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Quasilinear analysis of dispersion relation preservation for nonlinear schemes
Advances in Aerodynamics volume 4, Article number: 14 (2022)
Abstract
In numerical simulations of complex flows with discontinuities, it is necessary to use nonlinear schemes. The spectrum of the scheme used has a significant impact on the resolution and stability of the computation. Based on the approximate dispersion relation method, we combine the corresponding spectral property with the dispersion relation preservation proposed by De and Eswaran (J Comput Phys 218:398–416, 2006) and propose a quasilinear dispersion relation preservation (QLGRP) analysis method, through which the group velocity of the nonlinear scheme can be determined. In particular, we derive the group velocity property when a highorder Runge–Kutta scheme is used and compare the performance of different time schemes with QLGRP. The rationality of the QLGRP method is verified by a numerical simulation and the discrete Fourier transform method. To further evaluate the performance of a nonlinear scheme in finding the group velocity, new hyperbolic equations are designed. The validity of QLGRP and the group velocity preservation of several schemes are investigated using two examples of the equation for onedimensional wave propagation and the new hyperbolic equations. The results show that the QLGRP method integrated with highorder time schemes can determine the group velocity for nonlinear schemes and evaluate their performance reasonably and efficiently.
1 Introduction
Nonlinear approaches, such as the weighted essentially nonoscillatory (WENO) scheme, are widely used to calculate complex flow fields with discontinuities, such as shock waves. Because of the mathematics of such schemes, truncation errors can affect the convergence rate of the computation and the spectral properties affect the deviation of the Fourier modes of the numerical results from the exact solution. Therefore, studying the spectral properties of nonlinear schemes is important and meaningful when developing highorder nonlinear schemes.
Tam and Webb [1] suggested that it was necessary to consider the spectral properties besides using the standard Taylor series method in the construction of difference schemes. Specifically, by using the Fourier transformation of the governing equation, they proposed a basic method for analyzing the dispersion and dissipation relation of a linear scheme and developed a dispersion relation preservation (DRP) scheme. Moreover, this method can also be used to evaluate the spectral properties of different linear schemes and to construct new schemes. However, the method is inappropriate for analyzing the spectral properties of nonlinear schemes. To remedy this deficiency, Pirozzoli [2] studied the amplitude evolution of a disturbing mode after a tiny propagation period through a discrete Fourier transform (DFT), through which he proposed an approximate dispersion relation (ADR) analysis. ADR [2] gives an estimate of the total error generated by a nonlinear scheme and provides a method for analyzing its spectral properties. However, there are still some issues. For example, the time scheme can introduce errors, and if the grid number is not chosen carefully, there can be unexpected jumps in the spectral distribution [3].
Observing these problems, Mao et al. [3] proposed an improved ADR without time discretization called ADRNT. Using the assumption of a tiny time period, they transferred the effect of the temporal derivative into the contribution of the spatial derivative. The researchers reported [3] that ADRNT not only reduced the computational cost but also avoided the error due to the time discretization, as occurs in ADR. Mao et al. [3] also suggested ways to avoid jumping points when using ADR, i.e., the size of the grid should be twice some large prime or take the average over a large number of results with different initial phases.
