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### 知识点总结 #### 一、紧凑型有限差分方法概述本文介绍了一种在计算物理领域具有重大意义的算法——紧凑型有限差分方法（Compact Finite Difference Schemes）。这种方法由Sanjiva K. Lele于1992年提出，并发表在《计算物理学杂志》上。该论文主要探讨了如何通过这种新型的有限差分方法来提高对流体动力学问题中不同尺度特征的数值模拟精度。 #### 二、紧凑型有限差分方法的特点 ##### 2.1 高精度表示 Lele提出的紧凑型有限差分方案能够提供一种类似于谱方法的高精度表示，这意味着它们能够在计算流体力学问题时更准确地捕捉到较短波长的波动。这种高精度不仅限于一阶导数的计算，也适用于二阶及更高阶导数。 ##### 2.2 非均匀网格的应用这些紧凑型有限差分方案可以在非均匀网格上使用，这为解决复杂几何形状的流体流动问题提供了灵活性。在实际应用中，许多流体流动问题涉及复杂的边界条件和不规则的几何形状，非均匀网格的使用能够更好地适应这些问题。 ##### 2.3 多种边界条件的支持除了能够在非均匀网格上工作之外，紧凑型有限差分方法还支持多种边界条件的施加，这进一步扩展了其适用范围。例如，在处理流体动力学问题时，边界条件的选择对于模拟结果的准确性至关重要。 #### 三、应用场景与比较 ##### 3.1 流体力学问题的应用 Lele在其论文中详细讨论了这些紧凑型有限差分方法在流体力学问题中的应用。流体动力学是研究流体运动规律及其与固体相互作用的一门学科，包括湍流、边界层流动等现象。通过对这些现象进行直接数值模拟，可以更深入地理解其内在机制。 ##### 3.2 与其他方法的比较 Lele还将他的紧凑型有限差分方案与其他已知的方法进行了比较，特别是与传统的有限差分方法和谱方法。通过比较可以看出，紧凑型有限差分方法在保持谱方法的高精度的同时，克服了谱方法在处理复杂几何形状和边界条件方面的局限性。 #### 四、关键技术和创新点 ##### 4.1 中心位置的导数计算 Lele还提出了一些针对网格中心位置的导数计算方案，这对于精确模拟流体流动是非常重要的。此外，他还讨论了精确插值和谱似滤波等技术，这些技术同样对于提高数值模拟的准确性至关重要。 ##### 4.2 紧凑型有限差分方法的创新之处 Lele的方法相比传统有限差分方法的主要创新之处在于它能够更准确地表示短波长的波动。这对于处理包含多个空间尺度的现象特别有用，例如湍流流体流动。这种方法使得在保持计算效率的同时，能够更好地捕捉到流体流动中的小尺度结构。 #### 五、结论 Sanjiva K. Lele在1992年发表的这篇关于紧凑型有限差分方法的经典文献对计算物理领域产生了深远的影响。通过提出这种新型的有限差分方法，Lele不仅提高了数值模拟的精度，还拓宽了这类方法在处理复杂流体动力学问题中的应用范围。紧凑型有限差分方法因其在保持谱方法优点的同时又克服了其局限性而受到广泛关注，成为解决复杂流体力学问题的有效工具之一。

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JOURNAL OF COMPUTAl’lONAL PHYSICS

103, l&42 ( 1992)

Compact Finite Difference Schemes with Spectral-like Resolution

SANJIVA

LELE*

Center for Turbulence Research, NASA-Ames Research Center, MS 202A-1, Moffett Field, California 94035

Received February 24, 1990; revised August 2 1, 199 1

Finite difference schemes providing an improved representation of a

range of scales (spectral-like resolution) in the evaluation of first,

second, and higher order derivatives are presented and compared with

well-known schemes. The schemes may be used on non-uniform

meshes and a variety of boundary conditions may be imposed. Schemes

are also presented for derivatives at mid-cell locations, for accurate

interpolation and for spectral-like filtering. Applications to fluid

mechanics problems are discussed. 0 1992 Academic Press. Inc.

