1 Introduction

The characteristic-based Lagrangian models have been successfully applied to simulate the vortex merging, hydraulic jumps, and shear flows in the shallow water equations (Toro 1999; Wang and Yeh 2005, and others) with CFL ≥ 1, their results are quite accurate compared with the high-order finite volume models (Sun 2011; Sun and Oh 2022), but the former requires tremendous computing time. They are also difficult to apply to multi-dimensional flows. Hence, the semi-Lagrangian advections are popular in computational fluid dynamics (Staniforth and Cote 1991). It is noted that there is a big challenge to find the property of the fluid in the upstream regions in fluid dynamics (Kalnay et al 2000; Fournier 2005; Sun and Sun 2017). The conventional backward schemes may face similar challenges. On the other hand, the forward trajectory method can easily evaluate the fluid property in the downstream with great accuracy (Purser and Leslie 1991). However, the variables on a curvilinear Lagrangian grid are difficult to evaluate in the equations. Purser and Leslie (1991) proposed the ‘cascade interpolation method’, which was also used in Nair et al. (2003) and others. Sun et al. (1996) proposed the ‘split interpolation method’. Based on splitting the dimensions of the geometric space to simplify the interpolation formulae, both methods are very efficient with satisfactory accuracy. Because both methods hypothesize the same coordinate monotonicity, they are also subject to the restriction that neither the x- nor the y-projections of the Lagrangian curves should cross, which may not require CFL ≤ 1, but is still more stringent than the stability criterion of Sun and Yeh (1997) semi-Lagrangian scheme. It is noted that the numerical scheme applied to the flux-form model also requires CFL ≤ 1. A numerical scheme that allows CFL much greater than unity can be useful to simulate the fast-moving, high deformation flows inside hurricanes (Kurihara et al. 1998) and tornados (Fujita 1974, 1992), along the upper-level jet or the low-level jet, and strong vertical updrafts and downdraft inside the hot towers in the Equator trough (Rielh and Malkus 1958; Sun 2023), and the high-speed raindrop falling in the sky and the downslope density current (Setter and Kuo 1983; Sun 1993); or to ease the CFL constraint imposed by the unwanted acoustic waves in a compressible fluid and/or gravity waves in the Quasi-geostrophic phenomena (Sun and Sun 2015, 2019; Yeh et al. 2002). Because of high accuracy and simplicity, the current scheme is suitable to simulate the long-distance transport of dust by the upper-level-jet (Sun et al. 2013) and dispersion of pollution over the complex terrain (Wu et al. 2003; Sun 2021).

Sun and Yeh (1997) (will be referred to as SY) proposed a direct projection from the regular grids to the irregular Lagrangian grids in the 2D flow with velocity (u, v). Figure 1 shows that the fluid from the grid O(ξ, η) on \(\eta = y_{j}\) moves to P(X,Y).(indicated by red circle) on the Lagrangian curve \(\eta = y_{j}\) within ∆t, where X = ξ + u∆t, and Y = η + v∆t. The variable F (for example, water substance or pollutant) and the Lagrangian coordinates of the parcel remain the same, although the location changes from \(O(\xi ,\eta )\) to P(X,Y) in Euler coordinates. They:

  1. (A)

    Arranged the data in an orderly sequence,

  2. (B)

    Assumed that variables F and the coordinate of P(X,Y) on the Lagrangian curve \(\eta = y_{j}\), are given by the cubic polynomials of ξ:

    $$X(\xi ) = a_{3} \xi^{3} + a_{2} \xi^{2} + a_{1} \xi + a_{0} ,$$
    (1a)
    $$Y(\xi ) = b_{3} \xi^{3} + b_{2} \xi^{2} + b_{1} \xi + b_{0} ,$$
    (1b)
    $$F(\xi ) = c_{3} \xi^{3} + c_{2} \xi^{2} + c_{1} \xi + c_{0} .$$
    (1c)
Fig. 1
figure 1

A particle O in Euler grids (dashed lines) moves to P in Lagrangian grids (full lines), and the corresponding Eulerian coordinate lines x = xi and y = yj are transformed to the Lagrangian coordinate lines ξij and ηij, respectively

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Used 4-points values of ξ on \(\eta = y_{j}\) and at P(X,Y) to determine the coefficients of \(\, a_{k}\), \(\, b_{k}\), and \(\, c_{k}\) in the section of \(d\xi = \xi_{j + 1} - \xi_{j}\), where P(X,Y) is located.

  1. (C)

    If the curve of \(\eta = y_{j}\) intercepting the vertical line at X = Xk, as indicated by the blue diamond Q in Fig. 1, they solved and selected the appropriate \(\xi_{{X_{k} ,j}}\) among three roots of

$$a_{3} \xi^{3} + a_{2} \xi^{2} + a_{1} \xi + a_{0} = X_{k} .$$
(1d)

Then, put \(\xi_{{X_{k} ,j}}\) in Eqs. (1b) and (1c) to obtain the Y and F at Q; and applied same method to other Lagrangian curves to obtain entire interceptions of blue diamonds (Qs) on vertical line at X = Xk.

  1. (D)

    From the values and y-coordinates of (Qs) along the vertical line to derive the value at the regular grid C (purple star in Fig. 1), then repeated the same procedure for the grids in the entire domain

This method is as accurate as Purser and Leslie (1994), Nair et al. (2003) or Sun et al. (1996). It is not limited by the restriction of coordinate monotonicity, either. Hence, it can be applied to the flow with CFL > 1. Although it requires 1.7 to 2.3 times computation compared with the splitting method, it is still more efficient than the backward semi-Lagrangian scheme according to SY. The SY scheme has been applied in a 3D nonhydrostatic model to simulate the flows over mountains, and convective cloud bands develop over a warm ocean (Hsieh et al. 2010; Hsu et al. 2004, etc.) SY also showed solving coefficients of Eqs. (1a, 1b, 1c) and the roots of cubic Eq. (1d) consume most of their computing resources. Meanwhile, equations higher than the third degree are exceedingly difficult to solve, hence, they cannot be applied in SY.

Here, we propose a method that requires solving neither the coefficients nor the cubic equations. The new method not only significantly reduces the procedures, saving 2/3 or more of computing time, but also allows higher-order polynomials to be employed. To compare with SY results, we will show the simulations using the 3rd-order polynomials in the flow with large Courant Numbers.

2 Numerical scheme

As described previously, the fluid moves from \({\text{O}}_{i} {(}\xi ,\eta )\) to \(P_{i} (X,Y)\)(indicated by red circle) within Δt in the regular coordinates. We connect those red circles to form the Lagrangian curve (η = yj), which intercepts with the vertical line at X = Xk, indicated as blue diamond point Q. Since the variables on the Lagrangian curve are related to the cubic polynomials of the Lagrangian coordinates of the parcels, ξ and η, they can also be represented by the cubic polynomials of X and Y, the Euler coordinates of the parcels, because X and Y are well-defined functions of ξ and η. Consequently, F(Xk) and Y(Xk) can be derived from the coordinates of the neighboring reds by the Lagrangian polynomials:

$$\begin{aligned} F(X) = & F_{{i - 2,j}} \frac{{(X - X_{{i - 1,j}} )(X - X_{{i,j}} )(X - X_{{i + 1,j}} )}}{{(X_{{i - 2,j}} - X_{{i - 1,j}} )(X_{{i - 2,j}} - X_{{i,j}} )(X_{{i - 2,j}} - X_{{i + 1,j}} )}} \\ + & F_{{i - 1,j}} \frac{{(X - X_{{i - 2,j}} )(X - X_{{i,j}} )(X - X_{{i + 1,j}} )}}{{(X_{{i - 1,j}} - X_{{i - 2,j}} )(X_{{i - 1,j}} - X_{{i,j}} )(X_{{i - 2,j}} - X_{{i + 1,j}} )}} \\ + & F_{{i,j}} \frac{{(X - X_{{i - 2,j}} )(X - X_{{i - 1,j}} )(X - X_{{i + 1,j}} )}}{{(X_{{i,j}} - X_{{i - 2,j}} )(X_{{i,j}} - X_{{i - 1,j}} )(X_{{i,j}} - X_{{i + 1,j}} )}} \\ + & F_{{i + 1,j}} \frac{{(X - X_{{i - 2,j}} )(X - X_{{i - 1,j}} )(X - X_{{i,j}} )}}{{(X_{{i + 1,j}} - X_{{i - 2,j}} )(X_{{i + 1,j}} - X_{{i - 1,j}} )(X_{{i + 1,j}} - X_{{i,j}} )}} \\ \end{aligned}$$
(2a)
$$\begin{aligned} Y(X) = & Y_{{i - 2,j}} \frac{{(X - X_{{i - 1,j}} )(X - X_{{i,j}} )(X - X_{{i + 1,j}} )}}{{(X_{{i - 2,j}} - X_{{i - 1,j}} )(X_{{i - 2,j}} - X_{{i,j}} )(X_{{i - 2,j}} - X_{{i + 1,j}} )}} \\ + & Y_{{i - 1,j}} \frac{{(X - X_{{i - 2,j}} )(X - X_{{i,j}} )(X - X_{{i + 1,j}} )}}{{(X_{{i - 1,j}} - X_{{i - 2,j}} )(X_{{i - 1,j}} - X_{{i,j}} )(X_{{i - 2,j}} - X_{{i + 1,j}} )}} \\ + & Y_{{i,j}} \frac{{(X - X_{{i - 2,j}} )(X - X_{{i - 1,j}} )(X - X_{{i + 1,j}} )}}{{(X_{{i,j}} - X_{{i - 2,j}} )(X_{{i,j}} - X_{{i - 1,j}} )(X_{{i,j}} - X_{{i + 1,j}} )}} \\ + & Y_{{i + 1,j}} \frac{{(X - X_{{i - 2,j}} )(X - X_{{i - 1,j}} )(X - X_{{i,j}} )}}{{(X_{{i + 1,j}} - X_{{i - 2,j}} )(X_{{i + 1,j}} - X_{{i - 1,j}} )(X_{{i + 1,j}} - X_{{i,j}} )}} \\ \end{aligned}$$
(2b)

After obtaining F and Y of all intercepts (blue diamonds) on the vertical line at X = Xk, we apply Eq. (2a) again but on the vertical direction by replacing X by Y to get the value F at the Euler grid C. The same procedure is applied to the entire domain. There is no need to solve the cubic equations or the associated coefficients. Hence, this scheme is much simpler and more efficient than the one proposed by SY. It is noted that following the same procedure, the vertical Lagrangian curves can be interpolated from the fluids initially coming from the vertical lines, ξi. Interpolation from the Lagrangian curves projected by fluids coming from either a horizontal or vertical curve to the regular grids is referred to as ‘Economic Internet Interpolation.’ On the other hand, interpolation using both vertical and horizontal lines parcels came from is called ‘Complete Internet Interpolation’ (SY). They showed that Complete Interpolation is slightly more accurate than Economic Interpolation, but the Complete version requires twice computing resources as the Economic one. It is also noted that Eq. (2a, 2b) used in this model can be replaced by the 5th or higher order polynomials here, but not in SY.

3 Numerical results of Doswell’s idealized cyclogenesis

The idealized cyclogenesis of Doswell (1984) is used to demonstrate the performance of the scheme applied to a rotational flow with strong deformation. The governing equation is

$$\frac{\partial f}{{\partial t}} = - {\mathbf{V}}\cdot\nabla f\,{\text{in Euler forms}},{\text{ or}}\,\frac{Df}{{Dt}} = 0\,{\text{in Lagrangian form}}{.}$$
(3a)

The tangential velocity of the circular vortex is.

$$V\left( r \right) = A\ {\text{sech}}^{{2}} \left( r \right){\text{tanh}}\left( r \right),$$
(3b)

where r is the radius of the vortex; A = 2.598 is chosen so that the maximum value of V equals 1. The initial condition is.

$$f\left( {x,y,0} \right) = - {\text{tanh}}[(y - y_{c} )/\delta ],$$
(3c)

where δ is the characteristic width of the frontal zone, as defined in in Holm (1995). The analytic solution is.

$$f(x,y,t) = - \tanh \left[ {\frac{{(y - y_{c)} }}{\delta }\cos (\omega t) - \frac{{(x - x_{c)} }}{\delta }\sin (\omega t)} \right]$$
(3d)

where (xc, yc) is the center of rotation and ω = V/r is the angular velocity. The integration domain is 10 units long with resolutions of 65 × 65 or 129 × 129 grid points. An integration time of 5 units is chosen so that the analytic solution is still resolvable on the low-resolution grid. Nearly perfect results are obtained for the smooth cases with δ = 1, and we are interested only in the non-smooth cases with δ = 0.05. The analytic solution on the high-resolution grid is shown in Fig. 2a. The simulation from SY’s Complete Version with the filters (Sun and Sun 2004; Sun 2007) on 129 × 129 grids, 64-time steps, t = 5, CFL = 1, and Error = 0.083 shown in Fig. 2b is quite comparable with our Economic simulation (Error = 0.078) in Fig. 2c. They are also in good agreement with our numerical result with 16-time steps, CFL = 4, t = 5, and Error = 0.076 in Fig. 2d. The numerical result obtained from SY’s Complete Internet on 65 × 65 grids with CFL = 4, t = 5 and Error = 0.068 is shown in Fig. 3a. Our Economic simulation shown in Fig. 3b has Error = 0.147. A larger error indicates that twice of the data used in Complete Interpolation does improve the numerical results when the sharp edge of the vortex sheet is barely resolved by the coarse grids. When the phenomena can be well represented by the Lagrangian polynomials, the results do not have significant difference between our simple (Economic) model and SY’s Complete Model. Both models can handle flows with the Courant Number much larger than one. The simulation on 129 × 129 grids with CFL = 6.0 produces an Error = 0.132 at t = 9.843, 21-time steps (shown in Fig. 4). The Error = 0.073 at t = 5.15 at 11-time steps. The mass is also well conserved (Error of mass is 6.176 × 10–4 at t = 9.843) by using Sun (2007), and Sun and Sun (2004). Because of simplicity and efficiency of the model, the current model is suitable to simulate the multi-dimensional fast-moving flows or to ease the constrained by unwanted acoustic waves in compressible fluid or gravity waves in simulating the flows in the atmosphere or oceans. It can be applied to predict the transport of pollutants and dust along the jet (Sun et al. 2013; Wu et al. 2003) and the high-speed raindrops falling from convective clouds (Sun 1993), or strong downslope windstorms in a three-dimensional complex terrain model (Haines et al. 2019; Sun and Sun 2015; Sun 2021) because of simplicity and accuracy. This scheme can also incorporate the variation method to ensure the conservation of mass and/or energy of the entire domain (Sun and Sun 2004; Sun 2007).

Fig. 2
figure 2

The Idealized cyclogenesis test on 129 × 129 grids, δ = 0.05. a The analytic solution after 5-time units: b simulation at the same time from Complete Internet of SY (64 time-steps, CFL = 1, Error = 0.083). c Simulation from current Model (64 time-steps, CFL = 1, Error = 0.078), d simulation from current model with 16 time-steps (CFL = 4, Error = 0.076)

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Fig. 3
figure 3

Numerical simulation after 5-time units on 65 × 65 grids, δ = 0.05, and CLF = 4: a simulation from Complete version of SY(Error = 0.068), b simulation from current Economic model (Error = 0.147)

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Fig. 4
figure 4

Numerical simulation after 9.84-time units on 129 × 129 grids, δ = 0.05, and CLF = 6 (Error = 0.132) from current Economic model

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4 Conclusion

Here, we propose an efficient forward trajectory model, in which the property and position of the fluids can be accurately evaluated from the Euler coordinates advected to the Lagrangian coordinates. After sorting those fluid elements on the irregular Lagrangian curve, we apply the 3rd-order cubic polynomial to interpolate the coordinates and properties of the elements from the irregular curves to the regular grids. This scheme retains the advantages of the Internet Interpolation Model (SY) to allow a larger Courant number, but no need to solve the cubic equations and coefficients of a set of linear algebra equations as proposed by SY. The scheme is not only much simpler than SY’s models but also allows using higher-degree polynomials. Hence, it is suitable to simulate the multiple-dimensional fast-moving flows in hurricanes, tornados and jet-streams, storms, and jet-stream, or ease the CFL constraint imposed by the unwanted acoustic waves in a compressible fluid or gravity waves in simulating the flow in the atmosphere and ocean (Mesinger and Arakawa 1976; Haltiner and Williams 1980; Lin 2007; Coiffier 2011; Lin et al. 2018). The semi-Lagrangian method can also avoid the nonlinear instability and is easier to handle the open lateral boundary compared with the finite difference and finite volume as discussed on p. 89 and p. 142 in the review article of Sun (2023).