Multipoint Explicit Differencing (MED) for Time Integrations of the Wave Equation

Celal S. Konor Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado

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Akio Arakawa Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California

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Abstract

For time integrations of the wave equation, it is desirable to use a scheme that is stable over a wide range of the Courant number. Implicit schemes are examples of such schemes, but they do that job at the expense of global calculation, which becomes an increasingly serious burden as the horizontal resolution becomes higher while covering a large horizontal domain. If what an implicit scheme does from the point of view of explicit differencing is looked at, it is a multipoint scheme that requires information at all grid points in space. Physically this is an overly demanding requirement because wave propagation in the real atmosphere has a finite speed. The purpose of this study is to seek the feasibility of constructing an explicit scheme that does essentially the same job as an implicit scheme with a finite number of grid points in space. In this paper, a space-centered trapezoidal implicit scheme is used as the target scheme as an example. It is shown that an explicit space-centered scheme with forward time differencing using an infinite number of grid points in space can be made equivalent to the trapezoidal implicit scheme. To avoid global calculation, a truncated version of the scheme is then introduced that only uses a finite number of grid points while maintaining stability. This approach of constructing a stable explicit scheme is called multipoint explicit differencing (MED). It is shown that the coefficients in an MED scheme can be numerically determined by single-time-step integrations of the target scheme. With this procedure, it is rather straightforward to construct an MED scheme for an arbitrarily shaped grid and/or boundaries. In an MED scheme, the number of grid points necessary to maintain stability and, therefore, the CPU time needed for each time step increase as the Courant number increases. Because of this overhead, the MED scheme with a large time step can be more efficient than a usual explicit scheme with a smaller time step only for complex multilevel models with detailed physics. The efficiency of an MED scheme also depends on how the advantage of parallel computing is taken.

Corresponding author address: Dr. Celal S. Konor, Department of Atmospheric Science, Colorado State University, 200 West Lake St., Fort Collins, CO 80523-1371. Email: csk@atmos.colostate.edu

Abstract

For time integrations of the wave equation, it is desirable to use a scheme that is stable over a wide range of the Courant number. Implicit schemes are examples of such schemes, but they do that job at the expense of global calculation, which becomes an increasingly serious burden as the horizontal resolution becomes higher while covering a large horizontal domain. If what an implicit scheme does from the point of view of explicit differencing is looked at, it is a multipoint scheme that requires information at all grid points in space. Physically this is an overly demanding requirement because wave propagation in the real atmosphere has a finite speed. The purpose of this study is to seek the feasibility of constructing an explicit scheme that does essentially the same job as an implicit scheme with a finite number of grid points in space. In this paper, a space-centered trapezoidal implicit scheme is used as the target scheme as an example. It is shown that an explicit space-centered scheme with forward time differencing using an infinite number of grid points in space can be made equivalent to the trapezoidal implicit scheme. To avoid global calculation, a truncated version of the scheme is then introduced that only uses a finite number of grid points while maintaining stability. This approach of constructing a stable explicit scheme is called multipoint explicit differencing (MED). It is shown that the coefficients in an MED scheme can be numerically determined by single-time-step integrations of the target scheme. With this procedure, it is rather straightforward to construct an MED scheme for an arbitrarily shaped grid and/or boundaries. In an MED scheme, the number of grid points necessary to maintain stability and, therefore, the CPU time needed for each time step increase as the Courant number increases. Because of this overhead, the MED scheme with a large time step can be more efficient than a usual explicit scheme with a smaller time step only for complex multilevel models with detailed physics. The efficiency of an MED scheme also depends on how the advantage of parallel computing is taken.

Corresponding author address: Dr. Celal S. Konor, Department of Atmospheric Science, Colorado State University, 200 West Lake St., Fort Collins, CO 80523-1371. Email: csk@atmos.colostate.edu

1. Introduction

Selection of a time-integration scheme is one of the most fundamental issues in the development of an atmospheric model. In particular, the constraint on the length of time step required for computational stability can be a serious burden to the economy of the model, especially when the model is complex so that lengthy calculations are needed for each time step. For explicit time-differencing schemes applied to the wave or advection equation, the constraint is such that the Courant number for the fastest wave propagation must not exceed the order of 1. The alternative is the use of an implicit scheme (or a semi-implicit scheme in which an implicit scheme is partially applied) that is stable for arbitrarily large values of the Courant number.