1. INTRODUCTION

Many physical phenomena possess a range of space and

time scales, turbulent fluid flows being a common example.

Direct numerical simulations of these processes require all

the relevant scales to be properly represented in the numeri-

cal model. These requirements have led to the development

of spectral methods [l-2]. Some examples of the direct

simulation of turbulent flows by spectral methods may be

found in [3-51. The use of spectral methods is, however,

limited to flows in simple domains and simple boundary

conditions. These difficulties may be overcome by em-

ploying alternative numerical representations. For example,

finite difference schemes or spectral (finite) element schemes

may be used. Direct simulations of turbulent flows using

these alternative schemes is relatively new. Rai and Moin

[6, and references therein for earlier work] present simula-

tions of a turbulent channel flow using a high-order,

upwind-biased finite difference scheme. Work of Patera,

Karniadakis, and their co-workers [ 7-93 illustrates the use

of spectral element methods.

This paper presents finite difference schemes for use on

problems with a range of spatial scales. Compared to the

traditional finite difference approximations the schemes

presented here provide a better representation of the shorter

length scales. This feature brings them closer to the spectral

methods, while the freedom in choosing the mesh geometry

* Present atTliation: Department of Mechanical Engineering and

Department of Aeronautics and Astronautics, Stanford University.

and the boundary conditions is maintained. The emphasis

in this paper is on the resolution characteristics of the

difference approximations rather than their formal accuracy

(i.e., truncation error). By resolution characteristics we

mean the accuracy with which the difference approximation

represents the exact result over the full range of length scales

that can be realized on a given mesh. This notion of resolu-

tion is quantified by means of a Fourier analysis of the dif-

ferencing scheme. It is analogous to, but more general than,

the notion of

intervals per wavelength

used by Swartz and

Wendroff [l&13] and by Kreiss and Oliger [ 141 to com-

pare the

resolving power

of different schemes. The notion of

intervals per wavelength also uses Fourier analysis to quan-

tify phase errors. For very small phase errors the number of

intervals per wavelength needed by a differencing scheme is

sensitive only to the behavior of the longest waves repre-

sented on a mesh. This is precisely the same information as

obtained from the leading order truncation error (formal

accuracy) of the scheme. It should be stressed that the quan-

titative importance of correctly resolving a particular range

of length scales is dependent on the physical problem being

solved as well as on the nature of results being sought from

the numerical calculation.

The organization of the paper is as follows. Section 2

presents the basic schemes for approximating the first and

second derivatives. Schemes for higher derivatives are

described in Appendix A. Compact schemes on cell-

centered mesh are discussed in Appendix B and the

applications to interpolation and filtering are discussed in

Appendix C. Section 3 presents analysis of the schemes,

showing the associated dispersive errors and the anisotropy

of the schemes in multi-dimensions. Comparisons with

conventional finite difference schemes are made throughout

these sections. This analysis leads to a definition of the

resolving efficiency of the differencing schemes. Comments

are also made on the aliasing errors encountered with

nonlinear problems. Section 4 presents a treatment of

boundaries in the derivative approximations. Assessment of

the local boundary errors is presented. Its effect on the

overall scheme is analyzed by means of numerical tests. An

eigenvalue analysis of the complete scheme and the time-

0021~9991/92 35.00

All rights of reproduction in any form reserved.

COMPACT FINITE DIFFERENCE SCHEMES

stepping restrictions for stability are also described in this

section. General remaks on the application of the schemes

are made in Section 5 and some example applications from

fluid mechanics are presented.

