proposed discretization has the potential to provide accurate and stable nu-

merical solutions to fractional partial diﬀerential equations and exactly preserve fundamental

physics laws on the discrete level regardless of the mesh size.

Keywords: Fractional vector calculus, Discrete exterior calculus, Structure-preserving

discretization, Fractional partial diﬀerential equations

2020 MSC: 65M99, 65N99, 26A33, 35R11

1. Introduction

Fractional calculus generalizes the integer order integration and diﬀerentiation to non-

integer order. Unlike standard derivatives and integrals, fractional derivatives and integrals

are non-local operators, enabling them to model long-range dependence. In this work, we

focus on fractional vector calculus (FVC), which analogously extends the vector calculus to

fractional order. Fractional calculus and FVC are widely used in fractional partial diﬀerential

equations (FPDEs), which recently have a wide range of new scientiﬁc and engineering

applications. For example, fractional diﬀusion equations model anomalous diﬀusion [1, 2, 3,

4, 5], fractional Maxwell’s equations generalize Maxwell’s equations to fractional order [6,

7, 8], fractional advection-dispersion equations describe subsurface transport [9, 10, 11, 12],

fractional Laplacians are used in image processing [13], fractional diﬀerential equations are

used in ﬁnance [14], and a fractional gradient has been used for fractional backpropagation

in training neural networks [15].

∗Corresponding author

Email addresses: Alon.Jacobson@tufts.edu (Alon Jacobson), Xiaozhe.Hu@tufts.edu (Xiaozhe Hu)

arXiv:2210.11175v3 [math.NA] 26 Jan 2024

There are various deﬁnitions of FVC, each with their own strengths and weaknesses.

Many approaches use a fractional partial derivative in each coordinate direction to construct

a fractional nabla operator. Other approaches use an anisotropic mixture of fractional di-

rectional derivatives in each direction via an integral, while still other approaches use an

isotropic mixture of function values throughout Euclidean space to deﬁne the operators.

Due to the complexity of FPDEs, the solutions cannot usually be computed symbolically,

so numerical approximations are essential for solving them. Various ﬁnite-element and ﬁnite-

diﬀerence methods have been developed for the discretization of FVC to be used in solving

these FPDEs, with techniques including ﬁnite-diﬀerence methods [16], the discretization

of fractional directional derivatives [17, 18, 10], spectral decompositions [19] and physics-

informed neural networks [10].

When solving PDEs and FPDEs numerically, some consideration must be given to com-

putational eﬃciency (i.e., how much time or computer memory is required), as well as the

accuracy of the solution obtained (i.e., how close the numerical approximation is to the true

solution). Another property that is often desirable is to have chosen continuous-level prop-

erties of the model be satisﬁed exactly in its discretization. Such discretizations are termed

structure-preserving. One possible structure to preserve is to preserve the de Rham exact se-

quence, which essentially means preserving the vector calculus identities curl grad f=0and

div curl F= 0 exactly in the discretization. The de Rham exact sequence plays an important

role in many physical laws, such as incompressibility and Gauss’s law of magnetism.

One way to preserve this de Rham exact sequence is by using discrete exterior calculus

(DEC). DEC is a computational toolkit that creates discrete operators and deﬁnitions that

are analogous to the corresponding operators from multivariate calculus. It has recently

been gaining popularity as a tool for developing numerical methods for solving PDEs in

computational simulations, such as mechanics problems [20], Lie advection [21], and compu-

tational ﬂuid dynamics [22]. In addition to being used as a structure-preserving ﬁnite-element

method, DEC is also widely used in other areas such as computer graphics applications [23]

and geometry processing applications [24].

In DEC, the discrete exterior derivative operator,Dp, is the discrete version of grad, curl,

and div for p= 0, p= 1, and p= 2, respectively. Dpis a np+1 ×npmatrix, where npis the

number of p-cells in the complex (for more detail, see Section 2.3). DEC preserves the de

Rham exact sequence because the discrete exterior derivative operators satisfy Dp+1Dp= 0

for p≥0, which is the discrete version of curl grad f=0and div curl F= 0 for p= 0 and

p= 1, respectively.

Many types of fractional vector calculus possess an analogous exact sequence curlαgradαf=

0and divαcurlαF= 0. However, to the best of our knowledge, no discretization of FPDEs

or FVC preserves this exact sequence. Additionally, despite the usefulness of DEC and the

applicability of fractional calculus and fractional vector calculus, there is rarely any work

on formulating a fractional discrete exterior calculus (FDEC), which generalizes DEC to a

fractional order.

