Applied Mathematics and Computation 225 (2013) 610–621

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

The von Neumann analysis and modified equation approachfor finite difference schemes

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.09.046

⇑ Corresponding author.E-mail addresses: jiequan@bnu.edu.cn, jiequan@mail.cnu.edu.cn (J. Li), yangzhicheng@foxmail.com (Z. Yang).

Jiequan Li ⇑, Zhicheng YangSchool of Mathematical Sciences, Beijing Normal University, Beijing 100875, PR ChinaSchool of Mathematical Sciences, Peking University, Beijing 100871, PR China

a r t i c l e i n f o a b s t r a c t

Keywords:Finite difference schemesVon Neumann analysisModified equation approachAmplification factorFull equivalence

The von Neumann (discrete Fourier) analysis and modified equation technique have beenproven to be two effective tools in the design and analysis of finite difference schemes forlinear and nonlinear problems. The former has merits of simplicity and intuition in practi-cal applications, but only restricted to problems of linear equations with constant coeffi-cients and periodic boundary conditions. The later PDE approach has more extensivepotential to nonlinear problems and error analysis despite its kind of relative complexity.The dissipation and dispersion properties can be observed directly from the PDE point ofview: Even-order terms supply dissipation and odd-order terms reflect dispersion. In thispaper we will show rigorously their full equivalence via the construction of modified equa-tion of two-level finite difference schemes around any wave number only in terms of theamplification factor used in the von Neumann analysis. Such a conclusion fills in the gapbetween these two approaches in literatures.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

The von Neumann (discrete Fourier) analysis and the modified equation technique have been proven to be two effectiveand fundamental tools in the design and analysis of finite difference schemes (even finite element or finite volume methods)for differential equations. The link between them is seemingly well-known and clear, but only in smooth regions of solutionsor at low frequencies. The main purpose of this paper is to establish the full connection between them for all wave numbermodes, for which the modified equation is written in a very compact form just in terms of the amplification factor in the con-text of von Neumann analysis.

We consider time-dependent partial differential equations with constant coefficients,

@v@t¼ Lxv ; Lx ¼ Lð@xÞ ¼

XK

s¼1

as@sx; ð1:1Þ

where Lx represents linear differential operators with constant coefficients as. A conventionally used two-level differencescheme takes the form,

Bunþ1 ¼ Aun; ð1:2Þ

where the operator B is invertible; or explicitly (1.2) is expressed as

1 The

J. Li, Z. Yang / Applied Mathematics and Computation 225 (2013) 610–621 611

XN2

p¼�N1

Bpunþ1jþp ¼

XM2

p¼�M1

Apunjþp: ð1:3Þ

Here the discrete value unj on the grid point ðxj; tnÞ approximates the solution vðxj; tnÞ of (1.1), xj ¼ jh, tn ¼ ns, j 2 Z, n 2 N;h

represents the spatial mesh size and s is the length of time step. This scheme has the stencil sizes of the implicit and explicitparts N ¼ N1 þ N2 þ 1 and M ¼ M1 þM2 þ 1, respectively. The coefficients Ap and Bp are constant, satisfying a basic consis-tency condition,

XN2

p¼�N1

Bp ¼XM2

p¼�M1

Ap: ð1:4Þ

We are particularly concerned with explicit schemes, for which N1 ¼ N2 ¼ 0 and so N ¼ 1. In fact, in our analysis, the implicitscheme (1.3) can be converted into an explicit one by a linear transformation.

To analyze and use the discrete algebraic equation (1.3) in the approximation of the solution of the continuous equation(1.1), four most basic and fundamental issues are the consistency, stability, convergence and wave propagation. The consis-tency describes whether (1.3) approximates (1.1). Once (1.3) is consistent with (1.1), the Lax-equivalence theorem statesthat the solution of (1.3) converges to the solution of (1.1) if and only if the scheme (1.3) is stable [10]. The wave propagationcan be understood by using a so-called dispersion relation. All these issues can be addressed with the von Neumann analysis[2] and the modified equation technique [24]. The von Neumann analysis is relatively simple and intuitive, and the uniqueamplification factor implies the consistency, determines the stability of (1.3) and displays properties of wave propagations,despite the fact that it mostly applies to equations with constant coefficients and periodic boundary condition [16]. The mod-ified equation can be interpreted as the actual partial differential equation that the scheme (1.3) solves [24]. This approach isoriginated in [4] and can be generalized for linear problems with variable coefficients or even nonlinear problems [8], thedissipation and dispersion effects of (1.3) can be understood from the partial differential equation point of view: Even orderterms determines the dissipation and odd order terms represents the dispersion. This dissipative and dispersive informationderived from the modified equation approach is very heuristic, unfortunately just valid for solutions in smooth regions or atlow frequency modes [24]. Therefore the connection with the von Neumann analysis is only restricted there.