2.1. Approximation of First Derivative

Given the values of a function on a set of nodes the finite

difference approximation to the derivative of the function is

expressed as a linear combination of the given function

values. For simplicity consider a uniformly spaced mesh

where the nodes are indexed by

The independent variable

at the nodes is x, = h(

- 1) for 1 <

< N and the function

values at the nodes

(x,) are given. The finite difference

approximation

to the first derivative

(df/dx)(xi)

at the

node

depends on the function values at nodes near

For second- and fourth-order central differences the

approximation

i depends on the sets (h.-, , f,, i) and

(

fifi

fjfi

i, L+ , , fi, J, respectively. In the

spectral

methods, however, the value

off

I depends on all the nodal

values. The Pade or compact finite difference schemes

[ 15-191 mimic this global dependence. The schemes

presented here are generalizations of the Pade scheme.

These generalizations are derived by writing approxima-

tions of the form:

Pf:-*+cI +fl+olfi+1 +Bfi+2

=c~+~-~-~+bf,+2-.f;~2+afi+I-~-I

6h 4h

2h ’

(2.1)

The relations between the coefficients

a, b,

c and a, B are

derived by matching the Taylor series coefficients of various

orders. The first unmatched coefficient determines the

formal truncation error of the approximation (2.1). These

constraints are:

a+b+c=1+2a+2p

(second order)

22b

+ 32c = 2 G (a + 2’p) (fourth order)

24b

+ 34c = 2 z (a + 24/?) (sixth order)

a+26b+3”c=2~(a+2”B)

(eighthorder)

28b

+ 38c = 2 E (a + 28/?)

(tenth order).

(2.1.1)

(2.1.2)

(2.1.3)

(2.1.4)

(2.1.5)

If the dependent variables are periodic in x, then the

system of relations (2.1) written for each node can be solved

together as a linear system of equations for the unknown

derivative values. This linear system is a cyclic penta-

diagonal (tridiagonal) when fl is nonzero (zero). The

general non-periodic case requires additional relations

appropriate for the near boundary nodes. These are

described in Sections 4.1 and 4.2. The resulting linear

system is amenable to efficient numerical solution.

The relation (2.1), along with a mathematically defined

mapping between a non-uniform physical mesh and a

uniform computational mesh, provides derivatives on a

non-uniform mesh. It is also possible to derive relations

analogous to (2.1) for a non-uniform mesh directly (e.g.,

relations corresponding to the traditional Pad& scheme were

derived in [19-211). We now consider the various special

cases of (2.1). In the discussion below at least the first two

of the constraints (2.1.1)-(2.1.5) are imposed. Thus all the

schemes described have at least a fourth-order formal

accuracy.

In Section 3.1 an analysis of the dispersive errors of

schemes (2.1) is presented. This analysis shows the

improved representation of the shorter length scales (i.e.,

spectral-like resolution) of the schemes presented here. The

analysis also leads to schemes with very small dispersive

errors (almost spectral). These are also presented in Sec-

tion 3.1. In the present section we proceed in the traditional

way to classify the differencing schemes generated by (2.1)

in terms of the formal truncation error and the computa-

tional stencil required.

The general relation (2.1) with (2.1.1), (2.1.2) can be

regarded as a three-parameter family of fourth-order

schemes. If the schemes are restricted to /I = 0 a variety of

tridiagonal systems are obtained. For /I # 0 pentadiagonal

schemes are generated. If the additional constraint of sixth-

order formal accuracy is imposed, a two-parameter family

of sixth-order pentadiagonal schemes is obtained. These

may be further specialized into a one-parameter family of

eighth-order pentadiagonal schemes or a single tenth-order

scheme.

First the tridiagonal schemes are described. These are

generated by p = 0. If a further choice of c = 0 is made, a

one-parameter (a) family of fourth-order tridiagonal

schemes is obtained. For these schemes

B=O, a=f(a+2), b=f(4a-1), c=O.

(2.1.6)

The truncation error on the r.h.s. of (2.1) (unless stated

otherwise the term truncation error will be used in this sense

from here on) for this scheme and for other schemes to be

described below are listed in Table I. The stencil sizes

indicated in the table are the maximum stencil sizes needed

within a class of schemes.