The implicit time-integration schemes achieve stability at the expense of global calculation, in which information from all grid points is used at each time step. Thus, the use of an implicit scheme remains computationally advantageous only when the gain due to the use of a longer time step supersedes the loss due to the computational burden of global calculation. It has been known that, for the conventional resolution of global models, semi-implicit schemes provide better computational efficiency, especially in the spectral models with global basis functions. The reviews of these schemes and examples of their applications to various models can be found in Kurihara (1965), Holton (1967), Kwizak and Robert (1971), Robert et al. (1972), Bourke (1974), Burridge (1975), Tapp and White (1976), Simmons et al. (1978), Cullen (1990), Tanguay et al. (1990), Purser and Leslie (1994), and Yeh et al. (2002). A hybrid implicit/semi-Lagrangian scheme is also proposed by Israeli et al. (2000).

As the horizontal resolution of a model using an implicit scheme becomes higher, however, the computational burden due to global calculation becomes increasingly serious, particularly in view of parallel computing. The efficiency of an implicit scheme is closely related to the efficiency of the elliptic solver used in the integration, which is commonly based on the multigrid method (Brandt 1977; Fulton et al. 1986; Briggs et al. 2000). While this method is easily applicable and computationally economical for problems with regular boundaries, its generalization to handle arbitrarily shaped grids and/or boundaries can be extremely complicated. Under these circumstances, it is not straightforward to assess the merit of an implicit scheme over the explicit schemes.

There are studies that propose explicit schemes as an alternative to the implicit ones, which are stable for longer time steps than the traditional stability criteria require. Two noticeable examples of this approach are Verwer (1996) and Eriksson et al. (2003). In these examples, the equations are first discretized spatially using a fixed number of grid points in space and then a multilevel time-integration scheme is applied to stabilize solution for a desired large time step. In Eriksson et al. (2003), the multilevel scheme involves in a few stabilizing small time steps and a large time step, in which the length and number of small time steps are automatically adjusted to maintain stability for the desired large time step. However, these schemes are designed for the solutions of equations, in which decay or dissipative effects are dominant. Here we address a completely different problem: the propagation of waves with little or no dissipative effects. Frank et al. (2005) proposes a staggered time-integration scheme combined with a “regularization” of the continuous momentum equation to satisfy a neutral stability criterion for large Courant numbers with respect to wave. However, this method does not eliminate the global calculations because the regularization requires solution of an elliptic equation.

In this study, we seek the feasibility of constructing an explicit scheme that does essentially the same job as an implicit scheme but with a finite number of grid points in space. Any explicit scheme that does the same job as an implicit scheme must inevitably use a multipoint space differencing to maintain stability for large time steps. This can be illustrated considering the following one-dimensional linear equation for a component propagating in the x direction:
i1520-0493-135-11-3862-e1
where Ψ is an arbitrary quantity associated with the component and c is the propagation speed. Consider an Eulerian finite-difference scheme applied to the point on the xt plane shown by the large solid dot in Fig. 1. Here the horizontal and vertical lines show time levels and grid points in space, respectively; Δt and Δx are the time interval and the grid size, respectively. It is clear that the solution of the continuous equation [(1)] at the small solid dot is given by the value at the open dot, which is the intersection of the characteristics, xct = const, shown by the dashed line with the line representing the initial time. For an Eulerian discrete system to recognize this value with a stable algorithm (i.e., basically interpolation rather than extrapolation) for an arbitrary Courant number μcΔtx, it is obvious that multiple grid points are needed for space differencing. We further note that, because c is finite, the number of grid points can be finite, whereas an implicit scheme requires information at an infinite number of grid points unless the domain is bounded or periodic as is the case for the global atmosphere. As the horizontal resolution increases, however, this advantage due to the existence of a boundary or periodicity tends to vanish because a larger and larger number of grid points are needed to cover the entire globe.

Section 2 presents the linearized shallow-water equations and the corresponding wave equation in their continuous form. Sections 3 and 4 illustrate how an explicit scheme that mimics the trapezoidal implicit scheme can be constructed using a rectangular grid. Section 3 presents the trapezoidal implicit scheme with the simplest centered space differencing for the linearized shallow-water equations. Section 4 introduces multipoint explicit differencing (MED) schemes based on multipoint horizontal differencing with the forward time differencing. Section 4 then shows that a member of the MED schemes can be made equivalent to the trapezoidal implicit scheme presented in section 3. We call this the equivalent trapezoidal (ETZ) scheme. Section 4 also introduces truncated equivalent trapezoidal (TETZ) scheme, in which only a finite number of grid points are used, and discusses its stability. Section 5 presents a numerical method to construct a TETZ scheme applicable to any grid. Section 6 discusses the implementation of the TETZ scheme in the nonlinear shallow-water equations on a square grid and demonstrates its performance in the numerical simulation of gravity wave propagations in regularly and arbitrarily shaped domains. Finally, section 7 presents the summary and conclusions.