However, in many applications, for example, in the simulation of turbulent flows, aeroacoustic computations, or in theanalysis of numerical boundary conditions, the understanding of numerical (e.g., oscillatory) phenomena and many others[3,18], one is more concerned with solution behaviors for a large range of wave numbers, particularly, middle- or large-wave-number, which arises a question how to derive modified equations for any wave number Fourier mode, or establishthe full connection between these two approaches. This kind of missions has started earlier though just sporadic. For oscil-latory (large-wave-number) parts, a formal analysis is presented in [16] for some specific schemes (e.g., the box scheme) andin [12] for the stable three-point generalized Lax-Friedrichs schemes. The oscillatory behavior of solutions can be illustratedrather clearly via the associated modified equation.

As such, the modified equation approach is worth reinspecting. It was pointed out in [7]: ‘‘Because of the lack of any the-oretical foundation, this use has been accompanied by constant difficulties and results derived from modified equations havesometimes been regarded with apprehension. As a result a situation has arisen where authors either disregard entirely thetechnique or have an unjustified faith in its scope.’’ Nevertheless, this approach is very heuristic in the design of numericalschemes and applicable to nonlinear problems [5,17,20,6,22], and therefore we hope to clarify the connection with the clas-sical discrete Fourier analysis for the full range of wave number modes.

The full connection can be intuitively built as follows. Consider a Fourier mode unj ¼ kneijkh, i2 ¼ �1; k is real. The frequency

x and the amplification factor k are linked with k ¼ eixs, and thereby the dispersion relation x ¼ xðk; h; sÞ for (1.3). Then the(discrete) frequency x is a function of k, but also of h and s. We will simply write x as x ¼ xðkÞ, holding h; s fixed later on.Then this Fourier mode can be expressed as

unj ¼ e�xInseik½jh�vpðkÞns�; ð1:5Þ

where vpðkÞ ¼ �xRðkÞ=k is the phase speed, xRðkÞ and xIðkÞ are the real and imaginary parts of xðkÞ, respectively, i.e.,xðkÞ ¼ xRðkÞ þ ixIðkÞ. Hence this mode propagates with the phase speed vpðkÞ and is dissipated with the rate �xIðkÞ at eachstep if xI > 0, which is just the von Neumann criterion for stability1

jkðkÞj 6 1: ð1:6Þ

To understand how the overall shape of the wave amplitudes around a fixed wave number k0, let us think of k as the wavenumber around k0; k ¼ k0 þ ~k and thereby x ¼ x0 þ ~x, x0 ¼ xðk0Þ; ~k and ~x are small. Then with the smoothness assump-tion of ~x in terms of ~k, one obtains

xðkÞ ¼ xðk0Þ þ ~xðk0 þ ~kÞ ¼ x0 þX1s¼1

~xðsÞðk0Þs!

~ks; ~xðk0Þ ¼ 0: ð1:7Þ

von Neumann criterion becomes jkðkÞj 6 1þ Cs for some constant C if a zero term in the form C0u is present in (1.1).

612 J. Li, Z. Yang / Applied Mathematics and Computation 225 (2013) 610–621

In this paper we always use the notation xðsÞðkÞ to denote the s-th derivative of x in terms of k and similarly for others. Thenthis Fourier mode propagates, see [21], in the form,

unj � e�xIðk0Þns � eik0 ½jh�vpðk0Þns� � ei~k½jh�Gðk0Þns� ð1:8Þ

by ignoring higher order terms. Here Gðk0Þ ¼ �d ~xðk0Þ=dk is the group velocity at k0. Hence the principal part of this modepropagates with the phase speed vpðk0Þ and is damped with the rate �xIðk0Þ at each time step. The dispersion of (1.3) andother dissipation effects are reflected through ~xðk0 þ ~kÞ.

We can depict the propagation of the overall shape of this wave amplitudes (wave packet) in terms of modified equations.Set

unj ¼ an

j ~unj ; an

j ¼ eiðnx0sþjk0hÞ; ~unj ¼ eiðn ~xsþj~khÞ; i2 ¼ �1: ð1:9Þ

With the change of ~k; ~unj is the wave packet around ðk0;x0Þ. By comparing this with (1.7), and substituting i~k by @x and i ~x by

@t , the modified equation for ~unj is expressed in a compact form

@t ~u ¼1sX1s¼1

i�s

s!½ln k�ðsÞðk0Þ@s

x~u; ½ln k�ðsÞðk0Þ ¼ i ~xðsÞðk0Þs: ð1:10Þ

where kðsÞðk0Þ ¼ dskðk0Þdks . In terms of the group velocity Gðk0Þ, (1.10) is rewritten as

@t ~uþ Gðk0Þ@x~u ¼ �X1s¼2

i�ðs�1Þ

s!Gðs�1Þðk0Þ@s

x~u; ð1:11Þ

which implies that the wave packet of ~u propagate with the group velocity Gðk0Þ, and the behavior is totally determined interms of this unique quantity. This modified equation approach provides a PDE interpretation of the wave packet, throughwhich we clarify its full connection with the von Neumann analysis: (i) Take the von Neumann analysis for (1.3) to obtain theamplification factor kðk0Þ for a fixed wave number k0. Then we derive the modified equation (1.10) or (1.11) for the wave packet of~u. (ii) On the other hand, as long as all coefficients of the modified equation at a fixed wave number k0 are known, all derivatives ofthe amplification factor in the von Neumann analysis are obtained and thus the amplification factor kðk0Þ is known. Thus the vonNeumann analysis and the modified equation technique are unified in a common framework and we can conveniently adopteither the von Neumann analysis or the modified equation to analyze the dissipation (stability) and dispersion (oscillations)of the scheme (1.3). We point out particularly that as ðk0;w0Þ ¼ ð0; 0Þ, this modified equation (1.10) becomes that in [24] in acompact form,

