The von Neumann analysis and modified equation approach for finite difference schemes
Introduction
The von Neumann (discrete Fourier) analysis and the modified equation technique have been proven to be two effective and 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 solutions or at low frequencies. The main purpose of this paper is to establish the full connection between them for all wave number modes, for which the modified equation is written in a very compact form just in terms of the amplification factor in the context of von Neumann analysis.
We consider time-dependent partial differential equations with constant coefficients,where represents linear differential operators with constant coefficients . A conventionally used two-level difference scheme takes the form,where the operator B is invertible; or explicitly (1.2) is expressed asHere the discrete value on the grid point approximates the solution of (1.1), , , , represents the spatial mesh size and is the length of time step. This scheme has the stencil sizes of the implicit and explicit parts and , respectively. The coefficients and are constant, satisfying a basic consistency condition,We are particularly concerned with explicit schemes, for which and so . In fact, in our analysis, the implicit scheme (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 consistency describes whether (1.3) approximates (1.1). Once (1.3) is consistent with (1.1), the Lax-equivalence theorem states that 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 propagation can 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 unique amplification 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 modified equation can be interpreted as the actual partial differential equation that the scheme (1.3) solves [24]. This approach is originated in [4] and can be generalized for linear problems with variable coefficients or even nonlinear problems [8], the dissipation and dispersion effects of (1.3) can be understood from the partial differential equation point of view: Even order terms determines the dissipation and odd order terms represents the dispersion. This dissipative and dispersive information derived from the modified equation approach is very heuristic, unfortunately just valid for solutions in smooth regions or at low 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 the analysis 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 establish the full connection between these two approaches. This kind of missions has started earlier though just sporadic. For oscillatory (large-wave-number) parts, a formal analysis is presented in [16] for some specific schemes (e.g., the box scheme) and in [12] for the stable three-point generalized Lax-Friedrichs schemes. The oscillatory behavior of solutions can be illustrated rather 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 theoretical foundation, this use has been accompanied by constant difficulties and results derived from modified equations have sometimes been regarded with apprehension. As a result a situation has arisen where authors either disregard entirely the technique or have an unjustified faith in its scope.” Nevertheless, this approach is very heuristic in the design of numerical schemes and applicable to nonlinear problems [5], [17], [20], [6], [22], and therefore we hope to clarify the connection with the classical discrete Fourier analysis for the full range of wave number modes.
The full connection can be intuitively built as follows. Consider a Fourier mode , is real. The frequency and the amplification factor are linked with , and thereby the dispersion relation for (1.3). Then the (discrete) frequency is a function of k, but also of h and . We will simply write as , holding fixed later on. Then this Fourier mode can be expressed aswhere is the phase speed, and are the real and imaginary parts of , respectively, i.e., . Hence this mode propagates with the phase speed and is dissipated with the rate at each step if , which is just the von Neumann criterion for stability1To understand how the overall shape of the wave amplitudes around a fixed wave number , let us think of k as the wave number around and thereby , and are small. Then with the smoothness assumption of in terms of , one obtainsIn this paper we always use the notation to denote the s-th derivative of in terms of k and similarly for others. Then this Fourier mode propagates, see [21], in the form,by ignoring higher order terms. Here is the group velocity at . Hence the principal part of this mode propagates with the phase speed and is damped with the rate at each time step. The dispersion of (1.3) and other dissipation effects are reflected through .
We can depict the propagation of the overall shape of this wave amplitudes (wave packet) in terms of modified equations. SetWith the change of is the wave packet around . By comparing this with (1.7), and substituting by and by , the modified equation for is expressed in a compact formwhere . In terms of the group velocity , (1.10) is rewritten aswhich implies that the wave packet of propagate with the group velocity , and the behavior is totally determined in terms of this unique quantity. This modified equation approach provides a PDE interpretation of the wave packet, through which we clarify its full connection with the von Neumann analysis: (i) Take the von Neumann analysis for (1.3) to obtain the amplification factor for a fixed wave number . Then we derive the modified equation (1.10) or (1.11) for the wave packet of . (ii) On the other hand, as long as all coefficients of the modified equation at a fixed wave number are known, all derivatives of the amplification factor in the von Neumann analysis are obtained and thus the amplification factor is known. Thus the von Neumann analysis and the modified equation technique are unified in a common framework and we can conveniently adopt either 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 , this modified equation (1.10) becomes that in [24] in a compact form,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 stability or instability of the scheme (1.3) with variable coefficients or nonlinear problems. Indeed, we have developed a heuristic modified equation approach for nonlinear equations [13]. In a forthcoming contribution [14], we will investigate variable coefficient cases.
We organize this paper as follows. In Section 2 we will derive the modified equation rigorously. In Section 3 we present a throughout explanation about the full connection. In Section 4 we provide some simple discussions.
Section snippets
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 and the corresponding frequency . Hence we make an ansatz,This formulation somewhat corresponds to the setting in the Fourier analysisand extends the analysis in [12], [16] for highest frequency modes. As , and .
With the assumption that , it is shown in [21, p.
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 explanation 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 formwhere are constant, . The symbol of the operator is a polynomial, denoted byThen the elementary solution
Some extensions and discussion
The von Neumann analysis has the advantage of simplicity, intuition and easy implementation and it can supply at least local information about properties of dissipation and dispersion, even though there are the severe restriction over constant coefficients of PDE and periodic boundary conditions, and even counterexamples about the frozen coefficient method which leads to the instability of variable coefficients problems [15]. In contrast, the application of the modified equation approach may be
Acknowledgement
Jiequan Li is supported by NSFC (91130021, 11031001), BNU Innovation Funding (2012LZD08) and a special funding from Institute of Applied Physics and Computational Mathematics, Beijing. Both authors thank the reviewer for his helpful suggestions.
References (24)
A critical analysis of the modified equation technique of Warming and Hyett
J. Comput. Phys.
(1990)- et al.
Analysis of a new high resolution upwind compact scheme
J. Comput. Phys.
(2006) - et al.
The validity of the modified equation for nonlinear shock waves
J. Comput. Phys.
(1985) - et al.
Dispersion-relation-preserving finite difference schemes for computational acoustics
J. Comput. Phys.
(1993) Phase error and stability of second order methods for hyperbolic problems. I
J. Comput. Phys.
(1974)- et al.
On the method of modified equations. I: Asymptotic analysis of the Euler forward difference method
Appl. Math. Comput.
(1999) - et al.
The modified equation approach to the stability and accuracy analysis of finite difference methods
J. Comput. Phys.
(1974) - et al.
Numerical integration of the barotropic vorticity equation
Tellus
(1950) Estimation of the relaxation factor of small mesh sizes
Math. Tables Aids Comput.
(1956)- D. Gottlieb, E. Turkel, Phase error and stability of second order methods for hyperbolic problems, II, J. Comput. Phys....
On the scope of the method of modified equations
SIAM J. Sci. Statist. Comput.
On finite-difference approximations and entropy conditions for shocks
Commun. Pure Appl. Math.
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