The dispersion relation usually reflects the phase velocity of a spatial scheme. For practical problems with wave propagation, it is necessary to consider the group velocity, which is the one the energy propagates at [4]. De and Eswaran [5] analyzed the dispersion relation property of schemes from the perspective of the group velocity, which we refer to as the GRP method or just GRP. They pointed out that GRP was important when evaluating the spectral properties of linear schemes. In the space of the reduced wave number and frequency, group velocity preservation (GVP) occurs in the region where the ratio of the numerical group velocity to the theoretical group velocity is in [0.95, 1.05]. Obviously, the analyses of GRP proposed by De and Eswaran [5] are different from those of Tam and Webb [1] and Pirozzoli [2]. However, the GRP method can be used only for linear schemes and not for nonlinear schemes. Thus, we derived the modified wave number of nonlinear schemes with the ADR method, combined the aforementioned group velocity analysis with highorder Runge–Kutta schemes, and finally, derived a quasilinear method to analyze the spectral properties of nonlinear schemes. For brevity, the method is called the QLGRP method or just QLGRP. Its rationality was then verified with a numerical test and the DFT method.
As usual, a onedimensional wave propagation equation [5] is used in this study to explore the GVP of nonlinear schemes by QLGRP. As shown in [5], for this governing equation, the group velocity is the same as the phase velocity. Therefore, it is insufficient to investigate the GVP of difference schemes. To overcome this, we devised hyperbolic equations such that the solutions are synthetic waves where the group velocity and phase velocity are different. Using the equations, we numerically analyzed the GVP of different difference schemes and also verified the validity of QLGRP.
This paper is arranged as follows. The GRP and ADR methods are reviewed in Section 2. In Section 3, the GRP formula is derived for a highorder Runge–Kutta scheme, the QLGRP method is described, and the comparative results from QLGRP are given for typical schemes. In Section 4, the group velocities of selected cases are obtained computationally using DFT and compared with those from QLGRP, through which the rationality of QLGRP is verified. We construct hyperbolic equations for which the solutions are waves where the group velocity and phase velocity are different. In Section 5, the onedimensional wave propagation equation and the aforementioned hyperbolic equations are solved and analyzed. The conclusions are drawn in Section 6.
2 Review of GRP and ADR methods
Before we consider the QLGRP method, the GRP and ADR methods are reviewed first.
2.1 GRP method
Consider the onedimensional wave propagation equation [1,2,3, 5]:
For the initial distribution \(u\left(x,0\right)=\overset{\frown }{u}{e}^{ikx}\), Eq. (1) has the exact solution \(u\left(x,t\right)=\overset{\frown }{u}{e}^{i\left( kx\omega t\right)}\). For Eq. (1) it holds that ω = kc. Therefore, the exact group velocity is V_{g, exact} = dω/dk = c.
When using a time scheme, the following approximate relation is obtained:
where u^{n} is the value at the initial time t_{n}, u(t_{n} + Δt) is the exact solution at t_{n} + Δt, and u^{n + 1} is the corresponding numerical solution. Suppose the variable distribution on t_{n} takes the form \({u}^n=\overset{\frown }{u}{e}^{i\left( kx\omega t\right)}\). When the explicit Euler scheme is used for Eq. (1), Eq. (2) can be written as:
where f(t_{n}, u^{n}) denotes the contributions from the spatial derivative. When a spatial scheme is used, one can see [5] that
where κ^{′} represents the modified wave number. It can be further derived from Eq. (3) that:
Following the approach of De and Eswaran [5], the numerical value V_{g, num} of the Euler scheme can be derived by differentiating both sides of the above equation:
where ωΔt is the reduced frequency, \(\hat{\kappa}=k\Delta x\) is the reduced wave number, and c is the theoretical group/phase velocity. Equation (4) is exactly the same as that proposed by De and Eswaran [5].
When a linear scheme is employed, the spatial derivative at x_{i} can be approximated by:
where M and N are the numbers of nodes on the right and left sides of x_{i}, and a_{j} are the scheme coefficients. The modified wave number can be obtained by a Fourier transformation [1]:
Once the dispersion and dissipation relations are obtained, V_{g, num} for the firstorder explicit Euler scheme can be obtained from Eq. (4). The explicit Euler scheme usually yields relatively large temporal errors, therefore the highorder RungeKutta schemes are employed in computations especially for unsteady problems. Hence, it is necessary to develop GRP analysis for highorder RungeKutta methods. Thus, Sengupta et al. [6] assumed that a_{4} + ib_{4} = iκ^{′} and \({a}_5+{ib}_5=\frac{d{\kappa}^{\prime }}{d\hat{\kappa}}\), and derived the following GRP formula for the fourthorder Runge–Kutta scheme:
where σ = cΔt/Δx is the Courant–Friedrichs–Lewy (CFL) number. On checking, the following problems were found with this work:

1.
There are no details for the derivation. Therefore, it is difficult to check the correctness of the formula.

2.
When σ approaches 0, the formula does not regress to that of the firstorder explicit Euler scheme in Eq. (4).

3.
When ωΔt = π/2, the denominator cos(ωΔt) is 0, and there is a singularity. Therefore, the GRP formula needs further investigation for higherorder Runge–Kutta schemes.
Thus, we employ the idea of [5] and derive the GRP results for highorder Runge–Kutta schemes in Section 3.1.
2.2 ADR method
Despite its applicability to linear schemes, the spectral analysis method by Tam and Webb [1] is unsuitable for nonlinear schemes. Thus, Pirozzoli et al. [2] proposed a quasilinear spectral analysis method for nonlinear schemes or ADR. The process is as follows [2].
For Eq. (1), suppose the initial distribution is \(u\left(x,0\right)={\overset{\frown }{u}}_0{e}^{ikx}\) and consider a semidiscretized approximation of Eq. (1) on grids {x_{j} = jΔx} with the spacing Δx as
where v_{j}(t) ≈ u(x_{j}, t) and δv_{j} is for a specific numerical scheme, e.g., the linear difference scheme:
as before. From [2], Eq. (6) has the following exact solution: \({v}_j(t)=\overset{\frown }{v}(t){e}^{ij\hat{\kappa}}={\overset{\frown }{u}}_0{e}^{i\left( ct/\Delta x\right){\kappa}^{\prime }}{e}^{ij\hat{\kappa}}\). Furthermore, Pirozzoli [2] proposed ADR to derive κ^{′}:
In this equation, \({\overset{\frown }{v}}_0\left({\hat{\kappa}}_n\right)\) is the Fourier coefficient corresponding to the initial mode and \(\overset{\frown }{v}\left({\hat{\kappa}}_n,\tau \right)\) is the DFT of v_{j}(τ) at time τ where \({\hat{\kappa}}_n={k}_n\Delta x\). The operation of DFT is
where N_{x} is the number of grid number.
As indicated in the introduction, ADR depends on the choice of time step and number, and inappropriate choices can lead to temporal errors or unreasonable jump points in the spectral distributions [3]. Mao et al. [3] expanded the solution v_{j}(τ) of Eq. (6) with a Taylor series:
Substituting the above formula into Eq. (8) and the subsequent result into Eq. (7), Mao et al. [3] proposed ADR without time discretization, known as ADRNT, to derive \({\kappa}^{\prime}\left({\hat{\kappa}}_n\right)\):
where b_{j+l} are the coefficients of the nonlinear scheme.
In order to compare performances by ADR and ADRNT, the spectral property of WENO5JS is obtained by the two methods respectively. According to Pirozzoli’s advice [2], we set N_{x} = 422 and τ = 10^{− 8} for ADR and the results are shown in Fig. 1. The figure tells that the results of the two methods agree well with each other, which is consistent with that by Mao [2]. Compared with ADR, ADRNT has reduced the computation cost and the difficulties in deriving the numerical spectrum of schemes. For convenience, ADRNT is used in this paper.
3 QLGRP for nonlinear schemes
De and Eswaran [5] first analyzed and derived the numerical group velocity for linear schemes. However, their method can be used only for linear schemes and not for nonlinear ones. Considering the development of ADR [2] from DRP [1], which extends the analysis from linear to nonlinear, we introduce the influence of the nonlinearity in GRP [5] by using ADR to account for the spectral property of the nonlinear scheme. Next, employing a similar procedure as in GRP, the socalled QLGRP is derived for nonlinear schemes. Prior to a further discussion on QLGRP, we first derive the GRP formula for highorder Runge–Kutta schemes.
3.1 GRP formula for highorder Runge–Kutta schemes
Taking the thirdorder Runge–Kutta scheme as an example, we derive the GRP formula based on the method of [5]. For convenience, the nthorder Runge–Kutta scheme is referred to as RK_{n}.
RK_{3} here takes the form [7]:
where f(u^{n}), f(u^{(1)}), and f(u^{(2)}) are the derivatives on the subtime steps. From Eq. (10), u^{n + 1} can be reformulated as:
Let K_{1} = f(u^{(n)}), K_{2} = f(u^{(1)}), and K_{3} = f(u^{(2)}). Then, Eq. (2) can be rewritten as:
It is known that K_{1} in Eq. (10) corresponds to the derivative of u^{n} or K_{1} = f(t^{n}, u^{n}); similarly K_{2} = f(t^{n} + Δt, u^{(1)}) = f(t^{n} + Δt, u^{n} + ΔtK_{1}) and \({K}_3=f\left({t}^n+\frac{1}{2}\Delta t,{u}^n+\frac{1}{2}\Delta t\cdot \frac{1}{2}\left({K}_1+{K}_2\right)\right)\). Using κ^{′}, then K_{i} in Eq. (11) can be written as [5, 8]:
Substituting Eq. (12) into Eq. (11):
Applying d(.)/dk to the above equation and taking the real part as in [5], the group velocity for RK_{3} is
For a linear scheme, κ^{′} is found from Eq. (5) [5]. Similarly, when an RK_{4} scheme is used for time discretization, the GRP formula can be derived as:
From Eqs. (13) and (14), it can be seen that the numerical group velocity for a highorder Runge–Kutta scheme depends on σ, unlike the firstorder explicit Euler scheme in Eq. (4). For illustration, the group velocity of a linear fifthorder scheme, UPW5, namely:
was investigated for different CFL numbers. The velocity distributions are shown in Fig. 2 for σ = 0.01, 0.1, 0.3 and 0.5, respectively. The effect of σ is apparent through the variation of the GVP region, for which V_{g, num}/c ∈ [0.95, 1.05], as mentioned before.
As indicated by De and Eswaran [5], the group velocity distribution near the origin is of special concern. Because the region where V_{g}/c lies in (0.95,1.05) is of the major point, they [4, 5] divided the domain into three regions with the value ranges as < 0.95, 0.95 ~ 1.05, and > 1.05 to highlight the GVP region of schemes. In terms of the similar consideration, the same choice is used in this paper. Figure 2 shows that when the CFL number is small, the distributions of the firstorder explicit Euler scheme, RK_{3}, and RK_{4} are almost the same. With an increase of the CFL number, differences appear among the different time schemes. As shown in Fig. 2, when σ = 0.1, for the green area near the origin, RK_{4} ≈ RK_{3} > the explicit Euler scheme, which indicates that higherorder time schemes enhance the GVP. This outcome confirms the potential advantages of higherorder time schemes for unsteady problems. With the further increase of σ, GVP regions of RK_{3} and RK_{4} change apparently due to the inclusion of σ in Eqs. (13) and (14), which is qualitatively consistent with that in [5].
In summary, in this section the group velocities of RK_{3} and RK_{4} were found by applying the same procedure used for the explicit Euler scheme in [5]. The following remarks are given: (1) When the CFL number approaches 0, the group velocity distributions of higherorder Runge–Kutta schemes gradually degenerate into that of the firstorder explicit Euler scheme. Further, the results from GRP for different time schemes become the same under this condition. (2) No singularity arises in Eqs. (13) and (14), which indicates their rationality.
3.2 QLGRP
As mentioned in the introduction, QLGRP is proposed to evaluate the characteristics of the group velocity in a quasilinear manner. The main difference between QLGRP and GRP is that the modified wave number in the former considers the nonlinearity of schemes. Therefore, the formulas for QLGRP are the same as those for GRP, such as Eqs. (13) or (14), except that different evaluations of κ^{′} are implemented.
The concrete implementations are summarized as follows:

1.
Solve for the modified wave number \({\kappa}^{\prime}\left(\hat{\kappa}\right)\) of nonlinear spatial schemes.
Based on ADR, the dispersion and dissipation relations of the nonlinear scheme, \({\kappa}^{\prime}\left(\hat{\kappa}\right)\), are solved with Eq. (9).

2.
Solve for \(d{\kappa}^{\prime }/d\hat{\kappa}\).
When the spatial scheme is linear, \({\kappa}^{\prime}\left(\hat{\kappa}\right)\) can be obtained as \({\kappa}^{\prime}\left(\hat{\kappa}\right)=i\sum \limits_{j=N}^M{a}_j{e}^{ij\hat{\kappa}}\). Because the coefficient a_{j} is constant, \(d{\kappa}^{\prime }/d\hat{\kappa}\) can be derived analytically. However, when the spatial scheme is nonlinear, the coefficient b_{j + l} in Eq. (9) correlates with the initial variable distribution nonlinearly and \(d{\kappa}^{\prime }/d\hat{\kappa}\) is hard to resolve analytically. To overcome this difficulty, we use the difference method to evaluate it numerically. Taking the secondorder central difference as an example, \(d{\kappa}^{\prime }/d\hat{\kappa}\) can be evaluated as:
where j is the index of the reduced wave number and
for j = 0, …, N_{x}/2.