@tu ¼1sX1s¼1

i�s

s!½ln k�ðsÞð0Þ@s

xu: ð1:12Þ

This implies that the frequency defined by the dispersion relation for (1.3) be just the symbol of the modified equation (1.12).We remark that although the above analysis is made only for (1.1) with constant coefficients, it works for the local sta-

bility or instability of the scheme (1.3) with variable coefficients or nonlinear problems. Indeed, we have developed a heu-ristic modified equation approach for nonlinear equations [13]. In a forthcoming contribution [14], we will investigatevariable coefficient cases.

We organize this paper as follows. In Section 2 we will derive the modified equation rigorously. In Section 3 we present athroughout explanation about the full connection. In Section 4 we provide some simple discussions.

2. Modified equation for a wave packet around a fixed wave number

This section provides the derivation of a modified equation for a wave packet around a fixed wave number k0 and thecorresponding frequency x0 ¼ xðk0Þ. Hence we make an ansatz,

unj ¼ an

j~un

j ; anj ¼ eiðnx0sþjk0hÞ; x0 ¼ xðk0Þ: ð2:1Þ

This formulation somewhat corresponds to the setting in the Fourier analysis

unj ¼ einxseikjh; ~un

j ¼ ein ~xsei~kjh; x ¼ x0 þ ~x; k ¼ k0 þ ~k; ð2:2Þ

and extends the analysis in [12,16] for highest frequency modes. As ðk0;xðk0ÞÞ ¼ ð0;0Þ, anj � 1 and un

j ¼ ~unj .

With the assumption that ~k� 1, it is shown in [21, p. 76] that the solution with the phase velocity Cðk0 þ ~kÞ, in corre-spond to (2.1), has the form

uðx; tÞ ¼ cðx; tÞei~kðx�Gðk0ÞtÞeik0ðx�Cðk0ÞtÞ; ð2:3Þ

for some function jcðx; tÞj 6 1.In the following we will derive the modified equation for ~un

j , from which we can see clearly that ~u propagates with thegroup velocity Gðk0Þ and it is dissipated with the so-called numerical diffusion. Note that the principal part of (2.1) is damped

J. Li, Z. Yang / Applied Mathematics and Computation 225 (2013) 610–621 613

with the rate �xIðk0Þ, which is named as a numerical damping. It is evident that this is consistent with the traditional mod-ified equation at ðx0; k0Þ ¼ ð0;0Þ. See [24].

Proposition 2.1. The modified equation for the finite difference scheme (1.3), in terms of ~unj , is

@t ~u ¼1sX1N¼1

i�N

N!� ½ln k�ðNÞðk0Þ � @N

x ~u; i2 ¼ �1; ð2:4Þ

where the amplification factor kðk0Þ is expressed as

kðk0Þ ¼XN2

p¼�N1

Bpeik0ph

" #�1

�XM2

p¼�M1

Apeik0ph: ð2:5Þ

Note that the group velocity is defined as GðkÞ ¼ � dxðkÞdk ¼ i

sd ln kðkÞ

dk . Then the modified equation can also be expressed as (1.11).

Proof. Here we only prove for the explicit form of (1.3). For the general case, the theory of pseudo-differential operatorshould be applied [9], and the details are omitted here.

We substitute the ansatz (2.1) into (1.3) to obtain

eix0s~unþ1j ¼

XM2

p¼�M1

Apa0p ~un

jþp; a0p ¼ eipk0h: ð2:6Þ

Recall the operators

~uð�; t þ sÞ ¼ es@t ~uð�; tÞ; ~uðxþ ph; �Þ ¼ eph@x ~uðx; �Þ: ð2:7Þ

Then by ignoring the index j, (2.6) can be written as

eix0ses@t ~u ¼XM2

p¼�M1

Apa0peph@x

!~u: ð2:8Þ

Using the standard operator manipulation eph@x ¼P1

s¼0ðph@xÞs

s!and taking into account the value of kðsÞðkÞ at k ¼ k0, s ¼ 0;1; . . .

i�skðsÞðk0Þ ¼XM2

p¼�M1

ApðphÞseipk0h ¼XM2

p¼�M1

Apa0pðphÞs; ð2:9Þ

we find that (2.8) can be written as

ðes@t � 1Þ~u ¼ e�ix0sX1s¼1

i�s

s!kðsÞðk0Þ@s

x

!~u; ð2:10Þ

where we have used eix0s ¼ kðk0Þ. We proceed to use the operator manipulation again

s@t ¼X1m¼1

ð�1Þmþ1

mðes@t � 1Þm; ð2:11Þ

and obtain

s@t ¼X1m¼1

ð�1Þmþ1

me�ix0s

X1s¼1

i�s

s!kðsÞðk0Þ@s

x

!" #m

: ð2:12Þ

Now we denote

DðkÞ :¼X1m¼1

ð�1Þmþ1

m

X1s¼0

i�s

s!½e�ix0skðkÞ � 1�ðsÞ@s

x

" #m

: ð2:13Þ

Here if we use the relation between k and x, kðk0Þ ¼ eix0s such that e�ix0skðk0Þ � 1 ¼ 0, then Dðk0Þ ¼ s@t . Thus we want toprove