As a -+ 0 this family merges into the well-known fourth-

order central difference scheme. Similarly for a = $ the

classical Padt scheme is recovered. Furthermore, for

a = f

SANJIVA

TABLE I

Truncation Error

for the

First Derivative Schemes

Max. 1.h.s. Max. r.h.s.

Scheme stencil size stencil size Truncation error in (2.1)

(2.1.6)

(2.1.7)

(2.1.8)

(2.1.8)&a =;

(2.1.9)

(2.1.10)

(2.1.11)

(2.1.12)

(2.1.13)

(2.1.14)

5 $ (3a - 1) h4f”’

$ h6fc7’

F(-8a+3)hlf”’

$haf’P’

7 ~(-1+3a-l2/?+10c)h~“’

i (9a - 4) hqf”’

-;,(,I

(2a

- 1) h*f ‘9)

;hlOfilK

the leading order truncation error coefficient vanishes and

the scheme is formally sixth-order accurate. Its coefficients

are

a=$, j?=o,

a+, b+, c=Q

(2.1.7)

The specific tridiagonal schemes obtained for a = $ and

a = i

were given by Collatz [22, p. 5381.

With /3 = 0 and c #O the family of schemes (2.1.6) is

extended to a two-parameter family of fourth-order

tridiagonal schemes. Contained within these is a one-

parameter family of sixth-order tridiagonal schemes. For

this (sixth-order) family

P=O, a=$(a+9),

b=h(32a-9), c=&(-3a+l).

(2.1.8)

The sixth-order tridiagonal scheme (2.1.7) is a member of

this family (with c = 0,

= f). This sixth-order family can be

further specialized into an eighth-order scheme by choosing

= $. This is the tridiagonal scheme (p = 0) with the highest

formal accuracy within (2.1).

K. LELE

Pentadiagonal schemes are generated with j? ~0. In

general this fourth-order three-parameter

(a,

j?, and c)

family is given by

a=f(4+2a-168+5c),

b=$(-1+4a+228-8c).

(2.1.9)

Schemes of sixth-order formal accuracy contain two

parameters a and p. They are given by

a=i(9+a-208), b=&(-9+32a+62/?),

c=h(1-3a+12j).

(2.1.10)

The tridiagonal sixth-order family of (2.1.8) is a subclass

within (2.1.10). Another subclass is obtained with B # 0 and

c = 0. This sixth-order pentadiagonal family has

B=h(-1+3a), a=$(8-3a),

b=&(-17+57a), c=O.

(2.1.11)

This family limits to the sixth-order tridiagonal scheme

(2.1.7) as fi -+ 0 or

= $. The leading truncation error coef-

ficient for (2.1.11) vanishes for

= $ yielding an eighth-order

scheme. This eighth-order scheme has

a=$, p=$, a=%, b=g, c=O.

(2.1.12)

This particular scheme is also given by Collatz [22, p. 5381

and analyzed by Swartz and Wendroff [l&13].

By choosing p = $ ( - 3 + 8a) in (2.1.10) a one-parameter

family of eighth-order pentadiagonal schemes is generated.

This eighth-order family has

/I?=$(-3+8a), a=i(12-7a),

b=&(568a-183),

c=&(9a-4).

(2.1.13)

The specific eighth-order schemes obtained earlier viz., (a)

scheme (2.1.8) with

= 2 and (b) scheme (2.1.12), belong to

this one-parameter family.

By choosing

= i in (2.1.13) a tenth-order scheme is

generated. This is the scheme with the highest formal

accuracy amongst the schemes defined by (2.1). The

coefficients of this scheme are

a=;, j?=$j,

a=+$ b=s, c=&.

(2.1.14)

Among the class of derivative approximations repre-

sented by (2.1) those which achieve the highest possible

formal accuracy within each subclass of schemes (denoted

by a specified computational stencil on both the 1.h.s. and

r.h.s. of (2.1)) are precisely the schemes obtained by a

rational (or Padt) approximation of the first derivative

COMPACT FINITE DIFFERENCE SCHEMES

operator.’ This Pad& table is given by Kopal [23]. The

formal rational approximation generates only these specific

members of the multiparameter scheme (2.1.9).

An alternate and more effective way of classifying the

schemes presented here is provided by their Fourier

analysis. The Fourier analysis also quantifies the resolution

characteristics of the schemes. It also provides a way to

“optimize” the scheme from a multi-parameter family.

These issues are fully discussed in Section 3.1.

2.2. Approximation of Second Derivative

The derivation of compact approximations for the second

derivative proceeds exactly analogous to the first derivative.

Once again the starting point is a relation of the form

(2.2)

where

represents the finite difference approximation to

the second derivative at node i. The relations between the

coefficients a, b, c and ~1, /I are derived by matching the

Taylor series coefficients of various orders. The first

unmatched coefficient determines the formal truncation

error of the approximation (2.2). These constraints are:

a+b+c=1+2a+2B (second order) (2.2.1)

a+2*b+3’c=~(a+2*/3) (fourth order)

(2.2.2)

a + 24b + 34c = $ (a + 24/?) (sixth order)

(2.2.3)

a+26b+36c=~(a+2”/I) (eighth order)

(2.2.4)

a+28h+38c=F(a+2X/?) (tenthorder).

(2.2.5)

The form of these constraints is very close to those

derived for the first derivative approximations but the mul-

tiplying factors on the r.h.s. are different. In the following

discussion at least the first two of these constraints are

imposed resulting in schemes with at least a fourth-order

formal accuracy. For dependent variables which are

’ The author is grateful to Dr. K. ShariN for explicitly verifying this

equivalence (via the use of MACSYMA) and pointing out some errors in

the Pad& table of Kopal [23]. The coefficients of h4 and S6 in the expres-

sions for 0:“’ and 0:” (on p. 553) are in error, they should read as & and

A, respectively.

periodic in x the tridiagonal or pentadiagonal system

defined by (2.2) at each node may be solved to yield the

second derivatives. For the non-periodic case additional

relations are required at the boundary (presented in

Section 4.3).

By choosing B = 0 and c = 0 a one-parameter family of

fourth-order schemes is generated. This family has

B=O, c=o,

a=%-~), b=i(-l+lOcc).

(2.2.6)

The truncation error on the r.h.s. of (2.2) for this and other

schemes discussed in this section are listed in Table II. It

may be noted that as a + 0 this family coincides with the

well-known fourth-order central difference scheme. For

a = $ the classical Pade scheme is recovered. For a = & a

sixth-order tridiagonal scheme is obtained. This scheme has

a=&, b=O, a=g, b=$, c=O. (2.2.7)

The particular members obtained with a = h and a = h

were given by Collatz [22, p. 5383.

A three-parameter family of fourth-order schemes is

generated from (2.2) by considering /I # 0 and c # 0. These

satisfy

a=f(4-4a-408+5c),

b=f(-l+lOa+46P-8c).

(2.2.8)

This class of schemes can be further specialized into

a two-parameter family of sixth-order schemes, a one-

parameter family of eighth-order schemes or a single tenth-

TABLE II

Truncation Error for Second Derivative Schemes

Max. 1.h.s. Max. r.h.s.

Scheme stencil size stencil size Truncation error in (2.2)

(2.2.6) 3

(2.2.7) 3

(2.2.8)

(2.2.9)

(2.2.10)

(2.2.11) 5

5 $(ll,-2)hY’6’

5 s jqf’8’

7 ~(-2+ll~-124/?+20c)h~‘6’

7 g (9 - 38a + 214/l) h6f(”

7 *y&-$ j8f”O’

SANJIVA K. LELE

order scheme. The two-parameter sixth-order family is

defined by

6-9cr- 128

-3+24a-68

4 , 5 ’

2- lla+ 1248

(2.2.9)

When the eighth-order constraint is imposed (2.2.9)

reduces to a one-parameter family of eighth-order schemes.