3.
Substitute \(d{\kappa}^{\prime }/d\hat{\kappa}\) into the QLGRP formulas, such as Eqs. (13) or (14), to obtain V_{g, num}.
When solving for \({\kappa}^{\prime}\left(\hat{\kappa}\right)\) for nonlinear schemes with ADR, Δt in the time integration must be small enough [2], which means the corresponding reduced frequency ωΔt should also be small. Correspondingly, QLGRP can describe the GVP of nonlinear schemes effectively only for a small reduced frequency. When ωΔt is large, ADR does not accurately predict the spectral property, and therefore, QLGRP cannot accurately describe GVP either. However, QLGRP can give some insights into the group velocity of a nonlinear scheme. Hence, QLGRP can provide a useful reference for the overall characteristics of the group velocity for a nonlinear scheme.
For demonstration, we used QLGRP to find the group velocity for three typical spatial schemes: UPW5, WENO5JS [7], and WENO5M [9]. Besides, in order to show the difference of GVP regions more obviously, the optimized SLS scheme [8], \(0.0895\left({u}_{i2}^{\prime }+{u}_{i+2}^{\prime}\right)+0.57967\left({u}_{i1}^{\prime }+{u}_{i+1}^{\prime}\right)+{u}_i^{\prime}\approx \frac{1}{\varDelta x}\left[\left.0.00559\left({u}_{i+3}{u}_{i3}\right)+0.25154\left({u}_{i+2}{u}_{i2}\right)+0.6494\Big({u}_{i+1}{u}_{i1}\right)\right]\), which has smaller dispersion error and optimized spectral property is investigated. Adopting the aforementioned procedure, the corresponding distributions of V_{g}/c with the RK_{4} scheme are shown in Fig. 3.
The figure shows that:

1.
Comparing Fig. 3a with Fig. 2, one can see that the group velocity distributions for QLGRP with the UPW5 scheme are the same as those for GRP. This is because, for linear schemes, \({\kappa}^{\prime}\left(\hat{\kappa}\right)\) in ADR is the same as that in DRP [1]. In other words, QLGRP degenerates into GRP for a linear spatial scheme.

2.
The distributions of the group velocity are obviously different for the four schemes, which indicates that QLGRP can be used to distinguish between the GVP of different spatial schemes under the same time scheme. Specifically, the area of the GVP region (green) for the WENO5JS scheme is smaller than that for the UPW5 scheme, which indicates that the group velocities of the nonlinear scheme differ from that of the linear scheme. Moreover, the GVP area of the WENO5M scheme is larger than that of WENO5JS, which shows that the nonlinear optimization of the former enhances the GVP. Among the schemes in Fig. 3, SLS obviously yields the largest GVP region, which is consistent with its excellent spectral property than other three schemes.

3.
The GVP area near the origin, especially along the vertical axis, is the largest for SLS, followed by UPW5, then WENO5M and finally WENO5JS.
In short, QLGRP can be applied not only to analyze the group velocity of a linear scheme but also of a nonlinear scheme. Moreover, the method can clearly differentiate spatial schemes by considering the size of the GVP region.
Despite the presentation of QLGRP and its capability to tell the differences of nonlinear schemes on GRP, how to use it to optimize difference schemes is not indicated at present. One may observe that DRP can be well used in scheme optimizations as shown in [10, 11]. In this regard, an analogous occurrence can be noticed that ADR [2, 3], although its less use in scheme optimization, is widely accepted and applied to evaluate the spectral properties of nonlinear schemes, and hopefully QLGRP would play a similar role. Besides, as indicated in above, different nonlinear schemes yield different GVP regions, which might be used in scheme optimization and would be discussed in other investigations.
4 Numerical analysis and validation of QLGRP
The previous section introduced the derivation and implementation of QLGRP. In this section, the numerical group velocity is obtained numerically and the results are compared with those from QLGRP. Thus, the rationality of the method will be analyzed and verified. Moreover, we devise hyperbolic equations and a corresponding example to determine the numerical group velocity of the scheme directly, which further verifies the approach.
4.1 Numerical analysis and validation
In this section, the group velocity of different schemes is numerically obtained by solving the onedimensional wave propagation equation and by using DFT. By comparing with the analytic results from QLGRP, the numerical error is acquired and the rationality of QLGRP is evaluated.
First, we select several points P_{i} in the kΔx vs. ωΔt plane, then numerically compute the group velocity using DFT. The coordinates of P_{i} are the reduced frequency and wave number. Without loss of generality, we set kΔx as 1. Because ω = kc, then
Hence, once Δt and Δx are determined, the coordinates of P_{i} are defined.
Next, the derivation of numerical group velocity will be discussed. The solution of the semidiscrete Eq. (6) is
where \({\kappa}^{\prime }={\kappa}^{\prime}\left(\hat{\kappa}\right)\) [12]. Compared with the exact solution \({u}_j(t)={\overset{\frown }{u}}_0{e}^{i\left( kx\omega t\right)}\), the numerical frequency can be expressed as ω^{′} = cκ^{′}/Δx, which depends on κ^{′}. Hence, the reduced frequency at P_{i} can be expressed with the modified wave number as:
where L is the length of the computational domain. We consider Eq. (1) with c = 1 and the initial distribution u_{0} = cos(x) at x ∈ [0, L] with L = 2π. From [2], κ^{′} can be numerically found with ADR [2] for a spatial scheme and time scheme using DFT. Here, we use WENO5JS and RK_{4} for the spatial and time discretization. The number of grid cells and time step are given in Table 1. As shown in the table, four cases with respective (N_{x}, Δt) are chosen, and these correspond to P_{i}. Then, κ^{′} can be acquired with respect to \(\hat{\kappa}\), as illustrated in Fig. 4.
Besides P_{1} to P_{4}, we also investigated the case when the reduced frequency was larger. There were oscillations in the results. Hence, the frequency region for investigation was limited to ωΔt < 1. For illustration, the positions of P_{i} (i = 1, 2, 3, 4) are shown in the kΔx vs. ωΔt plane in Fig. 5. The colors indicate the group velocity calculated by QLGRP with WENO5JS and RK_{4}.
Thus far, the numerical solution of the group velocity can be found from:
where k_{1} and k_{2} represent two wave numbers with a small difference, \({\omega}_1^{\prime }\) and \({\omega}_2^{\prime }\) are the corresponding numerical frequencies, and Re denotes taking the real part. Substituting Eq. (16) into Eq. (17):
Under the assumption that \(\hat{\kappa}=1\) as above, we set \({\hat{\kappa}}_1\) and \({\hat{\kappa}}_2\) as \({\hat{\kappa}}_1=1\Delta \hat{\kappa}\) and \({\hat{\kappa}}_2=1+\Delta \hat{\kappa}\). The corresponding modified wave numbers Re(\({\kappa}_1^{\prime}\left({\hat{\kappa}}_1\right)\)) and Re(\({\kappa}_2^{\prime}\left({\hat{\kappa}}_2\right)\)) can be obtained from the distributions of Re(κ^{′}) in Fig. 4. Moreover, V_{g, num} is the slope of Re(κ^{′}) at the point \(\left(1,{\kappa}_i^{\prime }(1)\right)\).
Next, we compare the group velocities at P_{i} obtained by QLGRP and the numerical analysis. For QLGRP, the analytic group velocity V_{g, anal} at P_{i} can be obtained from Eqs. (9) and (14), whereas the numerical group velocity V_{g, num} at P_{i} can be obtained from Eq. (18). These are compared in Table 2.
Table 2 shows that when ωΔt is small, as for P_{1}, P_{2}, and P_{3}, then the group velocity given by QLGRP is quite close to the numerical value. This indicates that QLGRP can reasonably predict the group velocity at low reduced frequency. With an increase of ωΔt, the error between V_{g, anal} and V_{g, num} becomes large but is still within 21% when ωΔt < 1. Therefore, QLGRP can provide important insights for the group velocity at medium reduced frequency.
Note that QLGRP can provide an overview of the group velocity in the kΔx vs. ωΔt space, which is difficult for the numerical approach just mentioned. In addition, QLGRP avoids the tedious process of finding a numerical solution and is more convenient for practical analyses.
4.2 Constructing hyperbolic equations for a combination wave with different phase and group velocities
The distribution u = cos(kx − ωt) satisfies Eq. (1) when ω/k = c, where the group and phase velocities are V_{g, exact} = dω/dk = c and V_{p, exact} = ω/k = c, respectively. One can see that in such situations, the group velocity and phase velocity are indistinguishable, which is unfavorable for investigating the GVP property of the numerical scheme.
In textbooks, the combination wave u = cos(k_{1}x − ω_{1}t) + cos(k_{2}x − ω_{2}t) is usually used to explain the concept of group velocity, which can also be written as:
The envelope of Eq. (19) is
and the corresponding group velocity and phase velocity are
Without loss of generality, it is usually assumed that k_{1} = ω_{1}. When k_{2} ≠ ω_{2}, then V_{g} ≠ V_{p}.
Although the combination wave is used to explain the group velocity and phase velocity, the distribution does not satisfy Eq. (1). Therefore, this equation cannot be used to simulate the wave. We devised the following hyperbolic equations so that we could numerically study the GVP of different schemes:
It can be shown that Eq. (19) satisfies Eq. (20) and a = ω_{2}/k_{2} when k_{1} = ω_{1}. If a ≠ 1, the group velocity is different from the phase velocity. Hence, a measure is provided to study the combination wave with the cooccurrence of different group and phase velocities, which also provides a powerful way to verify the outcome of QLGRP.
5 Numerical examples
In the following, to compute the group velocity, the envelope of the wave is derived by a Hilbert transform.
5.1 Onedimensional wave propagation

(1)
Initial condition using sinuous distribution
Consider Eq. (1) with c = 1/8. The following initial distribution is chosen:
The exact solution is u = sin(8πx − πt) with V_{g, exact} = V_{p, exact} = 1/8.
Two tests were used to check the GVP: (1) using the same spatial scheme for different \(\hat{\kappa}\) and (2) using different spatial schemes for the same \(\hat{\kappa}\). Considering that \(\hat{\kappa}=k\Delta x= kL/{N}_x\), \(\hat{\kappa}\) can be adjusted by changing the size of N_{x}. In the computation, T = 2, Δt = 10^{−3}, and ωΔt = 10^{−3}π, and the computational domain was [− 1, 1].
First, the performance of WENO5JS was tested for different grids, as shown in Fig. 6.
When N_{x} was 48, 64, or 96, \(\hat{\kappa}\) was π/3, π/4, or π/6, respectively. Because the group velocity is equal to the phase velocity in this situation, the phase velocity can be computed by the phase change of the wave peak, and the group velocity can be obtained thereafter. Figure 6 shows that when \(\hat{\kappa}=\pi /3\), the numerical group velocity was about 0.1025 and the ratio V_{g, num}/V_{g, exact} = 0.82. When \(\hat{\kappa}=\pi /4\), V_{g, num} was about 0.11875 and V_{g, num}/V_{g, exact} = 0.95, which indicated that the group velocity was in the accurate region [0.95, 1.05]. With an increase of N_{x}, \(\hat{\kappa}\) decreased and the numerical group velocity gradually approached the theoretical value, i.e., V_{g, num} was 0.99 when \(\hat{\kappa}=\pi /6\).
Using QLGRP with the WENO5JS scheme and the RK_{4} scheme, for the same three points (π/3, 10^{−3}π), (π/4, 10^{−3}π), and (π/6, 10^{−3}π) in the kΔx vs. ωΔt plane, V_{g, num}/V_{g, exact} can be derived from Eq. (14) as 0.8259, 0.9592, and 0.9950, respectively. These values are nearly the same as those obtained numerically. Hence, QLGRP has been verified quantitatively, and its usefulness in analyses of the group velocity has been demonstrated.
Next, GVP was compared for UPW5, WENO5JS, and WENO5M for N_{x} = 48 and \(\left(\hat{\kappa},\omega \Delta t\right)=\left(\pi /3,{10}^{3}\pi \right)\). The results are shown in Fig. 7.
Figure 7 shows that the UPW5 scheme yields a relatively accurate group velocity, since V_{g, num}/V_{g, exact} = 0.96, whereas WENO5M and WENO5JS have relatively large errors, i.e., V_{g, num}/V_{g, exact} = 0.87 and 0.82, respectively. WENO5M performed better than WENO5JS. The rank of performance for the GVP is UPW5 > WENO5M > WENO5JS. The results for QLGRP are consistent with the numerical results, which demonstrates the validity and capability of QLGRP for nonlinear schemes.