DðkÞ ¼X1N¼0

ið�NÞ

N!½ln kðkÞ�ðNÞ@N

x ; ð2:14Þ

and then we obtain (2.4) by setting k ¼ k0.For the simplicity of presentation below, we introduce notations of index sets and indexed functions:Index set Mm,N. For a fixed pair ðm;NÞ, an ðm;NÞ non-negative index set Mm;N ¼ fðrN

0 ; . . . ;rNNÞg is so defined to satisfy:

614 J. Li, Z. Yang / Applied Mathematics and Computation 225 (2013) 610–621

(i) rNs P 0; for 0 6 s 6 N; and rN

s ¼ 0 for s > N.(ii)

PNs¼0rN

s ¼ m.(iii)

PNs¼0srN

s ¼ N.

Indexed function HN. For a smooth function f ðkÞ ¼ e�ix0skðkÞ � 1 with the index ðrN0 ; . . . ;rN

NÞ 2 Mm;N, we denote indexedfunctions

HNðs;rNs ÞðkÞ ¼ CrN

s

m�rNN�����r

Nsþ1

f ðsÞ

s!

� �rNs

ðkÞ; s ¼ 1; . . . ;N; ð2:15Þ

where Crm ¼ m!

r!ðm�rÞ! are the standard binomial coefficients.With the above notations, we have HNðs;0Þ ¼ 1 and write (2.13) as

DðkÞ ¼X1N¼1

X1m¼1

ð�1Þmþ1

m

XðrN

0;...;rN

NÞ2Mm;N

i�NYN

s¼0

HNðs;rNs Þ@

Nx : ð2:16Þ

Note that for N > 1 and k � k0,

k0ðkÞkðkÞ

� �ðN�1Þ

¼ lnð1þ e�ix0skðkÞ � 1Þ� �ðNÞ ¼X1

m¼1

ð�1Þmþ1

mðe�ix0skðkÞ � 1Þm� ðNÞ" #

; ð2:17Þ

where we have used the fact that there exists 0 < d� 1 such that je�ix0skðkÞ � 1j 6 d for k � k0. Therefore in order to prove(2.4), it suffices to show for each pair ðm;NÞ,

X

ðrN0;...;rN

NÞ2Mm;N

YN

s¼0

HNðs;rNs Þ ¼

1N!ðe�ix0skðkÞ � 1Þmh iðNÞ

: ð2:18Þ

Then we complete the proof immediately.We verify (2.18) by induction. In the proof process of (2.18), we denote f ¼ e�ix0skðkÞ � 1. It is easy to see that (2.18) holds

for N ¼ 1. This is because m ¼ 1 if N ¼ 1.Now we assume that (2.18) holds for N. Then we want to prove that it holds also for N þ 1, i.e.,

1ðN þ 1Þ! ½f

m�ðNþ1Þ ¼ 1ðN þ 1Þ!

ddk½f m�ðNÞ ¼

XðrNþ1

0 ;...;rNþ1Nþ1Þ2Mm;Nþ1

YNþ1

s¼0

HNþ1ðs;rNþ1s Þ: ð2:19Þ

For this purpose, we denote by DN‘ the differential operation for rN

‘ P 1; ‘ ¼ 0; . . . ;N,

DN‘

YNs¼0

HNðs;rNs Þ

" #:¼Ys–‘

HNðs;rNs Þ �

ddk

HNð‘;rN‘ Þ: ð2:20Þ

Observe that for 0 6 ‘ 6 N,

DN‘

YN

s¼0

HNðs;rNs Þ

" #¼Y

s–‘;‘þ1

HNðs;rNs Þ � ðrN

‘þ1 þ 1Þð‘þ 1Þ

� CrN‘�1

m�rNN�����ðrN

‘þ1þ1Þ

f ð‘Þ

‘!

� �rN‘�1

� CrN‘þ1þ1

m�rNN�����rN

‘þ2

f ð‘þ1Þ

ð‘þ 1Þ!

� �rN‘þ1þ1

:

ð2:21Þ

Denote

�rNþ1s ¼

rN‘þ1 þ 1; for s ¼ ‘þ 1;

rN‘ � 1; for s ¼ ‘;

rNs ; otherwise:

8><>: ð2:22Þ

Then so defined index ð�rNþ10 ; . . . ; �rNþ1

Nþ1Þ 2 Mm;Nþ1. In fact we calculate:

XNþ1

s¼0s � �rNþ1

s ¼X

s–‘;‘þ1

s � �rNþ1s þ ‘ � �rNþ1

‘ þ ð‘þ 1Þ � �rNþ1‘þ1

¼X

s–‘;‘þ1

s � rNs þ ‘ � ðrN

‘ � 1Þ þ ð‘þ 1Þ � ðrN‘þ1 þ 1Þ

¼XN

s¼0

s � rNs þ 1

¼ N þ 1;