To the best of our knowledge, the only existing work on FDEC is [25], which considered

the following “two-sided” fractional Caputo derivative of order α∈(0,1) of a function

f∈C1[a, b] in 1D:

Dαf(x) := 1

Γ(1 −α)Z[a,b]\{x}

f′(τ)

|x−τ|αdτ. (1)

[25] then deﬁned a fractional discrete exterior derivative by discretizing (1), and then gen-

eralizing the resultant discrete operator to higher dimensions. This results in the following

np+1 ×npmatrix,

Dα

p=W1−α

p+1 Dp,

where W1−α

p+1 ∈Rnp+1×np+1 is a straightforward generalization of the discretization of the

fractional integration of order 1−αpresent in (1) to higher dimensions. (Dα

0is a discretization

of (1) when the complex is one-dimensional.) Unfortunately, the FDEC introduced in [25]

does not satisfy the fractional generalization of the property Dp+1Dp= 0, i.e., Dα

p+1Dα

p̸=

0. Therefore, even if it is a discretization of some type of FVC, it cannot possibly be a

discretization that preserves a fractional de Rham exact sequence. Furthermore, in [25],

since the FDEC in the higher dimensional case is obtained directly from the 1D case, it is

not clear whether it is indeed a discretization of any FVC anymore in higher dimensions,

which may limit its potential in numerical simulations involving FPDEs in higher dimensions.

Indeed, while numerical experiments veriﬁed the expected result that Dα

0converges to Dα

in one dimension, convergence in higher dimensions was not shown; although the authors

compared Dα

0to a 2-sided Caputo gradient ﬁeld of a scalar-valued function in 2-d, convergence

with decreasing mesh size was not shown in this case. This is understandable, since due to

their deﬁnition of the fractional discrete exterior derivative, one would not expect that Dα

should converge to this fractional gradient ﬁeld.

1.1. Contributions

Our goal in this work is to deﬁne an FDEC that does not suﬀer the abovementioned

problems. Namely, we want our FDEC operators to (1) be direct discretizations of a type

of FVC that possesses the exact sequence curlαgradα=0and divαcurlα= 0, and (2) be

structure-preserving, by having the corresponding exact sequence Dα

p+1 Dα

p= 0. Both of

these properties are achieved by rewriting the operators from a type of FVC that does have

the exact sequence (namely, one deﬁned by a fractional nabla operator) as compositions of

fractional integration and exterior derivatives on the continuous level, and then discretizing

these composite operators using DEC on a regular cubical complex. This results in the

following fractional discrete exterior derivative operators,

Dα

p=I1−α

p+1 Dp(I1−α

p)−1, p = 0,1,2,0< α < 1,

where the matrix I1−α

pis a discretization of p-dimensional fractional integration of order 1−α

on the p-cells.

By discretizing a type of FVC, these operators can be implemented in software and

open doors for numerically solving FPDEs. Furthermore, since our approach is structure-

preserving – satisfying the continuous-level properties curlαgradα=0and divαcurlα= 0

exactly on the discrete level – our proposed FDEC operators have high accuracy in discretiz-

ing the corresponding continuous operators and behave similarly even at coarse mesh sizes,

and can potentially increase the ﬁdelity of numerical solutions to FPDEs. In addition, unlike

the usual dense matrices obtained from discretizing the fractional derivatives, the matrices

involved in our discretization are relatively sparse, which enables fast computations. Finally,

since these operators are extensions of DEC, they can provide fractional generalizations of

applications that use DEC.

1.2. Outline

An outline of the paper is as follows. In Section 2, we recall fractional calculus, fractional

vector calculus, and discrete exterior calculus. In Section 3, we present and prove our

reformulation of Tarasov’s FVC and describe its discretization. In Section 4, we show the

convergence of our FDEC to Tarasov’s FVC numerically, and ﬁnally we summarize our work

and suggest possible future work in Section 5.

2. Preliminaries

In this section, we brieﬂy recall fractional calculus ﬁrst, and then introduce the fractional

vector calculus and discrete exterior calculus, which are the building blocks of our FDEC.

2.1. Fractional calculus

First, we discuss fractional calculus. There are many diﬀerent deﬁnitions. This paper

focuses on the Riemann-Liouville fractional integral and the Caputo and Riemann-Liouville

fractional derivatives. Rather than deﬁning the usual fractional integrals and derivatives,

below we deﬁne “partial” fractional integrals and derivatives of a scalar-valued function of

multiple variables, f: Ω →R, where Ω = [x1

min, x1

max]× · · · × [xm

min, xm

max]⊂Rm. This

generalizes the usual deﬁnitions, in the sense that if m= 1, then these deﬁnitions reduce to

the usual one-dimensional deﬁnitions. Furthermore, deﬁnitions of partial fractional integrals

and derivatives that are essentially the same as the ones we will deﬁne can be found in [26].