(2)
Initial condition using wave packetlike distribution
Consider Eq. (1) with c = 1. The following initial distribution is chosen:
The period of the distribution can be found as π. For brevity, the “envelope” in the following figures is abbreviated as “envlp”. The initial distribution which imitates a wave packetlike structure and its envelope are shown in Fig. 8. Note that due to the employment of Eq. (1) as the governing equation, the group speed here is still the same as the phase velocity.
The same three spatial schemes as above, UPW5, WENO5M, and WENO5JS, are tested to further explore their GVP. In the computation, T = π, Δt = 10^{−3}, N_{x} = 48, and the computational domain was [π/2 , 3π/2]. The results are shown in Fig. 9 with envelopes derived.
From the figure, it can be seen that UPW5 yields the most accurate group velocity in three schemes, which is followed by WENO5M, whose peak value of envelope has decreased from the exact “1” to “0.4”. WENO5JS has the least performance considering its smallest and deformed envelope. The rank of GVP performance is still: UPW5 > WENO5M > WENO5JS, as indicated by previous QLGRP analysis.
5.2 Wave propagation with different group and phase velocities
In this section, the case described in Section 4.2 with different group and phase velocities is evaluated for the different spatial schemes, namely UPW5, WENO5M, and WENO5JS, which further demonstrates the validity of QLGRP.
For the distribution in Eq. (19), multiple waves were included in one wave packet for the purpose of better illustration. The initial condition is
in the domain [−3π, 3π]. We set k_{1} = ω_{1} = 6, k_{2} = 8, and ω_{2} = 12. The exact group and phase velocities were V_{g, exact} = 3 and V_{p, exact} = 9/7. The initial distribution and its envelope derived with a Hilbert transform are shown in Fig. 10. Here, N_{x} = 120, T = 1, and Δt = 5 × 10^{−4}.
The corresponding envelopes, which contain information about the group velocity, were derived by Hilbert transform [13], as shown in Fig. 11. By convention, the envelopes were drawn after taking the absolute value. The details to derive envelopes are shown in the Appendix. For the purpose of further comparison, the result of linear counterpart of WENO7JS, UPW7, which has lower dissipation, is also given in Fig. 11.
Figure 11 shows that UPW7 yields a result closest to the exact solution, which is due to its relatively smallest spectrum error such as low dissipation. Besides, UPW5 yields an envelope such that the numerical group velocity is closest to the theoretical one of the remaining three schemes, followed by WENO5M and then WENO5JS. The amplitudes of the envelope show that UPW5 had the least dissipation, followed by WENO5M and then WENO5JS. Recall that from Fig. 3, the area of GVP given by QLGRP was in the order: UPW5 > WENO5M > WENO5JS. Therefore, the results for QLGRP agree with the numerical results, which verifies its validity.
6 Conclusions
Because GRP [5] cannot be used for a group velocity analysis of a nonlinear scheme, QLGRP was proposed by combining ADR with GRP. Moreover, a detailed derivation and implementation of QLGRP were provided for highorder Runge–Kutta schemes. The conclusions are as follows:

1.
Since QLGRP can analyze the GVP for nonlinear schemes, it can distinguish between the group velocity of different schemes. When the spatial scheme is linear, QLGRP degenerates to GRP.

2.
The group velocities of typical schemes were derived numerically and the comparison confirmed the rationality of QLGRP.

3.
If the reduced frequency is small, the group velocity given by QLGRP is accurate. Moreover, its predictions are reasonable and meaningful at medium reduced frequency. Although QLGRP cannot accurately predict the GVP at high reduced frequency, it can provide a reference in an analysis of the group velocity of nonlinear schemes.

4.
Hyperbolic equations were devised for combination waves with different group and phase velocities. These enabled numerical investigations. Through two numerical examples, the validity of QLGRP was verified qualitatively and quantitatively.
As indicated in this study, distinct GVP regions exist among schemes with different performances regarding GRP, hence such information would be used to define the (free) parameters in optimized schemes and schemes with additional control parameters. Besides, it is known the multiple waves would yield nonlinear interaction through nonlinear schemes, which has drawn the attention in nonlinear spectrum analysis in [14]. It is natural to extend the current analysis to one that considers the interaction of multiple waves. Due to the limitation of space, such a topic will be discussed in other investigations.
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Acknowledgements
Thanks for Jianqiang Chen for his helpful discussion on the paper.
Funding
This study is sponsored by the National Numerical Windtunnel Project of China under grant number NNW2019ZT4B12.
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FX, PY and QL analyzed and provided data regarding the quasilinear analysis of dispersion relation preservation for nonlinear schemes; YY made a substantial revision to the manuscript. All authors read and approved the final manuscript.
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Appendix
Appendix
The main codes on deriving the envelopes by MATLAB are as follows:

U = abs(hilbert(a)); % “a” represents the numerical value.

plot(b, U); % “b” represents the xcoordinate, “U” represents the envelope.
Taking the distribution u = 2cos(3x − 5t)cos(x − 3t) as an example, the derived envelopes at different moments are depicted in Fig. 12, which shows that the numerical envelope obtained by the codes is the same as the exact one.
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Xu, F., Yan, P., Li, Q. et al. Quasilinear analysis of dispersion relation preservation for nonlinear schemes. Adv. Aerodyn. 4, 14 (2022). https://doi.org/10.1186/s42774022001042
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DOI: https://doi.org/10.1186/s42774022001042