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]:
$$\left\{\begin{array}{l}{u}^{(1)}={u}^n+\Delta t\cdot f\left({u}^n\right),\\ {}{u}^{(2)}=\frac{3}{4}{u}^n+\frac{1}{4}{u}^{(1)}+\frac{1}{4}\Delta t\cdot f\left({u}^{(1)}\right),\\ {}{u}^{n+1}=\frac{1}{3}{u}^n+\frac{2}{3}{u}^{(2)}+\frac{2}{3}\Delta t\cdot f\left({u}^{(2)}\right),\end{array}\right.$$
(10)
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:
$${u}^{n+1}={u}^n+\Delta t\left(\frac{1}{6}f\left({u}^{(n)}\right)+\frac{1}{6}f\left({u}^{(1)}\right)+\frac{2}{3}f\left({u}^{(2)}\right)\right).$$
Let K_{1} = f(u^{(n)}), K_{2} = f(u^{(1)}), and K_{3} = f(u^{(2)}). Then, Eq. (2) can be rewritten as:
$$\overset{\frown }{u}{e}^{i\left[ kx\omega \left(t+\Delta t\right)\right]}\overset{\frown }{u}{e}^{i\left( kx\omega t\right)}\approx \Delta t\cdot \left(\frac{1}{6}{K}_1+\frac{1}{6}{K}_2+\frac{2}{3}{K}_3\right).$$
(11)
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]:
$$\left\{\begin{array}{l}{K}_1=\left(c\right)\cdot \frac{i{\kappa}^{\prime }}{\Delta x}\overset{\frown }{u}{e}^{i\left( kx\omega t\right)},\\ {}{K}_2=\left[\left(c\right)\cdot \frac{i{\kappa}^{\prime }}{\Delta x}+\Delta t\cdot \left({c}^2\right){\left(\frac{\kappa^{\prime }}{\Delta x}\right)}^2\right]\overset{\frown }{u}{e}^{i\left( kx\omega t\right)},\\ {}{K}_3=\left[\left(c\right)\cdot \frac{i{\kappa}^{\prime }}{\Delta x}+\frac{\Delta t}{4}\cdot \left({c}^2\right){\left(\frac{\kappa^{\prime }}{\Delta x}\right)}^2+\frac{\Delta t}{4}\left(\left({c}^2\right){\left(\frac{\kappa^{\prime }}{\Delta x}\right)}^2+\Delta t\cdot {ic}^3{\left(\frac{\kappa^{\prime }}{\Delta x}\right)}^3\right)\right]\overset{\frown }{u}{e}^{i\left( kx\omega t\right)}.\end{array}\right.$$
(12)
Substituting Eq. (12) into Eq. (11):
$${e}^{ i\omega \Delta t}1\approx  i\sigma \cdot {\kappa}^{\prime }\frac{1}{2}{\left(\sigma \cdot {\kappa}^{\prime}\right)}^2+\frac{1}{6}i{\left(\sigma \cdot {\kappa}^{\prime}\right)}^3.$$
Applying d(.)/dk to the above equation and taking the real part as in [5], the group velocity for RK_{3} is
$${\left({V}_{g, num}\right)}_{RK_3}=c\cdot \operatorname{Re}\left(\left(1 i\sigma \cdot {\kappa}^{\prime }\frac{1}{2}{\left(\sigma \cdot {\kappa}^{\prime}\right)}^2\right){e}^{i\omega \Delta t}\cdot \frac{d{\kappa}^{\prime }}{d\hat{\kappa}}\right).$$
(13)
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:
$${\left({V}_{g, num}\right)}_{RK_4}=c\cdot \operatorname{Re}\left(\left(1 i\sigma \cdot {\kappa}^{\prime }\frac{1}{2}{\left(\sigma \cdot {\kappa}^{\prime}\right)}^2+\frac{1}{6}i{\left(\sigma \cdot {\kappa}^{\prime}\right)}^3\right){e}^{i\omega \Delta t}\cdot \frac{d{\kappa}^{\prime }}{d\hat{\kappa}}\right).$$
(14)
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:
$${\left( du/ dx\right)}_j\approx \frac{1}{\Delta x}\left(\frac{1}{30}{u}_{i3}+\frac{1}{4}{u}_{i2}{u}_{i1}+\frac{1}{3}{u}_i+\frac{1}{2}{u}_{i+1}\frac{1}{20}{u}_{i+2}\right),$$
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:
$${\left(\frac{d{\kappa}^{\prime }}{d\hat{\kappa}}\right)}_j\approx \frac{1}{60\Delta \hat{\kappa}}\left({\kappa}_{j3}^{\prime }+9{\kappa}_{j2}^{\prime }45{\kappa}_{j1}^{\prime }+45{\kappa}_{j+1}^{\prime }9{\kappa}_{j+2}^{\prime }+{\kappa}_{j+3}^{\prime}\right),$$
(15)
where j is the index of the reduced wave number and
$${\hat{\kappa}}_j={k}_j\Delta x=\frac{2\pi }{\lambda_j}\times \frac{L}{j}=\frac{2\pi j}{N_x},$$
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.