ð2:23Þ

J. Li, Z. Yang / Applied Mathematics and Computation 225 (2013) 610–621 615

and

XNþ1

s¼0

�rNþ1s ¼

XN

s¼0

rNs ¼ m: ð2:24Þ

Thus (2.21) is rewritten as

DN‘

YNs¼0

HNðs;rNs Þ

" #¼ ðrN

‘þ1 þ 1Þð‘þ 1ÞYNþ1

s¼0

HNþ1ðs; �rNþ1s Þ: ð2:25Þ

With careful check of the above process, we find that if ðrN0;j; . . . ;rN

N;jÞ 2 Mm;N satisfies,

rNs;j ¼

�rNþ1j � 1; for s ¼ jþ 1;

�rNþ1j þ 1; for s ¼ j;

�rNþ1s ; otherwise;

8><>: ð2:26Þ

there holds

DNj

YNs¼0

HNðs;rNs;jÞ

" #¼ ð�rNþ1

jþ1Þðjþ 1ÞYNþ1

s¼0

HNþ1ðs; �rNþ1s Þ: ð2:27Þ

We collect all terms with the indices ðrN0;j; . . . ;rN

N;jÞ 2 Mm;N satisfying (2.26) and sum up to obtain

mNj¼0DN

j

YN

s¼0

HNðs;rNs;jÞ

" #¼XN

j¼0

ðjþ 1Þ � �rNþ1jþ1 �

YNþ1

s¼0

HNþ1ðs; �rNþ1s Þ

¼XNþ1

j¼0

j � �rNþ1j �

YNþ1

s¼0

HNþ1ðs; �rNþ1s Þ

¼ ðN þ 1ÞYNþ1

s¼0

HNþ1ðs; �rNþ1s Þ:

ð2:28Þ

This means that the differentiation operation on ½f m�ðNÞ generates partial terms in ½f m�ðNþ1Þ.

On the other hand, for each term in ½f m�ðNþ1Þ, in each index ðrNþ10 ; . . . ;rNþ1

Nþ1Þ 2 Mm;Nþ1, we can find all indices

ðrN0 ; . . . ;rN

NÞ 2 Mm;N satisfying (2.26) such that (2.28) holds. That is, every term of ½f m�ðNþ1Þ can be obtained via the

differentiation operation on the corresponding terms of ½f m�ðNÞ.Thus we complete the proof of (2.19). Hence (2.18) holds. h

3. The Full connection between two approaches

In Section 2 we derived the modified equation (2.4) for the scheme (1.3). In this section we present a throughout expla-nation about its full connection with the von Neumann analysis in terms of consistency, stability and wave propagation,although it is quite clear from the previous discussions. We assume again that (1.1) takes an explicit form

Lxv ¼XK

s¼1

as@sxv; ð3:1Þ

where as are constant, s ¼ 1;2; . . . ;K. The symbol of the operator Lx is a polynomial, denoted by

SLðkÞ ¼ LðikÞ: ð3:2Þ

Then the elementary solution uðx; tÞ of (1.1), with the initial data v0ðxÞ ¼ eikx, is

vðx; tÞ ¼ eikx � eSLðkÞt : ð3:3Þ

We still discuss the explicit case of (1.3) only, and the implicit case can be explained in parallel but more tedious.

3.1. Consistency and accuracy

The concepts of consistency and accuracy in the present context illustrate how well the scheme (1.3) approximates Eq.(1.1), usually in smooth regions of the solutions. Using the von Neumann analysis, we just need to compare the solutionun

j ¼ kneikjh of the scheme (1.3) with the solution (3.3) at ðxj; tnÞ. Write unj ¼ eixnseikjh. Then we compare SLðkÞwith xðkÞ at k ¼ 0,

xðkÞ ¼X1s¼1

xðsÞð0Þs!

ks; ð3:4Þ

616 J. Li, Z. Yang / Applied Mathematics and Computation 225 (2013) 610–621

where we use the fact that xð0Þ ¼ 0. Thus we say the scheme (1.3) is consistent with Eq. (1.1) if there holds

iXK

s¼1

xðsÞð0Þs!

ks ¼ SLðkÞ: ð3:5Þ

If the scheme has accuracy of order q; q P 1, in addition that the consistency condition (3.5) is satisfied, there holds,

xðsÞð0Þ ¼ 0; s ¼ K þ 1; . . . ;K þ q� 1: ð3:6Þ

It is exactly the same using the modified equation (2.4) at small-wave-number k0 ¼ 0,

@tu ¼1sX1s¼1

i�s

s!� ½ln k�ðsÞð0Þ � @s

xu; ð3:7Þ

where we use the consistency condition (1.4) such that kð0Þ ¼ 1. Noting that

kðsÞð0Þ ¼ ishsX

p

psAp; ð3:8Þ

we obtain

½ln k�ðsÞð0Þ ¼ Csishs; ð3:9Þ

where CsðApÞ are real constants, depending on Ap , and they are easily calculated without using the tedious Taylor seriesexpansion in [24]. Then the modified equation (1.12) is written as

@tu ¼1sX1s¼1

Cs

s!hs@s

xu: ð3:10Þ

In comparison of (1.1) and (3.10), we have the following interpretation for consistency and accuracy.Consistency. The scheme (1.3) is consistent with (1.1) if

Cs

s� h

s

s!¼ as; s ¼ 1; . . . ;K; ð3:11Þ

are satisfied.Accuracy. The scheme (1.3) approximates (1.1) with the accuracy of order q; q P 1 in space if

Cshs ¼

s! � ass; 1 6 s 6 K;

0 K < s 6 K þ q� 1

ð3:12Þ

and CqþK hK=s is bounded.