These are defined by

38a-9

/I=-

’

696- 119la

214 ‘=

’

428

2454a - 294 1179a-344

(2.2.10)

535

) c=

2140

The particular eighth-order scheme with a = s corre-

sponding to c=O in (2.2.10) was given by Collatz [22,

p. 5391.

Finally, when the tenth-order constraint is imposed a

single tenth-order scheme is obtained. This scheme defined

j?=&$, a=%, a=$$, b=E, c=& (2.2.11)

has the highest formal accuracy within the class of schemes

defined by (2.2).

As before amongst the schemes defined by (2.2), those

which maximize the formal accuracy (with a prescribed

computational stencil) correspond precisely to the rational

or Padt approximation of the second derivative operator.

These have been given by Kopal [23, pp. 551-5521 in

operator notation. A comparison of schemes (2.2) by means

of Fourier analysis is presented in Section 3.1, where com-

parisons are also made with other well-known schemes.

This analysis brings out the spectral-like resolution of the

schemes described here and also leads to more “optimal”

schemes.

Compact schemes for evaluating higher derivatives are

described in Appendix A. Schemes involving a cell-centered

mesh in evaluating derivatives are described in Appendix B.

Compact schemes for interpolation and liltering are

presented in Appendix C.

3. FOURIER ANALYSIS OF ERRORS

This section presents a Fourier analysis of the errors

associated with the approximations introduced in the last

two sections. Comparison are made with the standard linite-

difference schemes to judge the improvement in the error

characteristics. Formal truncation error of the differencing

schemes were given in the preceding sections. The use of

Fourier analysis to characterize the errors of difference

approximations is described extensively in [24]. It is a

classical technique for comparing differencing schemes. It

was used by Roberts and Weiss [25], Fromm [26], Oliger

and Kreiss [14], Orszag [27-281 and by Swartz and

Wendroff [l&13]. Fourier analysis of the standard PadC

scheme was presented in [ 181 and comparisons were made

with the second- and fourth-order central differences.

The Fourier analysis provides an effective way to quantify

the resolution characteristics of the differencing approxima-

tions. This quantification may be used to further guide an

optimization of the differencing schemes. In the following

section the differencing errors are analyzed in terms of

dispersion or phase error and anisotropy (in multi-

dimensions). Comparisons are made throughout with the

standard difference formulae. Examples of optimization of

the schemes based on the resolution characteristics are also

presented. All differencing approximations (for the interior

nodes) studied here are of central difference form, thus there

are no dissipative errors (from the differencing of conser-

vative terms). The treatment of boundary errors (for non-

periodic problems) is presented in Section 4, where the

approximations appropriate for the near boundary nodes

are also introduced. Local errors introduced by the bound-

ary scheme are discussed along with the schemes and their

effect on the global accuracy is presented in Section 4.4. This

requires direct numerical tests on the performance of the

first derivative schemes. Analysis of the stability properties

of the overall scheme is deferred to Section 4.5 which is

followed by a summary of the time step restrictions for

stable explicit time advancement in Section 4.6.

3.1. Fourier Analysis of Differencing Errors

For the purposes of Fourier analysis the dependent

variables are assumed to be periodic over the domain [0, L]

of the independent variable, i.e.,fi =

fN+

, and

h = L/N.

The

dependent variables may be decomposed into their Fourier

coefficients

f(x)= ‘=c”:, fkexp(F),

k= -N/2

(3.1.1)

where

i=F

1. Since the dependent variables are real-

valued, the Fourier

COefflCientS

satisfy

fk =f?k

for 1 <

d N/2 and f0 = f,*,

where * denotes the complex

conjugate.

It is convenient to introduce a scaled wavenumber

w = 2nkhJL

2zk/N

and a scaled coordinate s =

x/h.

The

Fourier modes in terms of these are simply exp(iws). The

domain of the scaled wavenumber w is [0, rr]. The exact

first derivative of (3.1.1) (with respect to s) generates a func-

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