2. Linearized shallow-water equations

For the purpose of later use in this paper, we present the linearized shallow-water equations with no basic current and no Coriolis force, and the corresponding wave equation. The linearized continuity and divergence equations are given by
i1520-0493-135-11-3862-e2a
and
i1520-0493-135-11-3862-e2b
where h and H are the perturbation and mean height of the free surface, respectively; δ is the divergence given by · v, where v is the horizontal velocity; and g is the gravitational acceleration. For this system, the velocity potential χ and the divergent component of the velocity can be obtained from ∇2χ = δ and v = χ, respectively. The wave equation for h can be derived by eliminating the divergence between (2a) and (2b) as
i1520-0493-135-11-3862-e3
where c is the propagation speed of the waves given by cgH.

3. Discretization of the linearized shallow-water equations with the trapezoidal implicit time-integration scheme

Here we present a discrete version of (2a) and (2b) with the trapezoidal implicit scheme in time and the simplest centered scheme in space on a rectangular grid shown in Fig. 2. The discrete continuity and divergence equations are
i1520-0493-135-11-3862-e4a
and
i1520-0493-135-11-3862-e4b
respectively. Here the superscript denotes a time level, i and j are horizontal integer indices for the grid points increasing with x and y, respectively. In (4b), the discrete Laplacian operator is defined as
i1520-0493-135-11-3862-e4c
From (4a) and (4b), we can derive the discrete version of the wave equation [(3)] given by
i1520-0493-135-11-3862-e5
We now analyze the normal modes of the discrete system given by (4a) and (4b) seeking the solutions in the form
i1520-0493-135-11-3862-e6
where ĥ(n) and δ̂(n) are the complex amplitudes of h and δ at time level n, respectively; i = −1; and k and l are the wavenumbers in the x and y directions, respectively. Substituting (6) into (4a) and (4b) with (4c) and then eliminating δ̂(n+1) between the continuity and divergence equations, we obtain the equations that predict the complex amplitudes of the wave as
i1520-0493-135-11-3862-e7a
and
i1520-0493-135-11-3862-e7b
respectively. In (7a) and (7b), we used the definitions
i1520-0493-135-11-3862-e8
where
i1520-0493-135-11-3862-e9a
ξkΔx, ηlΔy, and the Courant number in the x and y directions are defined by
i1520-0493-135-11-3862-e9b
respectively. If we let ĥ(n+1) = λĥ(n) and δ̂(n+1) = λδ̂(n), where λ is the complex amplification factor of a normal mode, (7a) and (7b) with (8) yield
i1520-0493-135-11-3862-e10a
and
i1520-0493-135-11-3862-e10b
respectively. For nontrivial solutions of (10a) and (10b), the amplification factor λ must satisfy
i1520-0493-135-11-3862-e11
and thus
i1520-0493-135-11-3862-e12
where F and G are given by (8). Because the amplification factor λ has an imaginary part and |λ|2 = 1, the solution is a progressive neutral wave.

4. Discretization of the linearized shallow-water equations with MED

Our motivation in this paper is to construct an explicit scheme that mimics the trapezoidal implicit scheme. As discussed in the introduction, multipoint space differencing is needed to accomplish this task. Below we start with introduction of multipoint averaging, which is the essential part of the scheme to be constructed. Then, a family of MED schemes are discussed using a rectangular grid as an example.

a. Two-dimensional multipoint averaging

Let Ψ be an arbitrary variable. We express the two-dimensional multipoint weighted averaging of Ψ applied to the target grid point (i, j) by
i1520-0493-135-11-3862-e13
where A is an averaging operator and ai′,j is a weighting coefficient satisfying
i1520-0493-135-11-3862-e14a
For symmetric averaging, we have
i1520-0493-135-11-3862-e14b
The averaging is performed over the “domain of dependence” from i′= −J to +J and j′= −J to +J. Using (14a) in (13), we can obtain an alternative expression for the multipoint averaging as
i1520-0493-135-11-3862-e15
Because the operator A is independent of i and j, the multipoint finite difference Laplacian satisfies
i1520-0493-135-11-3862-e16
where ∇̃2 is defined by (4c).

b. An explicit scheme based on multipoint averaging

Expecting that a MED scheme based on the multipoint averaging described above can be made equivalent to the trapezoidal implicit scheme, we first write the discrete continuity and divergence equations as
i1520-0493-135-11-3862-e17a
and
i1520-0493-135-11-3862-e17b
respectively, where C is another multipoint averaging operator. In (17b), (16) has been used. To examine the stability of the solution of (17a) and (17b), we first use (6) in {C(ψ(n))}i,j and {A(ψ(n))}i,j, where ψ is either h or δ, to obtain
i1520-0493-135-11-3862-e18
Using (15) and a similar expression for the operator C, we find that and in (18) can be expressed as
i1520-0493-135-11-3862-e19
In (19), ci′,j and ai′,j are weighting coefficients to be determined while satisfying (14a) with (14b). Then, using (6) and (18), we can rewrite (17a) and (17b) as
i1520-0493-135-11-3862-e20a
and
i1520-0493-135-11-3862-e20b
These equations are similar to (7a) and (7b) for the trapezoidal scheme but now (ξ, η) and (ξ, η) are given by (19) instead of (8).
Following a procedure similar to the derivation of (12), we find the expression for the amplification factor for the MED scheme as
i1520-0493-135-11-3862-e21
For the solution to be stable,
i1520-0493-135-11-3862-e22
must be maintained.