We are particularly interested in the advection (hyperbolic) equation of the form

v t þ cvx ¼ 0; c > 0; ð3:13Þ

since most effective algorithms for hyperbolic systems or convective-dominated problems are related to this simple equa-tion. Assume that the explicit scheme (1.3) for (3.13) is symmetric, M1 ¼ M2 ¼ M. The maximal accuracy is of order 2M. Andit is of order q; q 6 2M if the consistency conditions

XM

p¼�M

pmAp ¼ ð�mÞm; m ¼ 0;1; . . . ; q; ð3:14Þ

are satisfied, where the Courant number is m ¼ cs=h. In fact, (3.14) is equivalent to

½ln k�ðsÞð0Þ ¼�ihm; s ¼ 1;0; 1 < s 6 q:

ð3:15Þ

For upwind-biased schemes, we just need to set Ap ¼ 0 for some p to derive the scheme with the restriction (3.15).

3.2. Stability

Eq. (1.1) is (strictly) dissipative if the real part of SLðkÞ is (negative) non-positive for all non-zero wave number k. Similararguments apply to the finite difference scheme (1.3) and the von Neumann criterion (1.6) put a constraint in order to makethe scheme (1.3) stable. We note that many schemes are not strictly dissipative for some wave number modes. In otherwords, jkðk0Þj ¼ 1 for some k0 2 ½�p=h;p=h�. For instance, the Lax-Friedrichs has the property that kð0Þ ¼ kðp=hÞ ¼ 1. SeeFig. 1. To make clearer the dissipative property of a difference scheme, the von Neumann analysis uses the Taylor seriesexpansion for kðkÞ at such wave numbers.

J. Li, Z. Yang / Applied Mathematics and Computation 225 (2013) 610–621 617

In terms of the modified equation (2.4), the dissipative property of the difference scheme (1.3) becomes obvious, and thestability requirement can be clarified as follows.

(i) Small-wave-number. At small-wave-number k0 ¼ 0; ~unj coincides with un

j and thus the modified equation is (1.12).The dissipation effect of the scheme (1.3) is reflected in terms of even-order derivative of (1.12) or (3.10),

Fig. 1.‘‘b’’; the

�½ln k�ð2sÞð0Þ > 0; or ð�1Þs�1C2s > 0; ð3:16Þ

which is just the explicit expression of lð2sÞ at the small-wave-number in [24, Page 167]. Indeed, the lowest even-orderterm is determinant. As s ¼ 1, it is the usual von Neumann numerical viscosity, which plays a decisive role in dissipatingsmall-wave-number modes.

(ii) Large-wave-number. In many applications, large-wave-number modes are much more concerned. In view of (2.1),the mode is divided into two parts an

j and ~unj , and thereby the dissipation effects are correspondingly distinguished into

two categories:

1. Strong dissipation. This type of dissipation comes from the fixed mode anj ¼ e�xI ðk0Þns � ei½jh�vpðk0Þns�, which is a traveling

wave solution of the PDE,

v t þ vpðk0Þvx ¼ �xIðk0Þv ; ð3:17Þ

where �xIðk0Þv plays a damping effect if xIðk0Þ > 0. Such an effect is thus termed as a numerical damping. For a dissipativescheme jkðk0Þj 6 1, this dissipation forces most Fourier modes to decay exponentially in time. However, it does not take anyeffect at small-wave-number k0 ¼ 0, or for neutrally stable schemes such as the box scheme with jkðkÞj � 1. Even for somestrongly stable schemes such as the Lax-Friedrichs scheme, in which this dissipation effect vanishes at the largest-wave-number since kðp=hÞ ¼ 1, as shown in Fig. 1. Then the weak dissipation described below will play a crucial role.2. Weak dissipation. This type of dissipation can be seen from the modified equation (1.10) for ~u. As usual, consider an ele-

mentary solution of (1.10) in the form, just similar to the small wave number case (1.12),

~uðx; tÞ ¼ ectei~kx: ð3:18Þ

Then c must satisfy

c ¼ 1sX1s¼1

½ln k�ðsÞðk0Þs!

~ks ¼ iX1s¼1

~xðsÞðk0Þs!

~ks: ð3:19Þ

Thus the weak dissipation effect is reflected with the real part of c,

Rec ¼ �X1s¼1

~xðsÞI ðk0Þs!

~ks: ð3:20Þ

This is just the imaginary part of xðkÞ at k ¼ k0, consistent with that by the Taylor series expansion in the von Neumannanalysis.

