1. Introduction
Semi-Lagrangian (SL) semi-implicit integration methods, first proposed by Robert (1981), have been extensively studied and widely incorporated into atmospheric numerical models. A comprehensive review of the applications of SL methods in atmospheric problems was given by Staniforth and Côté (1991). Other extensive studies have also been conducted to examine SL methods (e.g., Bonaventura 2000; White and Dongarra 2011). A three-time-level SL method was implemented operationally at the European Centre for Medium-Range Weather Forecasts in 1991, and was documented and described in Ritchie et al. (1995). Tanguay et al. (1990) generalized the use of a semi-implicit algorithm in order to integrate the fully compressible nonhydrostatic equations. Ritchie (1991) and Ritchie and Beaudoin (1994) applied the semi-Lagrangian semi-implicit method to a multilevel spectral primitive equation model.
McDonald and Bates (1987) and Temperton and Staniforth (1987) proposed two-time-level SL methods. This motivated many authors to use two-time-level SL methods instead of three-time-level schemes, since they reduce the number and impact of computational modes (e.g., McDonald and Haugen 1992; Temperton et al. 2001; Hortal 2002). The use of more than two time levels leads in general to computational modes having comparable amplitudes to those of physical modes, which can influence the accuracy of the solution if no efficient technique is used to damp those modes.
Hortal (2002) proposed a two-time-level SL method called the Stable Extrapolation Two-Time-Level Scheme (SETTLS), which has a region with absolute stability independently of the Courant number. Gospodinov et al. (2001) proposed a family of second-order two-time-level SL schemes that contain an undetermined parameter α. Durran and Reinecke (2004) showed that the size of the absolute stability region of the family of second-order schemes proposed by Gospodinov et al. (2001) varies dramatically according to the undetermined parameter, and that the optimal region is obtained for α = ¼, which corresponds to SETTLS. Still, SETTLS is not a perfect choice because the points on the imaginary axis are outside the domain of absolute stability, except for the origin point. Therefore, SETTLS can generate growth parasites for a purely oscillatory nonlinear term depending on the size of the time step.
The combination of the trapezoidal rule (TR) and the second-order backward differentiation formula (BDF2) is more commonly known as the TR-BDF2 method. Based on the analysis done by Dharmaraja (2007), the TR-BDF2 method performs well in terms of stability for solving partial differential equations. The method is able to compute the solutions using a reasonable step size for stiff problems. This motivated us to use this method to develop the semi-Lagrangian schemes.
In this paper, theoretical and numerical analyses are performed to study the properties of more complex methods. The aim is to try to avoid the problems of instability associated with the treatment of the nonlinear part of the forcing term. We propose a class of semi-Lagrangian semi-implicit schemes using a modified TR-BDF2 method. We use two stages as predictor and corrector in the trapezoidal method and one stage for the BDF2 method. The family of second-order approximations proposed in Gospodinov et al. (2001) is used in the predictor of the TR method. For the linear term we use the implicit trapezoidal method in the first step, the explicit trapezoidal method in the second step, and the implicit BDF2 method in the third step. Following Hortal (2002), the equation of the semi-Lagrangian trajectory used in the SETTLS method is obtained using an average approximation of the acceleration between the departure point and the arrival point. Since the middle point is in the interval where the average approximation of the acceleration is considered, in the proposed method we use the same average of the acceleration to obtain the position of the middle point. An explicit equation is used for this point and the only iterative equation is the one used to obtain the departure point. The potential practical application of the proposed schemes to a weather prediction model or any other atmospheric model is not analyzed in the current paper. The analysis of the application of the proposed class of schemes to these models will be considered in future studies.
The remainder of the paper is organized as follows. Section 2 introduces the notation and explicitly states the problem and expectations of the paper. Section 3 gives details on the proposed schemes for the nonlinear term as well as their linear stability. In section 4, the proposed schemes for the linear term are presented as well as their stability analysis. In section 5 the full semi-Lagrangian semi-implicit schemes are presented, and their accuracy, efficiency, and convergence are analyzed and compared to other existing schemes. Finally, section 6 provides some concluding remarks.