2.1.1. Fractional integrals and derivatives

For a real number α > 0 and a real-valued function g: [a, b]→R, the left-sided Riemann-

Liouville fractional integral of order αis deﬁned as follows:

aIα

x[x′]g(x′) = 1

Γ(α)Zx

dx′

(x−x′)1−αg(x′) (a≤x≤b)

where Γ(·) denotes the gamma function. Similarly, the left-sided Riemann-Liouville partial

fractional integral with respect to coordinate xjfrom ato bof order αof a function f: Ω →R

is deﬁned as follows:

aIα

b,xj[x′]f:= aIα

b[x′]f(x1, . . . , xj−1, x′, xj+1, . . . , xm)

Γ(α)Zb

f(x1, . . . , xj−1, x′, xj+1, . . . , xm)

(b−x′)1−αdx′,(xj

min ≤a≤b≤xj

max).

Also, we deﬁne

Iα

xjf(x1, . . . , xm) := xj

min

Iα

xj,xj[x′]f.

Next, we recall the left-sided Riemann-Liouville derivative and the left-sided Caputo deriva-

tive. If we let Dn

xjdenote the nth partial derivative with respect to coordinate xj(we drop

the superscript when n= 1), then the left-sided Riemann-Liouville fractional partial deriva-

tive of fwith respect to coordinate xjat a point (x1, . . . , xm) of order α≥0 is deﬁned as,

for n=⌊α⌋+ 1,

RLDα

xjf:= Dn

xjIn−α

xjf=1

Γ(n−α)∂

∂xjnZxj

min

f(x1, . . . , xj−1, x′, xj+1, . . . , xm)

(xj−x′)1−(n−α)dx′.

Naturally, RLDk

xjf=Dk

xjfif k∈N0:= {0,1,2, . . . }.

Similarly, for 0 < α /∈N0, the left-sided Caputo fractional partial derivative with respect

to coordinate xjof order αis deﬁned as, for n=⌊α⌋+ 1,

CDα

xjf:= In−α

xjDn

xjf=1

Γ(n−α)Zxj

min

xjf(x1, . . . , xj−1, x′, xj+1, . . . , xm)

(xj−x′)1−(n−α)dx′.

Otherwise, for integer orders, we deﬁne CDk

xjf:= Dk

xjffor α=k∈N0.

2.1.2. Fractional calculus identities

Here we will present some identities involving the fractional derivatives and integrals de-

ﬁned above that will be used in this work. The ﬁrst such identity is named the “fundamental

theorem of fractional calculus” (FTFC) by [6], which generalizes the fundamental theorem

of calculus to fractional order. Only the ﬁrst part of the FTFC will be presented, since the

other part is not necessary for the results of this paper.

The ﬁrst part of the FTFC states that both the Caputo and Riemann-Liouville derivatives

are left inverse operators of the Riemann-Liouville integration operator from the left. This

generalizes the well-known formula d

dx Rx

af(t)dt =f(x).

Lemma 1. Let f: Ω ⊂Rm→Rbe continuous and let α > 0. Then at any point

(x1, . . . , xm)∈Ωand any j= 1, . . . , m,

RLDα

xjIα

xjf=fand CDα

xjIα

xjf=f.

Proof. Since fis continuous, it is continuous in each variable separately. Then the ﬁrst

equality follows from Lemma 2.4 on page 74 and Lemma 2.9 (b) on page 77 of [26], and the

second equality follows from Lemma 2.21, part (a) on page 95 of [26].

We also have the following result which will be used later:

Lemma 2. Let f: Ω ⊂Rm→Rbe continuous and let α∈(0,1). Then at any point

(x1, . . . , xm)∈Ωand any j= 1, . . . , m,

Iα

RLDα

xjf=f.

Proof. The proof uses the FTFC (Lemma 1) presented above:

Iα

RLDα

xjf=Iα

xjDxjI1−α

xjf=CD1−α

xjI1−α

xjf=f.

The third identity that we will present is that the Riemann-Liouville fractional integral

satisﬁes the so-called semigroup property:

Lemma 3 (Theorem 2.2, [27]).Let f: Ω ⊂Rm→Rbe continuous and let α > 0,β > 0.

Then for any j= 1, . . . , m and any a, b ∈[xj

min, xj

max]with a≤b,

aIα

b[x′]aIβ

x′,xj[x′′]f=aIα+β

b,xj[x′]f.

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：

Structure-PreservingDiscretizationofFractionalVectorCalculususingDiscreteExteriorCalculusAlonJacobsona,XiaozheHua,∗aDepartmentofMathematics,TuftsUniversity,Medford,02155,MA,USAAbstractFractionalvectorcalculusisthebuildingblockofthefractionalpartialdifferentialequationsthatmodelnon-localorlong-rangep...

展开>> 收起<<

Structure-Preserving Discretization of Fractional Vector Calculus using Discrete Exterior Calculus.pdf

共25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！

Structure-Preserving Discretization of Fractional Vector Calculus using Discrete Exterior Calculus

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: