On the spectral and conservation properties of nonlinear discretization operators.

Transcription

1 On the spectral and conservation properties of nonlinear discretization operators. D. Fauconnier, E. Dick Department of Flow, Heat and Combustion Mechanics, Ghent University, St. Pietersnieuwstraat 41, B-9 Ghent, Belgium Abstract Following the study of Pirozzoli 1], the obective of the present work is to provide a detailed theoretical analysis of the spectral properties and the conservation properties of nonlinear finite difference discretizations. First, a Nonlinear Spectral Analysis (NSA) is proposed in order to study the statistical behaviour of the modified wavenumber of a nonlinear finite difference operator, for a large set of synthetic scalar fields with prescribed energy spectrum and random phase. Second, the necessary conditions for local and global conservation of momentum and kinetic energy are derived and verified for nonlinear discretizations. Because the nonlinear mechanisms result in a violation of the energy conservation conditions, the NSA is used to quantify the energy imbalance. Third, the effect of aliasing errors due to the nonlinearity is analyzed. Finally, the theoretical observations are verified for two simple, thought relevant, numerical simulations. Keywords: Nonlinear finite difference schemes, spectral properties, modified wavenumber, Nonlinear Spectral Analysis, conservation properties. 1. Introduction In Computational Fluid Dynamics (CFD) and Computational Aero-Acoustics (CAA), the use of nonlinear discretizations for the spatial derivative operators in the differential equations, is widespread. Although these nonlinear discretization operators come in various forms and may serve different purposes, they have a nonlinear mechanism in common that results in the application of different discretization stencils in different nodes throughout the computational domain. The particular selection of the discretization stencil in a node of interest, is typically based on a smoothness-criterion of the solution in that node and surrounding nodes. The oldest, and best known nonlinear discretization scheme is the upwind scheme, proposed by Courant et al. ] in 195. The upwind strategy involves a discretization stencil which is biased in the direction determined by the sign of the transporting velocity, i.e. the upwind direction. As long as the Courant-Friederichs-Lewy condition is satisfied, the scheme remains stable due to the intrinsic numerical dissipation of the eccentric stencils. Although this intrinsic stability is the most important asset of upwind schemes, the excessive dissipation is often considered undesirable for accurate numerical simulation of e.g. turbulent flows or acoustic wave propagation. Since the sign of the transporting velocity may alternate in the computational domain, the discretization stencil will alternate with it, making the upwind scheme de facto nonlinear. Another class of nonlinear schemes, introduced in the early 8 s by Harten 3], are the Total Variation Diminishing (TVD) schemes. These TVD schemes involve flux or slope limiters which must ensure the monotonicity of the solution at all times. If the solution is sufficiently smooth in a node of interest, a nd -order central discretization stencil is selected, which intrinsically lacks numerical dissipation. However, if the solution is not smooth on the computational grid, displaying spurious oscillations (e.g. Gibbs phenomenon, odd-even decoupling, etc.), steep gradients or discontinuities (e.g. shocks, etc.), an appropriate 1 st -or nd -order upwind discretization is enforced to ensure the monotonicity of the solution. Although this makes TVD schemes highly stable, these methods have the disadvantage address: dieter.fauconnier@ugent.be (D. Fauconnier) Preprint submitted to Journal of Computational Physics February 3, 11

2 that the 1 st -order upwind scheme is also engaged near smooth extrema, leading to unnecessary dissipation 4]. In order to obtain a higher order of accuracy, Harten et al. 5] introduced the Essentially Non-Oscillatory (ENO) discretization schemes. Here, the discretization stencils in any node of interest are constructed by selecting the smoothest interpolating polynomial of the solution, from a hierarchy of candidate interpolating polynomials with varying stencil support. Once the smoothest interpolant is known, a smooth approximation of the derivative is obtained readily. Later, Weighted Essentially Non-Oscillatory (WENO) schemes were developed, which use a convex combination of all candidate interpolating polynomials in the defined hierarchy 4]. Since the discretization stencils of both ENO and WENO schemes can differ from node to node, they are nonlinear by definition. Recently Fauconnier et al. 6] developed a class of Dynamic Finite Difference (DFD) scheme. In contrast to the previously discussed schemes, the obective is not to guard the smoothness of the solution, but to obtain higher accuracy by minimizing the dispersion error of the central discretization stencil. This is achieved by reformulating the basic central discretization stencil as a function of an adustable coefficient, which is then determined by a least square minimization on two grid resolutions. If the entire domain is taken into account in the minimization procedure, one obtains the same optimized discretization stencil in each node of the domain. However, if only the vicinity of the node of interest is taken as minimization region, then the optimized stencil varies from node to node, leading to a nonlinear discretization throughout the domain. We refer to 6, 7] for further information. Despite the fact that nonlinear schemes are widely used in CFD, e.g. in applications concerning compressible flows, shock capturing, shock-turbulence interaction and wave propagation, their nonlinear behaviour and its impact on the solution, has been discussed in only a few studies. In 1, Ladeinde et al. 8] compared the resolution of the small scale structures in the high-wavenumber region of the turbulent compressible energy spectrum, for three finite difference schemes: a linear compact scheme, an ENO scheme and a WENO scheme. They noticed that the nonlinear stencil adaptation of the ENO and WENO schemes leads to the generation of numerical small-scale turbulence, polluting the tail of the energy spectrum and limiting the useful resolution range of these schemes. Removing these spurious modes, improved the results. In 6, Pirozzoli 1] investigated in more detail the spectral properties of some shockcapturing schemes such as TVD, ENO and WENO. The author motivated that even though seldom stated explicitly, numerical tests show that the genuinely nonlinear mechanisms underlying shock-capturing schemes have a dramatic impact upon the computed solution, and their actual behavior may be very different from the one predicted on purely linear grounds. In order to study some effects of the nonlinearity on a solution in wavenumber space, an Approximate Dispersion Relation for nonlinear schemes was used, which relied on the extraction of a modified wavenumber from the convection of a sinusoidal wave with a certain wavenumber over a small distance. However, some caveats were reported in 1]. Indeed, this quasi-linear method did not consider solutions that contain an entire spectrum of scales. Further, the impact of the spurious modes, generated by the nonlinear schemes, and other nonlinear interactions on the evolution of a transported field is not considered. Based on this work, Bogey et al. 9] argued that in order to assess the quality of the solutions, in particular in aeroacoustic studies, there is an urgent need of analysing the spectral properties of the shock-capturing schemes in the Fourier space. In Fauconnier et al. 6], the spectral behaviour of the nonlinear DFD scheme was investigated. The authors showed analytically that the modified wavenumber of the nonlinear scheme equals that of the 6 th -order tridiagonal compact scheme for a single wave solution. A dual-wave analysis was presented in which the transfer functions of both schemes were calculated for a field with two independent wave components with equal amplitudes. A small loss of accuracy was noticed, due to the nonlinear interactions of the two waves. This was confirmed by the numerical study on the viscous Burgers equation. They concluded that the nonlinear mechanism produces spurious scales in the entire wavenumber range, severely polluting the solution and substantially reducing the accuracy of the Burgers solution. This is in accordance with the work of Ladeinde et al. 8]. Following the motivation of Pirozzoli 1] and Bogey et al. 9], the present work is an attempt to further analyze in detail, the spectral properties, as well as conservation properties, for a representative selection of nonlinear schemes. Moreover, the impact of the nonlinearity on the solution of a few elementary problems is investigated. First we analyze the spectral properties of a selection of nonlinear schemes, for a synthetic field containing a spectrum of Fourier modes up to a certain cutoff. The modes amplitudes satisfy a certain scaling law, whereas the modes random phases are uniformly distributed. This gives insight about the nonlinear interactions and their impact on the modified wavenumber and the energy spectrum. Second, we investigate analytically the conservation properties of nonlinear schemes. We verify whether the nonlinear schemes satisfy the necessary conditions for the conservation of momentum and kinetic energy in the simple, though essential and relevant, case of a one-dimensional linear convection equation.

3 We show that nonlinear schemes can be formulated in order to satisfy the requirements for momentum conservation. However, we prove analytically that due to the nonlinear mechanism, the conditions for the conservation of kinetic energy can never be satisfied. Since nonlinear schemes violate the conditions for conservation of kinetic energy, we also investigate the impact of the nonlinearity on the energy balance, by means of a nonlinear spectral analysis. A third topic, related to the previous one, concerns the occurrences of aliasing errors due to the nonlinear interactions. More specifically, we analyze the impact of aliasing on the energy balance. We emphasize that the nonlinear spectral analysis, used to assess the impact of nonlinearity on various properties, is only valid for a single evaluation (i.e. a single iteration) of the nonlinear discretization scheme for a predefined scalar field, and does not take into account nonlinear influences due to repeated evaluation (i.e. multiple iterations). Finally, we verify our findings in two fundamental simulations, i.e. the linear convection equation and a viscous Burgers equation with energy-conserving forcing.. A selection of Spatially Nonlinear Finite Difference Schemes Consider the one-dimensional hyperbolic conservation law for a scalar field u (x, t), x, L], u t + φ =, (1) x in which the flux φ is determined as φ = vu, with v (x, t) the transporting velocity. In the present work, 4 classes of nonlinear discretizations for the convective term φ x are investigated, i.e. upwind schemes, Total Variation Diminishing (TVD) schemes, Weighted Essentially Non-Oscillatory (WENO) schemes and nonlinear Dynamic Finite Difference (DFD) schemes. For each class, one or more representative examples are selected in order to investigate the nonlinearity in detail. In the current investigation, we select the 1 st, nd -and3 rd -order upwind scheme, the TVD scheme with the Chakravarthy limiter, the 5 th -order WENO scheme and the nd -order nonlinear DFD scheme, for further analysis. These nonlinear finite difference schemes are described below. 1. In general, the discretization of the convective term in a node i of the computational grid with uniform grid spacing Δx, can be expressed as the differential balance of fluxes at the intermediate node positions i ± 1, i.e. δφ δx = φ i+ 1 φ i 1. () i Δx Using the transport velocity v at the intermediate positions i ± 1, i.e. v i+ 1 = v i + v i+1 ; v i 1 = v i + v i 1, (3) the 1 st, nd and 3 rd -order upwind discretizations for the convective term in (1) are defined respectively as ] ] v + u i+ 1 i + v u i+ 1 i+1 v + u i 1 i 1 + v u i 1 i δφ δx i = δφ δx i = δφ δx i = (4) Δ ] ] v + (3u i+ 1 i u i 1 ) + v (3u i+ 1 i+1 u i+ ) v + (3u i 1 i 1 u i ) + v (3u i 1 i u i+1 ) Δ ] (5) v + (u i+ 1 i+1 + 5u i u i 1 ) + v (u i+ 1 i + 5u i+1 u i+ ) 6Δ ] v + (u i 1 i + 5u i 1 u i ) + v (u i 1 i 1 + 5u i u i+1 ), 6Δ (6) in which v + = max (v, ) and v = min (v, ) denote the positive and negative contributions of the transport velocity. It is well known that these upwind discretizations can be interpreted as a combination of a central 3

4 finite difference discretization with a dissipative correction that ensures stability. For example, the first order upwind scheme can be reformulated as δφ v i+ 1 δx = u i+1 + ( v i+ 1 v ) i 1 ui v i 1 u i 1 i Δ ( ) ( ) v + u i+ 1 i+1 v + + v + u i+ 1 i 1 i + v + u i 1 i 1 v u Δ + i+ 1 i+1 v + v u i+ 1 i 1 i + v u i 1 i 1 Δ. (7) Obviously, the nonlinearity in upwind schemes is only activated if the transporting velocity v has positive and negative values in the domain. Hence, for linear convection equations with constant transport velocity v, all upwind schemes reduce to standard linear backward (v > ) or forward (v < ) finite difference schemes.. Using the same definitions for v ±, v ± i+ 1 i 1 v+ i+ 1, the TVD discretization of the convective term is defined as ( ui + ψ(r i) (u i+1 u i ) ) ( ) + v i+ u 1 i+1 ψ 1 r i+1 (u i+1 u i ) δφ δx = i Δ ( v+ ui 1 + ψ(r i 1) i 1 (u i u i 1 ) ) ( ) + v i u 1 i ψ 1 r i (u i u i 1 ), (8) Δ in which the ratio of consecutive differences in a node i is r i = u i u i 1. (9) u i+1 u i The slope limiter function ψ (r) = max (min (r, ), ), (1) proposed by Chakravarthy et al. 1], is used in this work. If ψ (r) = (sharp gradient, opposite slopes or zero gradient), the scheme reduces to a 1 st -order upwind scheme, whereas for ψ (r) = r, the scheme is equivalent witha nd -order upwind scheme. In the particular case of r = 1 (smooth solution, equal slopes), the scheme turns to a nd -order central scheme, whereas for ψ (r) =, the scheme switches to a 1 st -order downwind scheme. Since this limiter function has a large interval in r, i.e. r, the scheme reduces to a nd -order upwind discretization most of the time. Remark that the Chakravarthy limiter resembles the minmod limiter, which is defined as ψ (r) = max (min (r, 1), ). Similarly to the upwind schemes, the TVD scheme can be reinterpreted as a central scheme in combination with a dissipative correction operator. Due to the limiter function, the dissipative correction is weaker than for a nd -order upwind scheme. Note that even for a constant transporting velocity v, this schemes remains nonlinear. 3. The 5 th -order Weighted Essentially Non-Oscillatory (WENO) scheme 4, 11] for the convective term, is constructed by first splitting the flux φ in expression () into a positive and negative contribution, φ + (u) and φ (u), such that ( ) ( ) δφ φ + + φ φ + + φ i+ δx = 1 i+ 1 i 1 i 1. (11) i Δx The flux splitting can be accomplished in different ways. However, often, the Lax-Friederichs flux splitting is used 4], i.e. φ + (u) = 1 (φ (u) + θu) = 1 (vu + θu) (1) φ (u) = 1 (φ (u) θu) = 1 (vu θu), (13) 4

5 ( φ ) with θ = max u = max ( v ), over the relevant range. The positive and negative numerical fluxes are obtained as a weighting between 3 basic flux definitions, giving φ + φ + = w + i 7φ + i 1 + ] 11φ+ i φ + i+ 1 + w + i 1 + 5φ + i + φ + ] i+1 φ w + i + 5φ + i+1 ] 1φ+ i+ (14) φ φ = w i+1 7φ i+ + ] φ i+3 φ i+ 1 + w i + 5φ i+1 ] φ i+ φ 1 + w i 1 + 5φ i + φ ] i+1 (15) φ + φ + = w + i 3 7φ + i + ] 11φ+ i 1 φ + i 1 + w + i + 5φ + i 1 + ] φ+ i φ w + i 1 + 5φ + i 1φ + ] i+1 (16) φ φ = w i 7φ i+1 + ] φ i+ φ i 1 + w i 1 + 5φ i φ ] i+1 φ 1 + w i + 5φ i 1 + ] φ i. (17) In order to achieve a convex combination of Essentially Non-Oscillatory (ENO) stencils, the normalized weighting coefficients w ± k are determined by w ± k = α ± k = α ± k α ± + α± 1 +, k =, 1, (18) α± c k ɛ + β ±, k =, 1,, (19) k in which the β ± k represent local indicators of the smoothness of the flux, and ɛ = 1 6 is a small positive number, necessary to avoid singularities in the method. For the 5 th -order WENO scheme, c = 1 1, c 1 = 6 1, c = 3 1. The smoothness indicators are determined by the relations β + = 13 φ + 1 i φ + i 1 + ] 1 φ+ i + 4 φ + i 1 φ + i + φ i+1] β + 1 = 13 1 β + = 13 1 β = 13 1 β 1 = 13 1 β = 13 1 φ + i 4φ + i 1 + 3φ+ i ] () ] φ + 4 i 1 φ + i+1 (1) φ + i φ + i i+] φ+ + 3φ + 4 i 4φ + i+1 + ] φ+ i+ () φ i+1 φ i+ + 1 i+3] φ + 3φ 4 i+1 4φ i+ + ] φ i+3 (3) φ i φ i ] i+] φ + φ 4 i φ i+ (4) φ i 1 φ i + φi+1] 1 ] + φ 4 i 1 4φ i + 3φ i+1. (5) Analogous relations are obtained for w ± k, α± k and β ± k. Note that in contrast to the upwind schemes, WENO schemes remain nonlinear even for constant transport velocity v. 4. The last nonlinear finite difference scheme that is investigated in this work, is the central nonlinear DFD scheme of Fauconnier et al. 6]. This scheme is defined by δφ φi+1 + c i+1 (φ i+ φ i+1 + φ i ) ] φ i 1 + c i 1 (φ i φ i 1 + φ i ) ] δx = + O ( Δ ), (6) i Δ with φ = vu. The dynamic coefficient c is determined via the transported scalar u (x) as 6] c i = 1 1 ( ( ) ). (7) 6 ui+ 4u i+1 + 6u i 4u i 1 + u i 1 + f max min,, 3 u i+1 u i + u i 1 5

6 The parameter f is obtained by calibrating the scheme such that it obtains maximum accuracy for a field with a prescribed inertial range spectrum defined in expression (8). This leads to values around f =.1. It was shown in 6], that for f =., the nonlinear DFD scheme has the same modified wavenumber as the standard 6 th -order tridiagonal compact scheme, for a single-wave field. Therefore, we will adopt f =. in this work. For more information about the mathematical derivation and ustification of the nonlinear DFD scheme, we refer to Fauconnier et al. 6]. This scheme does not depend on a transporting velocity, and thus remains nonlinear in each point for a constant transport velocity v. We emphasize that, in contrast to the upwind schemes, the TVD scheme and the WENO scheme, the DFD scheme (6) consists of a central discretization operator in combination with a dispersive correction operator. As mentioned in the introduction, the goal of the DFD scheme is to increase the accuracy of the discretization by adding a dispersive correction, instead of preserving the stability by adding a dissipative operator to the basic discretization. Hence, the spectral behaviour of the upwind schemes, the TVD scheme and the WENO scheme on the one hand, and the nonlinear DFD scheme on the other hand, will be substantially different and must not be compared. Here, the only obective is to examine the influence of the nonlinearity on e.g. the spectral characteristics, not comparison of the spectral characteristics between the various schemes defined above. 3. Nonlinear Spectral Analysis 3.1. Methodology For the selected nonlinear schemes, we first investigate the impact of the nonlinearity on the modified wavenumber and the energy spectrum, by means of an artificial or synthetic field for u (x), containing a predefined spectrum of Fourier modes. The one-dimensional synthetic field u (x), x < π is constructed such that its energy spectrum E u (κ) follows a predefined inertial range scaling { κ α <κ κ E u (κ) c, (8) κ c <κ κ max whereas the phase ϕ u (κ), κ κ c, of each mode is chosen randomly in the interval π, +π] according to a uniform distribution U π, +π], i.e. { U π, +π] <κ κc ϕ u (κ) =. (9) κ c <κ κ max In the previous expressions, κ max = π Δ denotes the maximum wavenumber of an equidistant computational grid with grid spacing Δ. Defining the Fourier transform of u (x) as the synthetic field is then determined by satisfying the conditions û (κ) = F {u (x)} = a (κ) + ib (κ), (3) E u (κ) = a (κ) + b (κ) = U κ α, <κ κ c (31) ( ) b (κ) ϕ u (κ) = arctan, <κ κ c, (3) a (κ) in which U is a proportionality factor that can be chosen freely. At κ = we impose E u () =, i.e. the mean of u (x) is zero. Extracting a (κ) and b (κ), leads to the expression of the synthetic field in Fourier space ± (1 U + i tan (ϕ u (κ))) κ α/ <κ κ c û (κ) = 1 + tan (ϕ u (κ)). (33) + i κ c <κ κ max Taking the inverse Fourier transform, one readily obtains the synthetic field in physical space u (x). Once the field u (x) is obtained, the modified wavenumber κ of any linear or nonlinear discretization δu δx of the first derivative, is determined as κ (κ) = 1 { δu } iû (κ) F, κ κ c. (34) δx 6

7 For linear schemes, analytical expressions can easily be derived for the modified wavenumber. For nonlinear schemes, however, such analytical expressions are prohibitively complex, involving the evaluation of convolution integrals. Hence, expression (34) is considered as a good and workable alternative. The phase function ϕ u (κ) in expression (33) must be considered as a particular k th random realization ϕ k (κ) of a uniform distribution of the phase in the interval π, π]. Hence, the modified wavenumber κ = κ (κ, ϕ k ) depends on that particular realization ϕ k (κ) of the phase function ϕ u (κ). Therefore, it is useful to provide statistical information about κ as well, such as the mean modified wavenumber and the variance. This is achieved by determining the modified wavenumber for N r realizations of the synthetic field u (x), and calculating respectively the mean modified wavenumber, and the standard deviation as μ κ (κ) = σ κ (κ) = 1 N r N r k= 1 N r 1 κ (κ, ϕ k ) (35) N r k= κ (κ, ϕ k ) μ κ (κ) ] 1/. (36) In the following, we consider the slope of the turbulent energy spectrum, i.e. α = 5/3. The cutoff wavenumber is chosen systematically as the maximum wavenumber of the computational grid, i.e. κ c = κ max = π Δ. In some particular situations, we also investigate κ c = 3 κ max, which matches the dealiasing limit of Orszag 1]. The statistical information is extracted, using N r = 1 5 realizations of the synthetic field. This is enough to ensure that the mean and the variation are representative. In other words, taking more realizations does not alter significantly μ κ (κ) nor σ κ (κ). Aside from the analysis of the modified wavenumber, it is also interesting to investigate the impact of the nonlinearity on the energy spectrum of the synthetic field. This allows to study the energetic impact of the nonlinear mechanisms and to visualize the spurious scales that are generated by the nonlinear schemes at the tail of the energy spectrum 8, 1]. In analogy with the analytical relation u = iκû, (37) x we can define a reconstructed synthetic field u r (x) as δu δx = iκû r. (38) Then the energy spectrum of the reconstructed field is obtained as E ur (κ, ϕ k ) = = 1 κ E δu (κ, ϕ δx k ), κ (39) κ κ E u (κ, ϕ k ) <κ κ c (4) spurious modes κ c <κ κ max Obviously, the energy spectrum of u r is directly proportional to the energy spectrum of the derivative. Remark that the reconstructed energy spectrum depends on the particular realization ϕ k of the phase function ϕ u (κ). Hence we determine again statistical information, i.e. a mean energy spectrum and the variance on the energy spectrum, using μ Eur (κ) = σ Eur (κ) = 1 N r N r k= 1 N r 1 E ur (κ, ϕ k ) (41) N r k= Eur (κ, ϕ k ) μ (κ)] Eur 1/. (4) In the following, we refer to the described methodology as a Nonlinear Spectral Analysis (NSA). 7

8 R(κ )/κmax I(κ )/κmax R(Δκ )/κmax I(Δκ )/κmax Figure 1: Modified wavenumber for the 3 rd -order upwind scheme: Dispersion characteristic (upperleft); dissipation characteristic (upperright). Error on the dispersion characteristic (lowerleft); error on the dissipation characteristic (lowerright). 1 st -order backward ( ); nd -order backward ( ); 3 rd -order backward ( ); κ (κ, ϕ k ) (-+-); μ κ (κ) ( ); μ κ (κ) ± σ κ (κ) ( ). 8

9 3.. Analysis Before the Nonlinear Spectral Analysis can be applied to the different nonlinear schemes, defined in section, the discrete derivative δu δx of the synthetic field u (x), used e.g. in expression (34), must be defined properly. Therefore, we choose the transporting velocity v in the flux definition φ of equation (1) to be unity in the entire domain. Indeed, the nonlinear TVD scheme, the 5 th -order WENO scheme and the DFD scheme for δφ δx, given by expressions (8), (11) and (6), reduce to the equivalent nonlinear discretizations for δu δx. However, since the nonlinearity in upwind schemes (4), (5) and (6) is only activated if the transporting velocity v has positive and negative values in the domain, the choice v = 1, reduces all upwind schemes to standard linear backward finite difference schemes. As a consequence, the Nonlinear Spectral Analysis, proposed in section 3.1, is redundant for upwind schemes when assuming a constant v, because the modified wavenumber can then be obtained analytically. For the Nonlinear Spectral Analysis of the upwind schemes in this work, we intentionally assume that v + = ( ) max sgn (u), and v = ( ) min sgn (u),,inorder to have an idea about the impact of the nonlinearity on the modified wavenumber. In the current analysis, we only focus on the 3 rd -order upwind scheme. The NSA results for the 3 rd -order upwind scheme, the TVD-scheme, the 5 th -order WENO-scheme and the nonlinear DFD scheme are shown in Figures 1,, 4 and 5. The results show the instantaneous modified wavenumber κ (κ, ϕ k ) for the arbitrary k th synthetic field realization with κ c = κ max, the mean modified wavenumber μ κ (κ) for N r = 1 5 synthetic field realizations, and the spreading of the results bounded by the lines μ κ (κ) ± σ κ (κ). Further, also the error on the modified wavenumber, and the spreading of the error around the mean is shown on a logarithmic scale, in order to focus on the asymptotic behaviour of the schemes in the low wavenumber range. For the 3 rd -order upwind scheme, the TVD-scheme and the 5 th -order WENO scheme, the spectral characteristics are compared with those of the 1 st -, nd -and3 rd -order backward finite difference schemes, whereas for the DFD scheme, we compare the characteristics with those of the 4 th -and6 th -order central schemes and the 6 th -order tridiagonal compact scheme. Figures, 4 also show the quasi-linear Approximate Dispersion Relation (ADR) of Pirozzoli 1], respectively for the TVD-scheme with the minmod and Chakravarthy limiters, and for the 5 th -order WENO-scheme. First, we consider the 3 rd -order upwind scheme in figure 1. It is seen that both the real and imaginary part of the modified wavenumber κ (κ, ϕ k ) display a very erratic behaviour, which increases with increasing wavenumber. Looking at the dispersion property, the mean modified wavenumber μ κ (κ) collapses, as expected, with the modified wavenumber of the 3 rd -order backward scheme in almost the entire wavenumber range. However, the error on the real part reveals that 3 rd -order scaling is lost if κ. In fact, both κ (κ, ϕ k ) and μ κ (κ) tend to the scaling of the 1 st - and nd -order backward schemes. For the dissipation property, the mean modified wavenumber μ κ (κ) is nearly zero in the entire wavenumber range, although the discretization stencil is never unbiased. This somewhat misleading result stems from the choice to define synthetic fields, with zero mean component, and use it for this particular upwind case as the transporting velocity. In other words, since the field u (x) has a zero mean, the 3 rd -order forward discretization stencil is applied as much as the backward discretization stencil, resulting in an average zero-dissipation. Hence, if the mean component was slightly positive, then, the mean modified wavenumber μ κ (κ) would shift more towards that of the 3 rd -order backward scheme. The error on the imaginary part reveals that in the asymptotic range κ, κ (κ, ϕ k ) and μ κ (κ) obtain the scaling of the 1 st -and nd -order backward schemes. Based on Figure 1, one is tempted to assume that the dispersion and the dissipation characteristics are quasi randomly distributed around the mean modified wavenumber μ κ (κ). Based on the standard deviation, however, it is clear that for approximately 95% of all realizations, κ (κ, ϕ k ) is confined to the area between the curves μ κ (κ) ± σ κ (κ). The spreading is quite large, indicating that occurrences of a negative dispersion relation is possible in the range κ>.6κ max. The spreading on the dissipation relation, shows that the dissipation can be higher than that of a 3 rd -order forward or backward scheme, in the entire wavenumber range. Moreover, in the low wavenumber range, the dissipation is close to that of a 1 st - order forward or backward scheme. The conclusions for other upwind schemes, which are not shown, are very similar. Next, we investigate the TVD-scheme in figure. Again, we notice that both the real and imaginary parts of the modified wavenumber κ (κ, ϕ k ) display erratic behaviour, similarly to the upwind schemes. Looking at the dispersion property, the mean modified wavenumber μ κ (κ) (almost) collapses with the modified wavenumber of the 3 rd -order backward scheme in the medium and high wavenumber range. However, the error on the real part reveals that the 9

10 R(κ )/κmax.4. I(κ )/κmax R(Δκ )/κmax I(Δκ )/κmax Figure : Modified wavenumber for the TVD scheme: Dispersion characteristic (upperleft); dissipation characteristic (upperright). Error on the dispersion characteristic (lowerleft); error on the dissipation characteristic (lowerright). 1 st -order backward ( ); nd -order backward ( ); 3 rd -order backward ( ); ADR of TVD + minmod limiter 1] ( ); ADR of TVD + Chakravarthy limiter ( ); κ (κ, ϕ k ) (-+-); μ κ (κ) ( ); μ κ (κ) ± σ κ (κ) ( ). 1

11 pdf(κ/κmax = 1/4 ) pdf(κ/κmax = /3 ) I( )&R( )) I( )&R( )) Figure 3: Distribution of modes for the TVD scheme: distribution at wavenumber = 1/4 (left) and = /3 (right). Histograms represents the distribution of the dispersion characteristic (light grey), and the distribution of the dissipation characteristic (dark grey). accuracy is not maintained in the asymptotic range κ. In fact, both κ (κ, ϕ k ) and μ κ (κ) tend to the scaling of the 1 st -and nd -order backward schemes. Remark that the quasi-linear ADR results of Pirozzoli 1] for the dispersion relation of the TVD-scheme with limiter (1), are clearly different from the mean dispersion relation obtained with the NSA. For the dissipation property, the mean modified wavenumber μ κ (κ) follows a specific path in between the modified wavenumbers of the 1 st -and3 rd -order backward schemes in the medium and high wavenumber range. Again, the error on the imaginary part reveals that in the asymptotic range κ, κ (κ, ϕ k ) and μ κ (κ) tend to a 1 st -order scaling. Note that for the dissipation, the quasi-linear ADR results of Pirozzoli 1] are surprisingly close to the results obtained with the NSA. Further, the spreading around the mean modified wavenumber is significant, indicating that occurrences of a negative dispersion relation are possible for κ>.6κ max. Moreover, the spreading shows that in the high wavenumber range, the numerical dissipation could become smaller than that of a 3 rd -order backward scheme, but could also become significantly larger than that of a 1 st -order backward scheme. In order to have a good idea of the real distribution of κ (κ, ϕ k ) around its mean, the Probability Density Function is shown in Figure 3 for two wavenumbers, i.e. = 1/4 and = /3. The bell-shaped curves clearly indicate a normal or Gaussian distribution of the modified wavenumber around the mean. We conclude that the mean dispersive and dissipative properties of the TVD scheme behave as expected in the medium and high-wavenumber range, despite reduction of the order of accuracy for κ. The latter, caused by the nonlinearity of the scheme, is in contrast with the formal nd -order accuracy, aimed at in the construction of the TVD-scheme. For the 5 th -order WENO scheme, erratic behaviour is again observed for both the real and imaginary parts of the modified wavenumber κ (κ, ϕ k ). Looking at the dispersion property, the mean modified wavenumber μ κ (κ) does not reach the accuracy of the 3 rd -order backward scheme in the entire wavenumber range. Moreover, the error on the real part reveals that in the asymptotic range κ, the scaling of κ (κ, ϕ k ) and μ κ (κ) tend to only the scaling of the 1 st -and nd -order backward schemes. These results are in contrast to the quasi-linear ADR results of Pirozzoli 1] for the 5 th -order WENO scheme, which show a 5 th -order scaling for κ. Note that the scheme is constructed to have a 5 th -order accuracy. In the high wavenumber range, the NSA predicts a much better behaviour of the dispersion relation than the quasi-linear ADR, although it does not reach the quality of the 3 rd -order scheme. For the dissipation property, the mean modified wavenumber μ κ (κ) lies between the modified wavenumbers of the 1 st -and 3 rd -order backward schemes in the medium and high wavenumber range. The error on the imaginary part shows that 11

12 R(κ )/κmax.4. I(κ )/κmax R(Δκ )/κmax I(Δκ )/κmax Figure 4: Modified wavenumber for the 5 th -order WENO scheme: Dispersion characteristic (upperleft); dissipation characteristic (upperright). Error on the dispersion characteristic (lowerleft); error on the dissipation characteristic (lowerright). 1 st -order backward ( ); nd -order backward ( ); 3 rd -order backward ( ); ADR of 5 th -order WENO scheme 1] ( ); κ (κ, ϕ k ) (-+-); μ κ (κ) ( ); μ κ (κ) ± σ κ (κ) ( ). 1

13 for κ, κ (κ, ϕ k ) and μ κ (κ) tend to a 1 st -order scaling, which is in contrast with the quasi-linear ADR results of Pirozzoli 1]. Indeed, the ADR results, show a different behaviour of the dissipation characteristics in comparison with the NSA results, in the entire wavenumber range. This emphasizes the importance of contributions of nonlinear interactions to the spectral properties. The spreading around the mean modified wavenumber indicates that negative dispersion can occur for κ>.6κ max. The dissipation can become lower than that of a 3 rd -order backward scheme, but can also be significantly larger than that of a 1 st -order backward scheme. We conclude that the dispersive and dissipative properties of the 5 th -order WENO scheme are strongly determined by the nonlinear mechanism, resulting in different spectral characteristics in the entire wavenumber range, than intended by construction, or expected from linear analysis. The Probability Density Function shows a normal distribution of the modified wavenumber around the mean, similarly to that of the TVD scheme. Therefore it is not shown. Finally, we consider the nonlinear DFD-scheme in figure 5. Once more, one observes the erratic behaviour of the modified wavenumber κ (κ, ϕ k ), that increases with increasing wavenumber. In comparison with the upwind schemes, the TVD scheme and the WENO scheme, the amplitude of these disturbances is much smaller. Looking at the dispersion property, the mean modified wavenumber μ κ (κ) does not collapse with the characteristic of the 6 th -order tridiagonal compact scheme, which is in contrast to the conclusion from the analytical single-wave analysis in the work of Fauconnier et al. 6]. In the high wavenumber range, the mean dispersion relation μ κ (κ) displays a pretty good behaviour, as it is better than that of the 6 th -order central scheme, although, it does not reach the accuracy of the 6 th -order tridiagonal compact scheme. However, the error on the real part reveals once more that the accuracy is not maintained in the asymptotic range κ. Whereas μ κ (κ) tends to nd -order scaling, κ (κ, ϕ k ) seem to indicate only 1 st -order scaling. For the dissipation property, the mean modified wavenumber μ κ (κ) is very close to zero, which is not surprising since the nonlinear DFD has a central discretization stencil, regardless the value of the preceding coefficient c. Hence in the mean, no dissipation should occur. However, the small symmetric spreading predicts an equal probability for a small amount of dissipation or anti-dissipation of the scheme for any Fourier mode. In the asymptotic range κ, κ (κ, ϕ k ) and μ κ (κ) display 1 st -order scaling. The Probability Density Function learned again that the modified wavenumber is normally distributed, and is therefore not shown. For the nonlinear DFD scheme we conclude that the dispersion properties are of significant lower quality for small wavenumbers than one would theoretically expect from the construction of this scheme. Again this is due to the nonlinearity. As discussed in section 3.1, the energy spectra of the reconstructed fields give us some more information about the impact of the nonlinearity on the magnitude of the different Fourier modes, and the generation of spurious scales above the cutoff wavenumber. Figure 6 shows these energy spectra for respectively the TVD-scheme, the 5 th -order WENO scheme and the DFD scheme, using cutoff wavenumbers κ c = κ max and κ c = κ max /. All plots show that the nonlinearity is responsible for spurious disturbances in the entire spectrum of Fourier modes. As a consequence, some modes are energized, whereas others are damped. Moreover, the scattering of these disturbances is relatively large around the cutoff wavenumber κ c, as indicated by the curves μ Eur (κ) ± σ Eur (κ). Hence, the smallest regular modes are affected the most by this nonlinear behaviour. In the case κ c /κ max = 1/, one clearly notices the creation of spurious modes above the cutoff wavenumber κ c. This supports the conclusion of both Ladeinde et al. 8], who stated that ENO and WENO schemes generate numerical turbulence, and Fauconnier et al. 6] who found similar results for the nonlinear DFD scheme. For the TVD scheme and the WENO scheme, these spurious scales are approximately two magnitude smaller than the smallest scales at the cutoff wavenumber, whereas for the DFD scheme, the spurious modes are approximately four magnitudes smaller than the smallest scales at the cutoff wavenumber. Hence, the relative importance of these spurious modes in comparison with the regular modes, seem to depend strongly on the type of discretization. Note that in case of a linear or nonlinear convection equation, these spurious modes interact with all other modes. This may have a severe impact on the evolution of the solution. Moreover, the interactions of the spurious scales in the wavenumber region κ c κ max inevitably lead to aliasing errors, affectingevenmorethe evolution of the solution. These issues will be investigated later in this work. 4. Conservation properties of nonlinear schemes In the discussion above, it was shown that the nonlinear mechanisms have a strong impact on the modified wavenumber and the energy spectrum of the reconstructed synthetic field. Not only does it seem to affect all ex- 13

14 1 R(κ )/κmax I(κ )/κmax R(Δκ )/κmax I(Δκ )/κmax Figure 5: Modified wavenumber for the DFD scheme: Dispersion characteristic (upperleft); dissipation characteristic (upperright). Error on the dispersion characteristic (lowerleft); error on the dissipation characteristic (lowerright). 4 th -order central ( ); 6 th -order central ( ); 6 th -order tridiagonal compact ( ); κ (κ, ϕ k ) (-+-); μ κ (κ) ( ); μ κ (κ) ± σ κ (κ) ( ). 14

15 1 1 Eδuδx(κ)/κ Eδuδx(κ)/κ Eδuδx(κ)/κ Eδuδx(κ)/κ Eδuδx(κ)/κ Eδuδx(κ)/κ Figure 6: Energy spectrum of u r : κ c /κ max = 1(left column); κ c /κ max = 1/ (right column). TVD scheme (1 st row); 5 th -order WENO scheme ( nd row); DFD scheme (3 rd row). Symbols for the TVD and WENO schemes: κ 5/3 ( ); 1 st -order upwind ( ); nd -order upwind ( ); 3 rd -order upwind ( ). Symbols for the DFD-scheme: κ 5/3 ( ); 4 th -order central ( ); 6 th -order central ( ); 6 th -order tridiagonal compact ( ). Scheme of interest: E ur (κ, ϕ k ) (-+-); μ Eur (κ) ( ); μ Eur (κ) ± σ Eur (κ) ( ). 15

16 isting scales, i.e. large scales as well as small scales, it also creates spurious modes that were not present in the original synthetic field. This immediately raises some questions about the conservation of momentum, and more importantly, the net conservation of energy during a simulation. Therefore, it is interesting to verify the effect of the nonlinear mechanism on the conservation properties, in order to know its impact on the solution in a real simulation. In particular the net energy contribution of the nonlinearity to the energy balance is of concern. In the following, we investigate the conservation properties of nonlinear schemes. We restrict ourselves in this work to the analysis of the elementary, though relevant, one-dimensional linear convection equation. Analysis of the conservation properties in the one-dimensional and linear case is an essential prerequisite before analyzing conservation for more-dimensional and/or nonlinear cases. If conservation is violated in a linear, one-dimensional setup, conservation will certainly fail in nonlinear and higher-dimensional setups. Note that because the linear convection equation is assumed, the current analysis is not directly representative for upwind schemes. In this work we follow the approach of Wissink 13], as it is convenient for our purposes One-dimensional linear convection equation Consider the following linear convection equation for the periodically transported field u (x, t), x, π], t u t + v u =, (43) x in which v represents the constant convective transport velocity. The global conservation of momentum u (x, t) is expressed as d dt π u (x, t) dx = whereas the global conservation of the kinetic energy u (x, t) /isgivenby d dt π u (x, t) dx = Using expression (43), both expressions are reformulated as and π π π π u dx =, (44) t u u dx =. (45) t v u dx =, t, (46) x vu u dx =, t. (47) x Assuming a non-uniform grid with N nodes x i, i N, the continuum expressions are written discretely as v δu δx Δx i =, t, (48) i i and δu vu i δx Δx i =, t. (49) i i 16

17 where Δx i denotes the integration weight which gives an approximation of the grid spacing at node i, i. In order to derive the conservation properties of nonlinear schemes, we define the discrete derivative δu δx in a node i as δu r δx = β i, u i+, (5) i = r in which the coefficients β i, determine the stencil in a particular node i. Obviously, this formulation accounts for the nodal variation of the discretization stencils, which is the main issue of interest in this work. The only condition stencil β i, is assumed to satisfy, is that it gives an approximation for the first derivative. This implies that the trivial condition that the derivative of a constant field should be exact, i.e. r= r β i, =, i. It is emphasized that the summation over = r r does not necessarily imply that the discretization is central, since the integer r merely denotes the maximum of the right and left stencil widths. We further define the stencil width as w = r + 1. Using definition (5), the discrete conservation expressions (48) and (49) are written as i r β i, u i+ Δx i =, t, (51) = r and r β i, u i+ u i Δx i =, t. (5) i = r Working out (51) by collecting the factors for each node i, that is, changing the index notations in expression (51) by replacing index i + with i (and as a consequence i with i ), and replacing afterwards the index with, one obtains the expression for the global conservation of momentum r β i+, Δx i+ u i =, t. (53) i = r Repeating the latter procedure for expression (5), yields i r β i+, u i u i+ Δx i+ =, t. (54) = r Obviously, the conditions (5) and (54) for the conservation of kinetic energy must be satisfied at the same time, since they share the mutual term u i u i+. Hence, their sum must also be equal to zero, leading to following expression for the global conservation of kinetic energy i r ( ) βi, Δx i + β i+, Δx i+ ui u i+ =, t. (55) = r First we focus on expression (53) for the global momentum conservation. In order to satisfy expression (53) for any arbitrary realization of u (x i ), the vector with elements r = r β i+, Δx i+ must satisfy any linear combination of the Null Space of u (x i ), i.e. Null (u 1,..., u n ]), where the Null Space is a set of n 1 basis vectors with size n 1, each satisfying (53). Thus, for n 1 arbitrary weighting coefficients, the corresponding linear combination of Null (u 1,..., u n ]), results in n conditions for the coefficients r = r β i+, Δx i+, i. Hence, this implies that in order to satisfy global conservation of momentum, the momentum deficit in n 1 computational cells must be compensated in the remaining single cell. In other words, any linear combination of the Null Space, gives a sufficient condition for global momentum conservation. However, local conservation of momentum requires that each cell has a zero loss of momentum. In other words, only the trivial solution, in which n 1 weighting coefficients are chosen exactly zero, gives a sufficient and necessary condition for local momentum conservation. Obviously, local conservation also 17

18 guarantees global conservation in a periodic domain. Hence, we conclude that the necessary and sufficient condition for local conservation of momentum is given by r β i+, Δx i+ =, i. (56) = r Applying the same reasoning to expression (55), leads to the conclusion that the necessary and sufficient condition for local conservation of kinetic energy is given by Assume now a coefficient matrix γ β i, Δx i =, i (57) β i, Δx i + β i+, Δx i+ =, i,. (58) γ i,i+ = β i, Δx i, i, r r. (59) Note that in a periodic domain with n nodes, positions i + < 1, correspond to n + i + whereas positions i + > n correspond to i + n. Then, condition (56) for the momentum conservation expresses that the sum of each column in this coefficient matrix γ must be zero. This is known as the telescoping property. Condition (57) for the energy conservation expresses that the diagonal of the coefficient matrix γ must be zero, whereas condition (58) states that matrix γ must be skew-symmetric, i.e. γ = γ T. (6) Although the stencil coefficients β i, can vary from node to node, conditions (56), (57) and (58) impose certain restrictions, as shown hereafter. Considering again the nonlinear schemes defined in section (), it is understood that all of these schemes can be reinterpreted somehow as a spatial weighting of a series of linear (node-independent) flux definitions. In analogy with this observation, we continue with the very general assumption that a general discretization stencil β i, of the derivative could be interpreted as a spatial weighting of a series of node-independent stencils, defined on an uniform grid. For the sake of simplicity, we restrict ourselves to a spatial weighting of two basic discretizations β (1) and β (), giving β i, = p i,i+ β (1) + q i,i+ β (), (61) in which p i,i+ and q i,i+ represent the weighting functions that depend on the node index i as well as on the discretization index. At this point, the only condition to which the standard, Taylor-based finite difference discretizations β (1) and β () are subected, is that their combination (61) must result into a valid approximation for the first derivative. Substituting (61) into expression (56) for the momentum conservation, yields r = r ( pi+,i β (1) + q i+,iβ () ) Δxi+ =, i. (6) Analogously, substitution of (61) into expressions (57) and (58) for the conservation of kinetic energy yields p i,i β (1) Δx i + q i,i β () Δx i =, i (63) p i,i+ β (1) Δx i + p i+,i β (1) Δx i+ + q i,i+ β () Δx i + q i+,i β () Δx i+ =, i,. (64) Momentum conservation First we focus on the condition (6) for the momentum conservation. Assume that the standard linear discretization stencils β (1) and β () both satisfy condition (56) for the conservation of momentum on an equidistant grid, such that r r β (1) =, β () =, i. (65) = r = r 18

19 Condition (6) is then satisfied, if the factors p i+,i Δx i+ and q i+,i Δx i+ are independent of index, i. In other words, for a particular index i, the weightings p i+,i Δx i+ and q i+,i Δx i+ must be constant in. However, both weightings do not have to be constant in i. Assuming that one obtains the following relation for the weighting functions p i+,i Δx i+ = p i,i (66) q i+,i Δx i+ = q i,i, (67) p i+,i = q i+,i = p i,i Δx i+ (68) q i,i,. Δx i+ (69) Transformation of these expressions to index (i, i + ), i.e. replacing index i with i and afterwards with, results finally into the conditions p i,i+ = p i+,i+ Δx i (7) q i,i+ = q i+,i+ Δx i. (71) Obviously, the weighting functions p i,i+ and q i,i+ in definition (61) must include the contribution Δxi 1 which compensates for the grid non-uniformity, since the basic discretization stencils β (1) and β () do not account for this. Apart from that, the remaining weighting of the two basic discretizations must be written as β i, = 1 p Δx i+,i+ β (1) i in order to conserve momentum. This means that the coefficient matrix is given by ] + q i+,i+ β(). (7) γ i,i+ = p i+,i+ β(1) + q i+,i+ β(), i, r r. (73) Hence, all weightings p i+,i+ and q i+,i+ in a specific column i +, must have a constant value, equal to that of the diagonal element at position (i +, i + ). In conclusion, any weighting of two or more basic finite difference approximations results in a loss of momentum conservation, unless it is expressed as (7). Obviously, the nonlinear finite difference schemes (4), (5), (6), (8), (11) and (6), defined in section (), are expressed in the divergence formulation, and thus satisfy the conditions for conservation of momentum Energy conservation For the analysis of the conditions (63) and (64) for energy conservation, we assume that the standard finite difference discretization stencils β (1) and β () both satisfy conditions (57) and (58) for the conservation of kinetic energy on an equidistant grid, such that = β (1) = β (), (74) = β (1) + β (1) = β () + β (). (75) Note that this implies that both standard discretizations are central and skew-symmetric. It is obvious that under this assumption, condition (63) is satisfied automatically. However, the only reasonable condition to satisfy (64) i,is p i,i+ Δx i = p i+,i Δx i+ i, (76) q i,i+ Δx i = q i+,i Δx i+ i,. (77) 19