Comparison of several 3-point first and second order upwind scheme: The Lax–Wendroff scheme labeled ‘‘a’’; the first order upwind scheme labeledWarming-Beam scheme labeled ‘‘c’’ and the Lax–Friedrichs scheme labeled ’’d’’. The Courant number m ¼ cs=h ¼ 0:8, c ¼ 1.

618 J. Li, Z. Yang / Applied Mathematics and Computation 225 (2013) 610–621

In Fig. 1 (a), we display the modulus of the amplification factors for several popular schemes: (a) The Lax-Wendroff scheme;(b) the upwind scheme; (c) the second order Warming-Beam scheme and (d) the Lax-Friedrichs scheme. As is well-known,the Lax-Friedrichs and the upwind schemes are of first order and have strong numerical viscosities at small wave numbers;while the Lax-Wendroff [11] and the Warming-Beam schemes [23] are of second order and the dissipation effect at smallwave numbers are reflected through four-order terms in their respective modified equations. However, at large wave num-bers, the situations change: The Lax-Wendroff scheme has the strongest numerical damping, while the Lax-Friedrichs doesnot have any amplitude error there. Once the data is polluted by the highest frequency mode of the form ð�1Þjþn, the Lax-Friedrichs solution will exhibit oscillatory phenomena, see [12]. These examples highlight the fact that it is necessary to ana-lyze the amplitude error for all wave number modes when investigating the dissipative property of a finite differencescheme.

As an example, the neutrally stable box scheme has no amplitude error at all. That is to say, both the strong and weakdissipation vanish. This scheme takes the compact form for the linear advection equation (3.13), see [16, Page 117],

Fig. 2.respect

ðlxdt þ mltdxÞunþ1=2jþ1=2 ¼ 0; ð3:21Þ

where lxunjþ1=2 ¼ 1

2 ðunj þ un

jþ1Þ, dxunjþ1=2 ¼ un

jþ1 � unj , and similarly for lt and dt . It can be written in a purely transport form

½Dt þ cDx�unþ1=2jþ1=2 ¼ 0; ð3:22Þ

where Dt ¼ l�1t dt=s and Dx ¼ l�1

x dx=h. The amplification factor is

kðkÞ ¼cos 1

2 kh� im sin 12 kh

cos 12 khþ im sin 1

2 kh; ð3:23Þ

and jkðkÞj � 1. In order to stabilize the scheme, the time-average technique is used, see [16, Page 176],

½Dt þ cMtDx�unþ1=2jþ1=2 ¼ 0; ð3:24Þ

with Mt ¼ 1þ ðh� 12ÞsDt . Now the amplification factor of (3.24) is

kðkÞ ¼ ð1� 2mþ 2mhÞeikh þ ð1þ 2m� 2mhÞð1þ 2mhÞeikh þ ð1� 2mhÞ : ð3:25Þ

Such a manipulation actually adds some artificial viscosity at small wave numbers and numerical damping at large wavenumbers, see Fig. 2. The modified equation for (3.24) can be obtained through calculating the amplification factor kðkÞ in(3.25). At k ¼ 0, we have kð0Þ ¼ 1, ½ln k�0ð0Þ ¼ �csi and ½ln k�00ð0Þ ¼ ðihÞ2m2ð2h� 1Þ; . . ., and thus the modified equation is

@tuþ c@xu ¼ c2sðh� 1=2Þ@2x uþ � � � : ð3:26Þ

Hence as h > 1=2, we introduce artificial viscosity which suppresses oscillations caused by low frequency modes. For k – 0,we also have jkðkÞj < 1 uniformly if h > 1=2, as shown in Fig. 2, which provides the numerical damping (strong dissipation).We do not write out the explicit expression of jkðkÞj, which is tedious but does not help too much.

Amplitude errors of the modified box scheme. The Courant number is taken to be m ¼ 0:8 and the parameter h is taken to be 0.5, 0.58, 0.66 and 0.74,ively, for a, b, c and d.

J. Li, Z. Yang / Applied Mathematics and Computation 225 (2013) 610–621 619

3.3. Dispersion and wave propagation

To understand the behavior of the solution by (1.3), it is necessary to look at the dispersion relation between the wavenumber k and the frequency x;x ¼ xðkÞ. For this, there are two main elements: Phase speed and group velocity [19].The phase speed is the rate at which the phase of the wave propagates in space while the group velocity is the velocity withwhich the wave packet propagates through space. The von Neumann analysis works on this well, as we pointed out in theintroduction section that the finite difference solution can be expressed in terms of the phase speed and group velocity (cf.(1.8)),

unj ¼ e�xIðk0Þns � eik0 ½jh�vpðk0Þns� � ei½~kjhþ ~xðk0þ~kÞns�; ð3:27Þ

where we set k ¼ k0 þ ~k, xðkÞ ¼ xðk0Þ þ ~xðk0 þ ~kÞ and vpðk0Þ ¼ �xRðk0Þ=k0. Note again that

~xðk0 þ ~kÞ ¼X1s¼1

~xðsÞðk0Þs!

~ks; and � ~x0ðk0Þ ¼ Gðk0Þ: ð3:28Þ

Thus (1.8) is just (3.27) by ignoring higher order terms of ~x in (3.28). This provides the interpretation of phase speed andgroup velocity through the von Neumann analysis (cf. [21]).

The modified equation (2.4) is obviously obtained via transforming i ~x into @t and i~k into @x formally, or it is written as

@t ~uþ Gðk0Þ@x~u ¼X1s¼2

i�s

s!½ln k�ðsÞ@x~u; ð3:29Þ

i.e., (1.11). Thus it provides a PDE interpretation of group velocity and ~u can be understood as the wave packet around thewave number k0.

Thus we unify the two approaches to depict the wave propagation, particularly the propagation of wave packets.

4. Some extensions and discussion

The von Neumann analysis has the advantage of simplicity, intuition and easy implementation and it can supply at leastlocal information about properties of dissipation and dispersion, even though there are the severe restriction over constantcoefficients of PDE and periodic boundary conditions, and even counterexamples about the frozen coefficient method whichleads to the instability of variable coefficients problems [15]. In contrast, the application of the modified equation approachmay be much more extensive. For example, the modified equation can be derived for the monotone scheme for nonlinearhyperbolic conservation laws [8].

In the sequel, we discuss several aspects concerning the von Neumann analysis and the modified equation approach.

4.1. Multidimensional extension

It is straightforward to extend the one-dimensional case to the following multidimensional case

v t ¼ LxðvÞ ¼XK

s¼1

as � rsxv ð4:1Þ

with periodic boundary conditions, where x ¼ ðx1; . . . ; xdÞ 2 ½�p;p�d,rx ¼ ð@x1 ; . . . ; @xdÞ, as are constant vectors andrs

x is de-fined as convention. With a uniform partition of the domain so that xj ¼ ðj1; . . . ; jdÞh, j‘ 2 f0;1; . . . ; Jg, and h ¼ 2p=J is themesh size, a linear scheme is

X

q2N 1

Bqunþ1j ¼

Xp2N 2

Apunjþp; ð4:2Þ

where Bq;Ap are constants depending on s=h, andN 1 andN 2 are the finite sets to index the neighbors of the point xj (we usebold letters to denote vectors). Then just as one-dimensional case, we use the von Neumann analysis to obtain accuracy, sta-bility and dispersion results by setting the Fourier mode un

j ¼ knðkÞeik�jh to study the amplification factor,

kðkÞ ¼X

q2N 1

Bqeik�ph

" #�1 Xp2N 2

Apeik�ph: ð4:3Þ

The only difference is that we use multiple indices here.The modified equation for (4.2) is also the same with multiple indices by setting

unj ¼ an

j ~unj ; an

j ¼ eiðxðk0Þnsþk0 �jhÞ; ~unj ¼ eið ~xnsþ~k�jhÞ: ð4:4Þ

Then we derive a modified equation for ~unj , which is similar to (2.4) via replacing @x by rx.

620 J. Li, Z. Yang / Applied Mathematics and Computation 225 (2013) 610–621

4.2. System cases

Consider system cases,

vt ¼ LxðvÞ ¼XK

s¼1

as � rsxv ¼: PðrxÞv; ð4:5Þ

where v is a vector-valued function, and as are constant matrices in addition to the aforementioned rx. The von Neumannanalysis can proceed, almost the same as before, by setting

unj ¼ unðkÞeik�jh; ð4:6Þ

as we consider the following finite difference scheme for (4.5),

Xq2N 1

Bqunþ1jþq ¼

Xp2N 2

Apunjþp; ð4:7Þ

where unj is the numerical solution approximating vðxj; tnÞ; unðkÞ is a vector-valued function of k. Then the amplification ma-

trix is

Gðk; sÞ ¼X

q2N 1

Bqeik�qh

" #�1 Xp2N 2

Aqeik�ph: ð4:8Þ

Thus the von Neumann analysis of consistency, stability and dispersion property of (4.7) is just how to compare the ampli-fication matrix (4.8) with the Fourier symbol PðikÞ of (4.5), or more precisely, we need to compare Gðk; sÞ with esPðikÞ. Werefer to [15] for details.

However, the derivation of the modified equation for (4.7) becomes much more involved. This is left for the future.

4.3. Multi-level methods

Multi-level methods, such as the leap-frog scheme, can be considered similarly. The difference is, as pointed out in [1],that in the von Neumann analysis, the amplification factor associated with a wave number generally has ðL� 1Þ roots foran L-level scheme, while the coefficients of the modified equation provide information for only the principal root.

4.4. Variable coefficients and nonlinear cases

For problems with variable coefficients or even nonlinear cases, the von Neumann analysis does not seem straightforwardto apply. A usual strategy is to adopt the frozen coefficient method at a point, even though there are some counterexamplesto display the invalidity of this approach [15]. On the other hand, the modified equation approach seems much flexible toapply for such cases and even for discontinuous solutions (shocks) [5,8]. In this sense, the modified equation has more po-tential in practical applications. In [13] we have developed a heuristic modified approach for nonlinear problems, but justrestricted to conservation laws. For a general problem, we still work in progress [14].

Acknowledgement

Jiequan Li is supported by NSFC (91130021, 11031001), BNU Innovation Funding (2012LZD08) and a special funding fromInstitute of Applied Physics and Computational Mathematics, Beijing. Both authors thank the reviewer for his helpfulsuggestions.

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