c. ETZ scheme

In this subsection, we present a procedure to determine the coefficients ai′,j and ci′,j in such a way that the explicit scheme given by (17a) and (17b) is equivalent to the trapezoidal implicit scheme given by (7a) and (7b). For this purpose, we require that (20a) and (20b) share the same ĥ(n+1) and δ̂(n+1) with (7a) and (7b), respectively, from the same initial condition, ĥ(n) and δ̂(n). This can be achieved by choosing
i1520-0493-135-11-3862-e23
Then, in view of (19), ci′,j and ai′,j are the coefficients in the cosine Fourier expansion of the functions F(ξ, η) and G(ξ, η) given by (8), respectively. We calculate the coefficients ci′,j and ai′,j with a sufficiently large value of J (=250) so that the approximate series given by and are virtually identical to F and G, respectively. The sum of ci′,j and ai′,j between −250 ≤ i′ ≤ 250 and −250 ≤ j′ ≤ 250 are approximately unity, automatically satisfying (14a). We present the coefficients obtained in this way in Fig. 3 for j′ = 0 and selected values of μ. In the figure, we see that the coefficients ai′,0 monotonically and rapidly decrease away from the target point (i′ = 0) and finally become negligible for all Courant numbers. The coefficients ci′,0 start with a minimum value at the target point (i′ = 0), negative for the Courant numbers larger than 1, then become positive and maximum at the surrounding grid points (i′ = 1 and i′ = −1), and then monotonically and rapidly decrease away from those points. The negative weighting of c0,0 for the Courant numbers larger than 1 indicates that the contribution of the target point to the averaging is through an extrapolation.

In conclusion, the scheme given by (17a) and (17b) is equivalent to the trapezoidal scheme discussed in section 3 if the coefficients given by (23) are used. Figure 3 presents examples of such coefficients. We call this scheme the ETZ scheme.

d. TETZ scheme

The equivalent trapezoidal scheme presented in section 4c is an explicit scheme, but it behaves as the implicit trapezoidal scheme does. This is achieved by using information from practically all grid points in the domain because the scheme uses a very large J. As stated earlier, our objective is to construct explicit schemes that do essentially the same job as the implicit trapezoidal scheme but using a finite number of grid points. Here we present the TETZ scheme, which accomplishes this task. The prospect for such a scheme can be seen, for example, in the distribution of the coefficients ai′,0 and ci′,0 in Fig. 3, which shows that the coefficients become negligible at grid points sufficiently far from the target point.

In truncating the coefficients, we are guided by the stability analysis presented in section 4b. We first define T(ξ, η) and T(ξ, η) in a way similar to (19) but with the coefficients ãi′,j and i′,j truncated at a finite J and renormalized so that their sums over −Ji′ ≤ J and −Jj′ ≤ J are unity. The criterion for neutral solutions (|λ|2 = 1) obtained from (22) but applied to the truncated system is satisfied if we choose
i1520-0493-135-11-3862-e24a
In choosing the signs in (24a), we are guided by the definition of F given by (8). With this choice of T(ξ, η), the stability criterion is reduced to
i1520-0493-135-11-3862-e24b
The first step of the truncation and renormalization process is to find the minimum J for a given Courant number μ that satisfies (24b) while maintaining
i1520-0493-135-11-3862-e25
To find the coefficients in (25), we use a least squares method, which minimizes the mean-square error between the continuous function G given by (8) and its approximation T, which is a truncated version of given by (19), for the intervals −πξπ and −πηπ. Thus, we let
i1520-0493-135-11-3862-e26
which yields
i1520-0493-135-11-3862-e27
To derive (27), we treat ã0,0 as a function of the rest of the coefficients as suggested by (25). From (19) and (25), we obtain ∂T/∂ã0,0 = ¼, ∂T/∂ãi′,j = cos(ξi′) cos(ηj′), and ∂ã0,0/∂ãi′,j = −4. Using these relations, ∫ππππ T dξ dη = π2ã0,0 and ∫ππππ T cos(ξi′) cos(ηj′) dξ dη = π2ãi′,j in (27), we obtain
i1520-0493-135-11-3862-e28a
and, similarly,
i1520-0493-135-11-3862-e28b
and
i1520-0493-135-11-3862-e28c
Note that, in (28a) to (28c), G is given by (8). Finally, ã0,0 is obtained by using (28a)(28c) in (25).

To study the stability of the truncated system, we first obtain the coefficient ãi′,j from (28a–c) and (25) for different Js for a given Courant number μ. Then we calculate (8μ*T)2max ≡ max{[8μ*T(ξ, η)]2} in the interval of −πξπ and −πηπ. Recall that the stability criterion given by (24b) requires (8μ*T)2max ≤ 1. Figure 4 illustrates (8μ*T)2max as a function of J for selected Courant numbers. The figure shows that (8μ*T)2max asymptotically approaches to unity from above as J increases. Because the stability is gradually reached with increasing J, we tabulate the minimum Js that keep (8μ*T)2max under 1.05 and 1.01 for the Courant numbers between 1 and 10 in Table 1. In practice, we use (8μ*T)2max ≤ 1.01 as the stability limit and the corresponding minimum Js in the truncation process. In Table 2, we show the truncated weighting coefficient ãi′,j obtained with J given in the lower part of Table 1. Using a similar procedure, we find the truncated weighting coefficient i′,j. The coefficients obtained in this way are tabulated in Table 3 for the Courant numbers presented in Table 2. Although not shown, these coefficients are virtually identical to those for the ETZ scheme. As expected, this implies that the truncation does not substantially modify the coefficients.

In summary, we have shown that a scheme stable for a wide range of the Courant number can be constructed for a rectangular grid following the MED approach, in which the coefficients in the scheme are analytically determined. For an irregular grid, however, we must find a more general method to construct a MED scheme that is applicable to any grid. This problem is discussed in the next section.

5. TETZ scheme for an arbitrary grid

To develop a TETZ scheme for a more general model, in this section we present an algorithm to numerically determine the weighting coefficients for an arbitrary grid. The algorithm assumes that a discrete linearized shallow-water equation model with the trapezoidal implicit scheme is available and uses the results of single-step integrations of the model starting from a selected set of initial conditions. It then finds the coefficients of a MED scheme that produces identical results to those obtained by the trapezoidal scheme from the same set of initial conditions.

To perform the one-step integrations, the implicit trapezoidal scheme given by (4a) and (4b) is manipulated to the form
i1520-0493-135-11-3862-e29a
where m2 ≡ 4/[gHt)2] and ∇̃2 is the discrete Laplacian defined for the grid used, and
i1520-0493-135-11-3862-e29b
respectively. In (29a) and (29b), we use a single index j to identify the grid points on the 2D domain. The MED scheme for the shallow-water equations given by (17a) and (17b) are also generalized to an arbitrary grid as
i1520-0493-135-11-3862-e30a
and
i1520-0493-135-11-3862-e30b
where j and j + j′ denote a target grid point at which predictions are made and an arbitrary grid point surrounding the target point, respectively, while J denotes the number of all grid points used in the scheme, the determination of which will be discussed later. While the scheme is required to be identical to the ETZ and TETZ schemes for a very large J and a finite J, respectively. For consistency, the weighting coefficients must satisfy
i1520-0493-135-11-3862-e31
To find the coefficients for the target point located at j = j0, we use the following initial condition:
i1520-0493-135-11-3862-e32a
i1520-0493-135-11-3862-e32b
on a grid shown in Fig. 5. In (32a) and (32b), σ is the area of the grid cell represented by the grid point. Note that the initial condition satisfies Σj δ(0)jσj = 0, where the sum is taken over the entire domain. This particular initial condition is selected because it provides a relatively simple way of determining the coefficients for the TETZ scheme as described below.
After the single-step integration of the implicit scheme given by (29a) and (29b), we obtain h(1)j and δ(1)j, which are predicted values of h and δ at the grid point j. We now require that the TETZ scheme given by (30a) and (30b) produce identical values of h(1)j and δ(1)j with the same initial condition. Using the values obtained by the implicit scheme and the initial condition given by (32a) and (32b) in (30a) and (30b) for given j0 and j′, we obtain
i1520-0493-135-11-3862-e33a
and
i1520-0493-135-11-3862-e33b
respectively. We repeat this procedure for all possible j′ while still fixing j0. To determine aj0,0 and cj0,0, we write ΣJj′≠0(33a)σj0+j and ΣJj′≠0(33b)σj0+j as
i1520-0493-135-11-3862-e34a
and
i1520-0493-135-11-3862-e34b
respectively. Because the left-hand sides of (34a) and (34b) are equal to 1 − aj0,0 and 1 − cj0,0 from (31), respectively, we can write the equations that determine aj0,0 and cj0,0 as
i1520-0493-135-11-3862-e35a
and
i1520-0493-135-11-3862-e35b
respectively. Then the coefficients aj0,j and cj0,j can be obtained from (33a) and (33b), respectively.

For a square grid, the numerical procedure described above produces coefficients virtually identical to those shown in Tables 2 and 3, which have been obtained with the analytical method. For a more general grid, the procedure must be repeated for all possible j0 unless we can take advantage of the geometrical symmetry that may exist in the grid. The choice of the minimum Js should be made requiring that the scheme be stable. For a quasi-regular grid such as a geodesic grid, the choice can be guided by the stability for a similar regular grid such as the hexagonal grid, whereas it must be done empirically for a fully irregular grid.

For comparison purposes, we performed numerical integrations of linearized shallow-water equations using explicit schemes with small time steps, the implicit trapezoidal (TZ) and TETZ schemes with different Courant numbers. The explicit integration uses a two-level forward–backward time-integration scheme with a second-order centered special differencing. The implicit TZ scheme is given by (29a) and (29b) with a second-order centered Laplacian. In solving (29a), we used an iteration method. The coefficients of the TETZ scheme are obtained by the procedure described by (29a)(30b) on a Cartesian grid using Js for the given Courant numbers tabulated in the bottom row of Table 1. The explicit, TZ and TETZ models described above are applied to the simulation of gravity wave propagations on a square grid (100-km grid distance in both directions) in a square domain (200 grid points in both directions) with cyclic boundary conditions. The initial surface height consists of a uniform component of 10 km deep and a cone-shaped perturbation with a 100-m-high maximum centered at the middle of the domain. Figure 6 shows the surface height perturbation at t = 320 min obtained by the explicit with a small time step, the implicit TZ and TETZ schemes for the Courant numbers, 1, 3, and 10. The simulations with the TZ and TETZ schemes are virtually identical and both show reasonable propagation speed but dispersion and dissipation errors increase with the Courant number. This type of errors is expected for the large Courant numbers with an implicit and the mimicking MED schemes, which should be selected for their superior stability characteristics rather than the accuracy they provide.

We also quantitatively studied the errors of the simulated surface height by calculating the root-mean-square (RMS) error between the solutions with the TZ and TETZ schemes obtained by various horizontal grid distances, namely 12.5, 25, 50, 100, and 200 km, and a reference solution for the fixed Courant numbers 3 and 5. The TZ solution with 12.5-km grid distance for the respective Courant number is selected as the reference solution. The simulations use the same initial condition discussed in the previous paragraph and the model is integrated for 80 min. Because the solutions are virtually axisymmetric, we analyzed the one-dimensional surface height data on a row at the middle of the domain. Interpolations from lower horizontal resolutions to the reference resolution is made through a running fifth-order polynomial fit using information from six grid points. In Fig. 7, the resulting RMS errors are displayed as functions of the grid distance in a log–log diagram for two Courant numbers. The figure shows that the RMS errors with the TZ and TETZ schemes are virtually identical and the slope of the line connecting the scattered points indicates approximately a second-order error convergence in time and space.

6. Application of the MED to the nonlinear shallow-water equations on a square grid and numerical simulations of surface waves

To demonstrate the performance of the MED scheme on a square grid, we have constructed a nonlinear 2D shallow-water model based on the ETZ and TETZ schemes using the coefficients ai′,j and ci′,j obtained through the procedure described in section 4. For the tests we present here, we assume that the fluid depth is quasi-uniform, there is no Coriolis force, and the fluid is at rest initially. The MED scheme for the nonlinear shallow-water equations can be written as
i1520-0493-135-11-3862-e36a
and
i1520-0493-135-11-3862-e36b
where the superscript asterisk denotes a provisional value at time level (n + 1), vχ is the divergent component of the velocity, and K is the kinetic energy ½v2χ. The predictor we used to determine h(*), vχ(*), and K(*) is described in the appendix.

The model is used to simulate propagation of waves with different values of the Courant number. The model domain is a 30 000 km × 30 000 km with cyclic or rigid lateral boundaries. The horizontal grid size is Δx = Δy = 100 km. There is no topography at the lower boundary. The initial height of the free surface consists of a uniform height H (=10 km) and a cone-shaped perturbation field with a maximum height of 100 m. The fluid is initially at rest. The time step is determined from the specified value of the Courant number by using c ≈ 313 m s−1 and the grid size; for example, Δt = 319 s and Δt = 1595 s for μ = 1 and μ = 5, respectively.

a. Simulations with cyclic boundaries

In these simulations, the coefficients given in Tables 2 and 3 are used. The initial height of the free surface for these simulations is shown in Fig. 8. The snapshots of simulated waves are shown in Fig. 9 at t = 7, 14, 20, and 27 h from the initial for the Courant numbers 1, 3, 5, and 10 with corresponding Js 3, 7, 12, and 24. These results show propagation of waves as expected from the initial location of the perturbation. The solution is stable and close to neutral for all Courant numbers. The simulations with very large Js (not shown), for which the scheme becomes the ETZ scheme, are virtually identical to those shown in Fig. 9, demonstrating that solutions with the TETZ scheme are close approximations to those with the implicit trapezoidal scheme.

b. Simulations with rigid boundaries

For a domain with cyclic boundaries such as the one used in section 6a, the scheme has an identical form at all grid points. For a rectangular or irregularly shaped domain with rigid boundaries, however, the coefficients in the scheme vary, especially at grid points close to the boundaries. For such a domain, we calculate the coefficients using the numerical method described in section 5 ignoring contributions from the grid points outside of the domain. The algorithm then automatically adjusts the coefficients on the inner grid points.

To demonstrate the performance of the TETZ scheme with rigid boundaries, we perform two simulations, first of which is similar to those presented in section 6a but with the initial perturbation placed off-center in the domain. The snapshots of simulated waves are shown in Fig. 10 at t = 7, 14, 20, and 27 h from the initial for Courant numbers 3 and 5 with corresponding Js 7 and 12. These results show propagation of waves as expected from the initial location of the perturbation with reflections at the boundaries. The second simulation we present here uses an irregularly shaped domain with rigid boundaries. The initial condition and the domain are shown in Fig. 11, in which the green patches indicate land. The snapshots of simulated waves are shown in Fig. 12 at t = 4, 8, 12, and 16 h from the initial for Courant numbers 3 and 5 with corresponding Js 7 and 12. These results show no computational difficulties due to the irregular boundary.

7. Summary and conclusions

The purpose of this study is to seek the feasibility of constructing an explicit scheme that does essentially the same job as an implicit scheme with a finite number of grid points in space. In this paper, we have selected the TZ scheme as the target implicit scheme to be mimicked. A family of MED schemes has been introduced to do the job.

To illustrate how a TETZ scheme as a member of the MED family can be constructed, we first considered the case of a rectangular grid with cyclic boundary conditions, for which the coefficients in the scheme can be analytically determined. We then presented a numerical method to construct such a scheme applicable to any grid with arbitrary boundary conditions including irregular rigid boundaries. The results of the analytical method for the square grid were used to verify the numerical method. Performance of the TETZ scheme with the nonlinear shallow-water equations was demonstrated with cyclic and irregular rigid boundaries. These results show that the TETZ scheme can produce stable and near-neutral solutions even when the Courant number is much larger than unity.

The issue we have addressed in this paper is how to construct a nonglobal scheme that is applicable over a wide range of the Courant number. We do not necessarily claim that the scheme presented here is more economical than usual explicit schemes with smaller time steps or any particular semi-implicit scheme. In a MED scheme, the number of grid points necessary to maintain stability and, therefore, the CPU time needed for each time step increase as the Courant number increases. The overall economy of a MED scheme, however, depends on many other factors, such as the code structure for parallel computing and the sophistication of model dynamics and physics. We do claim, however, that with a MED scheme the Courant number for the fast-moving waves does not limit the stability of a model. This is advantageous for complex multilevel models with detailed physics, in which the fraction of the overhead due to the use of multiple points in a MED scheme can be negligibly small. It is shown that the MED approach can easily be applied to arbitrary grids with irregular boundaries, and does not require information from all the grid points in the domain.

In this paper, we assumed that the height of the free surface is quasi-uniform and the amplitude of the disturbance is sufficiently smaller than the mean height. This allowed us to use the linearized system to determine the coefficients in the TETZ scheme. Although the performance of the system was tested with the nonlinear system, the initial conditions were chosen in such a way that the system is quasi-linear. We are currently working on generalization of the approach so that it can also be applied to the models with highly varying layer thickness. This is the case, for example, with the hybrid isentropic-sigma vertical coordinate model of Konor and Arakawa (1997). We are also working on implementation of this approach in 3D atmospheric models, applying multipoint averaging to those terms that would vanish if the anelastic approximation were used in the model. In this way, the model can be made stable for large Courant numbers with respect to acoustic waves, leaving anelastic motions unaffected.

In principle, the MED approach can be applied to any implicit or partially implicit scheme other than the implicit trapezoidal scheme. Also, we may be able to rewrite the MED scheme as a multistep iterative scheme with fewer grid points in space. If this is possible, it brings more flexibility to the MED method by introducing a choice between the multiple steps with fewer spatial grid points and fewer steps with more spatial grid points without changing the stability property of the solutions. In practical applications, the choice between these and any hybrid approaches can be made to maximize the computational efficiency for the specific computer architecture.

The MED approach can also be applied to the advection equation. We are working on the construction of a family of MED schemes that are stable for a wide range of the Courant number for advection while using an Eulerian approach.