2. Problem and notation
We assume that the two terms depend on the field variable ψ(x, t), x, and time. At each time tn = nΔt, where Δt is the time step, the values of the parameters ψ and the velocity V are known for all times tp = pΔt with p = 0, 1, 2, …, n. The proposed method requires for each arrival point xj the evaluation of the position at time tn of the air parcel arriving at a point (xj, tn+1) and the position of the middle point on the semi-Lagrangian trajectory. The details of the computation of the departure point and the position of the middle point along the semi-Lagrangian trajectory are presented in appendix B.
3. Explicit scheme for the nonlinear term
a. The proposed explicit predictor corrector scheme
The terms in the right-hand side of each stage correspond to the explicit form for the nonlinear operator. In the first stage, we calculate the predictor value ψ* by approximating the value of the nonlinear term Nn+1/4. This value is obtained by using a trapezoidal treatment of Nn and the α approximation of the nonlinear term denoted by 
A family of SL schemes is obtained based on the values of the undetermined parameter α and the decentering parameter θ. As will be confirmed in the next section, the use of the corrector in the second stage improves the stability of the proposed schemes. We will demonstrate that α = ¼ is the optimal choice with the decentering θ in the interval [0.5, 0.75] to provide good accuracy.
b. Stability analysis of the proposed scheme
The region of stability depends on the Courant number, which is proportional to the parameter s. The region on the λΔt − ωΔt plane where the scheme, for the case θ = 0.7, is absolutely stable is plotted in Fig. 1 for several values of the parameter of approximation α. Since the solutions of (8) are periodic in ks, the curves plotted in Fig. 1 are given for several discrete values of ks covering the whole periodic domain [0, 2π]. The regions of absolute stability are only slightly sensitive to the values of α, but the part of the imaginary axis inside the region of absolute stability is variable, and the most interesting is for α = ¼, which corresponds to ωΔt ∈ [−0.708, 0.708]. This has a positive influence for stability and, especially, for solutions that have a purely oscillatory character. The decentering acts directly on the size of the stable imaginary part, as shown in Fig. 2. Table 1 shows the values of (ωΔt)max of the segments [−(ωΔt)max, (ωΔt)max] on an imaginary axis, which are included in the regions of absolute stability depending on the values of the approximation parameter α and the decentering parameter θ. To obtain the proposed class of SL schemes, we set α = ¼, which improves stability. As shown in Table 1, the use of this value ensures a large stability zone along the imaginary axis for all non-high-decentering parameter θ.
Region of absolute stability for six values of the parameter α plotted for 15 Courant–Friedrichs–Lewy (CFL) number values. The central white region corresponds to the absolutely stable region independent of the CFL number for a decentering parameter θ = 0.7.
Citation: Monthly Weather Review 142, 12; 10.1175/MWR-D-13-00302.1
Region of absolute stability of the proposed schemes (α = ¼) for four values of the decentering parameter θ plotted for 15 CFL number values. The central white region corresponds to the absolutely stable region independent of the CFL number.
Citation: Monthly Weather Review 142, 12; 10.1175/MWR-D-13-00302.1
Value of (ωΔt)max of the segment along an imaginary axis [−(ωΔt)max, (ωΔt)max] included in the region of absolute stability depending on the values of the approximation parameter α and the decentering parameter θ.
To conduct a comparison and in order to show the usefulness of using the trapezoidal corrector iteration, the regions of absolute stability are analyzed in the same way as the proposed class of schemes, but without using any corrector value for the variable ψn+1/2. We will use the two-step method and, first, we examine the case in which the value 
Region of absolute stability of the scheme without a corrector step and using the value 
Citation: Monthly Weather Review 142, 12; 10.1175/MWR-D-13-00302.1
In the case of the two-step method in which we consider the value of Nn+1/2 by using ψn+1/2 obtained from the first stage, a similar form of (8) is obtained with 
c. Accuracy of oscillatory solutions
Solutions to the oscillatory equation using the proposed schemes for the nonlinear term. (a) Module of the errors of the amplification factors |Ak − Aan|. (b) Damping of the solution. (c) Relative phase change. (d) Module of the computational mode.
Citation: Monthly Weather Review 142, 12; 10.1175/MWR-D-13-00302.1
The proposed class of schemes presents the advantage of having a damped computational mode as shown in Fig. 4d compared to the computational modes of the class of semi-implicit predictor–corrector schemes developed by Clancy and Pudykiewicz (2013). Note that more comparisons in terms of accuracy between those schemes and the proposed full semi-implicit semi-Lagrangian schemes will be presented in detail in section 5.
4. The proposed scheme for the linear term
a. Treatment of the linear term
b. Stability analysis and choice of parameter μ
In Fig. 5a we plot the module of the amplification factor, which is obtained by using (11) for the oscillatory equation (9) and without decentering (θ = 0.5), as a function of the parameters μ and ωΔt. In this case the schemes are stable for μ ≤ 0.5 and we do not observe any damping in the case of oscillatory solutions for μ = 0.5. For general equation (5), in Fig. 5b we plot for the case θ = 0.5 the module of the amplification factor |Ak| of the solutions for the proposed schemes (μ = 0.5). The schemes are stable and the oscillatory solutions are perfectly preserved. The analytical solutions that correspond to |Aan| = eλΔt are represented in the same figure by the vertical lines.
Treatment of the linear term. (a) Module of the amplification factor, for (9) using θ = 0.5, plotted as function of μ and ωΔt. (b) Module of the amplification factor for (5) using μ = 0.5 and θ = 0.5 plotted along the λΔt – ωΔt plane.
Citation: Monthly Weather Review 142, 12; 10.1175/MWR-D-13-00302.1
In Fig. 6a, we plot the module of the amplification factor for (5) for θ = 0.55, and the results confirm that the scheme is stable. In the same way, if we use θ = 0.45 for (5), the results are not stable, as shown in Fig. 6b. Finally, we use μ = 0.5 for the full semi-implicit semi-Lagrangian combination, and θ is the only variable parameter, with 0.5 ≤ θ ≤ 0.75.
Treatment of the linear term. Module of the amplification factor for (5), plotted along the λΔt – ωΔt plane. (a) Example of a stable case using μ = 0.5 and θ = 0.55. (b) Example of an unstable case using μ = 0.5 and θ = 0.45.
Citation: Monthly Weather Review 142, 12; 10.1175/MWR-D-13-00302.1
5. Numerical experiments of the semi-Lagrangian semi-implicit combination
a. The full semi-Lagrangian semi-implicit scheme
b. Linear stability and error analysis of the two-frequency system
In our analysis we will consider a parameter of decentering θ = 0.7 for the proposed schemes. In Fig. 7a we plot the module of the amplification factor Ak corresponding to the physical solution for five values of the parameter ks on the ωΔt − ΩΔt plane. The dashed lines are the limit defined by Ω = |ω| of the domain subject of our analysis. The results confirm the stability of the proposed schemes. In Figs. 7b, 7c, and 7d, we plot the module of the errors |Ak − Aan| of the amplification factors, respectively, for the three values ks = 2π/3, ks = π, and ks = 4π/3. Figure 8 shows the module of the errors for the same system (17) for the four semi-implicit predictor–corrector schemes proposed in Clancy and Pudykiewicz (2013), which perform well for this test. The authors used the notations AM2*–LFT, AM2*–ABM, T–ABT, and AM2*–ABT for their schemes, where they used the leapfrog trapezoidal (LFT) method, the Adams–Bashforth trapezoidal (ABT) scheme, and the Adams–Bashforth–Moulton (ABM) method as explicit predictor–corrector schemes. For the implicit schemes, the authors used the trapezoidal method (T) and a method denoted by AM2*. The schemes proposed in this paper perform very well in terms of accuracy compared to the three schemes AM2*–LFT, AM2*–ABM, and AM2*–ABT, as shown in Figs. 7 and 8. Note that the best method in Clancy and Pudykiewicz (2013) for this test in terms of amplitude is the method T–ABT, but it presents some phase errors and it is less accurate than the proposed schemes, as shown in Figs. 7 and 8.
The proposed full semi-Lagrangian semi-implicit schemes using μ = 0.5 and θ = 0.7. (a) Module of the amplification factors for five CFL number values. (b) Module of the errors of the amplification factors |Ak − Aan| for ks = 2π/3. (c) Module of the errors for ks = π. (d) Module of the errors for ks = 4π/3.
Citation: Monthly Weather Review 142, 12; 10.1175/MWR-D-13-00302.1
Module of the errors of the amplification factors |Ak − Aan| for the schemes proposed by Clancy and Pudykiewicz (2013) for the (a) AM2–LFT, (b) AM2–ABM, (c) T–ABT, and (d) AM2–ABT methods.
Citation: Monthly Weather Review 142, 12; 10.1175/MWR-D-13-00302.1
In Fig. 9a we plot the module of the errors |Ak − Aan| of the amplification factor for the method studied by Cullen (2001). As can be seen in Figs. 9 and 7, for positive and large values of ωΔt the proposed schemes are more accurate than the method studied by Cullen (2001). Note that the errors of this method are mainly due to the phase errors, as shown in Fig. 9b.
The scheme studied by Cullen (2001). (a) Module of the errors of the amplification factors |Ak − Aan|. (b) Relative phase change.
Citation: Monthly Weather Review 142, 12; 10.1175/MWR-D-13-00302.1
c. Accuracy, efficiency, and convergence of the proposed class of schemes
1) Error analysis using one mode
Following the expressions of (20), (21), and (22), respectively, for the three steps, the terms k2Δt, c2Δt, and δ can be considered to be dimensionless parameters. These parameters will be used to conduct a more general analysis of the characteristics of the proposed schemes and their comparison with those of the scheme studied by Cullen (2001). The three steps of the temporal discretization of the proposed schemes can be rewritten in matrix form using the vector variable Up = (Dp, ϕp)T at each step p. The vector value U* of the first step is substituted into the second step, and the result obtained for Un+1/2 is substituted into the third step to obtain a fourth-degree polynomial equation for the amplification factor Ak. In appendix B we offer further explanations of the matrix form of the proposed schemes and the characteristics of the polynomial equation of the amplification factor. This polynomial has one root equal to zero. Therefore, it can be reduced to a three-degree polynomial, which has one real root corresponding to the computational mode. The two other roots are complex conjugate numbers and correspond to the inertio-gravity waves. The computational mode is largely damped, as shown in Fig. 10 for the two cases δ = 0.75 and δ = 1.25. Note that one of the advantages of the proposed schemes is that they have a damped computational mode, as opposed to the method studied by Cullen (2001), which has one meteorological mode that is undamped (equal to unity).
Module of the computational mode for the proposed schemes (θ = 0.7).
Citation: Monthly Weather Review 142, 12; 10.1175/MWR-D-13-00302.1
Module of the errors of the amplification factors |Ak − Aan| for the proposed schemes (θ = 0.7).
Citation: Monthly Weather Review 142, 12; 10.1175/MWR-D-13-00302.1
Table 2 shows the values of the errors E corresponding to the amplification factors, which are obtained by using the method studied in Cullen (2001), and those obtained by using the proposed schemes with the decentering parameters θ = 0.7 and θ = 0.75 for five values of the dimensionless parameter δ. Following the results of this test, we conclude that the proposed schemes provide more accurate results compared to those obtained by using the method studied by Cullen (2001). To further verify the resolution quality, the efficiency, and the convergence of the proposed schemes, in the following section several tests are performed by using other solutions of (19), which are the combination of two different modes.
Errors of the amplification factors using the proposed schemes and the method studied by Cullen (2001).
2) Numerical test using two modes
Evolution of the errors until time T = 50 using the proposed schemes and the method studied by Cullen (2001) for an initial condition that combined two modes (νR = 1, νI = −1) with Δt = 0.03. Shown are errors for parameters (a) D and (b) ϕ.
Citation: Monthly Weather Review 142, 12; 10.1175/MWR-D-13-00302.1
3) Efficiency of the proposed schemes
The most expensive parts in terms of computational cost are the dynamical part, which is related to the implicit resolution of the linear term, and the physical part, related to the explicit resolution of the nonlinear term. The computational cost of the explicit resolution of the linear term, which is the case in the second step in the proposed schemes, is negligible compared to the computational costs of the dynamical and physical parts.
For the proposed schemes we have one dynamical part and one physical part in the first step, one physical part in the second step, and one physical part and one dynamical part in the third step. The method studied by Cullen (2001) has one dynamical part and one physical part in each step. The efficiency of the proposed schemes and the efficiency of the method studied by Cullen (2001) will be compared by using the time steps satisfying the condition Δtm = 1.25Δtc, where Δtm and Δtc are the time steps used for the proposed schemes and the method studied by Cullen (2001), respectively. This condition is sufficient to give a reliable comparison of the efficiency, depending on the number of physical and dynamical parts in each method.
Equation (19) with the gravity speed c = 0.01 is solved, for two cases of analytic solutions of the form (24), using the proposed schemes and the method studied by Cullen (2001). We consider k = 0.01, and in the first case we use the initial conditions with νR = 1 and νI = 1 and in the second case we consider νR = 1 and νI = 4. In Fig. 13 we plot the evolution of the errors of the solutions (D, ϕ)T, which are obtained by using the proposed schemes with a time step Δt = 0.030 and the method studied by Cullen (2001) with a time step Δt = 0.024. The results of the proposed schemes remain more accurate in comparison to those obtained by using the method studied by Cullen (2001). Therefore we conclude that the proposed schemes perform well in terms of efficiency.
Evolution of the error of the solution (D, ϕ)T until time T = 50 using the proposed method with Δt = 0.030 and the method studied by Cullen (2001) with Δt = 0.024 for an initial condition that combined two modes, for cases (a) νR = 1 and νI = 1 and (b) νR = 1 and νI = 4.
Citation: Monthly Weather Review 142, 12; 10.1175/MWR-D-13-00302.1
Our numerical tests show that the method used to compare the competitive efficiency by considering the time steps that satisfy the condition Δtm = 1.25Δtc is valid for the large interval of time steps that meet the Lipschitz criterion. In the following we give another numerical test to study the efficiency of the proposed method using the two-frequency system in (17). In this example we will consider the initial value of the trajectories x(0) ∈ [0, 1] and the analytical solution ψ(x, t) = (x + ρxt)e(iΩ+iω)t, where ρ = 0.8, ω = 1.0, Ω = 1.8, and the trajectories are given by x(t) = x(0)/(1 + ρt).
Figure 14 (right) shows the error as a function of the variable time step Δt for the proposed method. The efficiency of the proposed method and the efficiency of the method studied by Cullen (2001) are compared by using the time steps that satisfy the condition Δtm = 1.25Δtc. The comparison is performed using the large interval of the time steps that meet the Lipschitz criterion. Figure 14 (left) shows the error for the method studied by Cullen (2001) using a time step Δtc and the error for the proposed method using a time step Δtm = 1.25Δtc. The results of the proposed method remain accurate, which confirms that the proposed method performs well in terms of efficiency.
(a) The error function for the method studied by Cullen (2001) using the time step Δt and the error function for the proposed method using the time step 1.25Δt. (b) The error function for the proposed method using the time step Δt.
Citation: Monthly Weather Review 142, 12; 10.1175/MWR-D-13-00302.1
4) Convergence of the proposed schemes
6. Conclusions
This paper presents theoretical and numerical analyses of the properties of more complex methods. We try to avoid the problems of instability associated with the treatment of the nonlinear part of the forcing term. A class of semi-Lagrangian semi-implicit schemes is proposed that uses a modified TR-BDF2 method based on the trapezoidal rule and the second-order backward differentiation formula. For the nonlinear term we used two stages as the predictor and corrector in the TR method and one stage for the BDF2 method. A family of second-order approximations derived by Gospodinov et al. (2001) is used in the first stage as the predictor of the TR method. For the linear term we used the implicit trapezoidal method in the first step, the explicit trapezoidal method in the second step, and the implicit BDF2 method in the third step. The use of the corrector stage improves the stability of the proposed schemes, and a large stable imaginary part is obtained. Following the numerical tests, the second-order approximation using α = ¼, which corresponds to the approximation of SETTLS, is the appropriate choice that guarantees the large domain of absolute stability and the optimal intersection of the region of absolute stability with the imaginary axis. This intersection is improved by using a decentering in the second step to obtain the schemes that perform well for purely oscillatory solutions. The numerical analysis demonstrates that the proposed class of semi-Lagrangian semi-implicit schemes performs well in terms of stability, accuracy, convergence, and efficiency. The potential practical application of the proposed family of schemes to a weather prediction model or any other atmospheric model is not analyzed and could be the subject of other forthcoming studies.
Acknowledgments
The authors are grateful to the anonymous reviewers for their valuable comments and suggestions, which helped to improve the quality of the paper significantly. The research was supported by Natural Sciences and Engineering Research Council of Canada (NSERC) and Environment Canada.
Appendix A
TR-BDF2 Method
In our case we use γ = ½. Equations (A2) and (A3) can be solved by using implicit or explicit formulas for the source term.
APPENDIX B
Matrix Form of the Proposed Scheme and Trajectory Evaluation
a. Matrix form of the proposed scheme
b. Trajectory evaluation
In the proposed method, for each arrival point xj at time t + Δt, the departure point 
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