Acknowledgments

We thank Prof. David Randall for his encouragement and support of this work. Dr. Ross Heikes’s help in plotting some of the figures is greatly appreciated. We also thank two anonymous reviewers for their constructive comments and suggestions. This work was funded by the U.S. DOE under Grants DE-FG02-04ER63848, DE-FG02-02ER63370, and DE-FC02-06ER64302 to Colorado State University, CSU Contracts G-3816-3 and G-3818-1 to UCLA, and NSF under Grant ATM-0415184 to Colorado State University.

REFERENCES

  • Bourke, W., 1974: A multi-level spectral model. I. Formulation and hemispheric integrations. Mon. Wea. Rev., 102 , 687701.

  • Brandt, A., 1977: Multi-level adaptive solutions to boundary value problems. Math. Comput., 31 , 333390.

  • Briggs, W. L., V. E. Henson, and F. McCormick, 2000: A Multigrid Tutorial. 2d ed. SIAM, 193 pp.

  • Burridge, D. M., 1975: A split semi-implicit reformulation of the Bushby–Timpson 10-level model. Quart. J. Roy. Meteor. Soc., 101 , 777792.

    • Search Google Scholar
    • Export Citation
  • Cullen, J. P., 1990: A test of a semi-implicit integration technique for a fully compressible non-hydrostatic model. Quart. J. Roy. Meteor. Soc., 116 , 12531258.

    • Search Google Scholar
    • Export Citation
  • Eriksson, C., C. Johnson, and A. Logg, 2003: Explicit time stepping for stiff ODE’s. SIAM J. Sci. Comput., 25 , 11421157.

  • Frank, J., S. Reich, A. Staniforth, A. White, and N. Wood, 2005: Analysis of a regularized, time-staggered method and its link to the semi-implicit method. Atmos. Sci. Lett., 6 , 97104.

    • Search Google Scholar
    • Export Citation
  • Fulton, S. R., P. E. Ciesielski, and W. Schubert, 1986: Multi-grid methods for elliptic problems: A review. Mon. Wea. Rev., 114 , 943959.

    • Search Google Scholar
    • Export Citation
  • Holton, J. R., 1967: A stable finite difference scheme for the linearized vorticity and divergence equation system. J. Appl. Meteor., 6 , 519522.

    • Search Google Scholar
    • Export Citation
  • Israeli, M., N. H. Naik, and M. A. Cane, 2000: An unconditionally stable scheme for the shallow water equations. Mon. Wea. Rev., 128 , 810823.

    • Search Google Scholar
    • Export Citation
  • Konor, C. S., and A. Arakawa, 1997: Design of an atmospheric model based on a generalized vertical coordinate. Mon. Wea. Rev., 125 , 16491673.

    • Search Google Scholar
    • Export Citation
  • Kurihara, Y., 1965: On the use of implicit and iterative methods for the time integration of the wave equation. Mon. Wea. Rev., 93 , 3346.

    • Search Google Scholar
    • Export Citation
  • Kwizak, M., and A. J. Robert, 1971: A semi-implicit scheme for grid point atmospheric models of the primitive equations. Mon. Wea. Rev., 99 , 3236.

    • Search Google Scholar
    • Export Citation
  • Purser, R. J., and L. M. Leslie, 1994: An efficient semi-Lagrangian scheme using third-order semi-implicit time integration and forward trajectories. Mon. Wea. Rev., 122 , 745756.

    • Search Google Scholar
    • Export Citation
  • Robert, A., J. Henderson, and C. Turnbull, 1972: An implicit time integration scheme for baroclinic models of the atmosphere. Mon. Wea. Rev., 100 , 329334.

    • Search Google Scholar
    • Export Citation
  • Simmons, A. J., B. J. Hoskins, and D. M. Burridge, 1978: Stability of the semi-implicit method of time integration. Mon. Wea. Rev., 106 , 405412.

    • Search Google Scholar
    • Export Citation
  • Tanguay, M., A. Robert, and R. Laprise, 1990: A semi-implicit semi-Lagrangian fully compressible regional forecast model. Mon. Wea. Rev., 118 , 19701980.

    • Search Google Scholar
    • Export Citation
  • Tapp, M., and P. W. White, 1976: A non-hydrostatic mesoscale model. Quart. J. Roy. Meteor. Soc., 102 , 277296.

  • Verwer, J. G., 1996: Explicit Runge–Kutta method for parabolic partial differential equations. Appl. Num. Math, 22 , 359379.

  • Yeh, K-S., J. Côté, S. Gravel, A. Méthot, A. Patoine, M. Roch, and A. Staniforth, 2002: The CMC–MRB Global Environmental Multiscale (GEM) Model. Part III: Nonhydrostatic formulation. Mon. Wea. Rev., 130 , 339356.

    • Search Google Scholar
    • Export Citation

APPENDIX

The Predictor Used in the Nonlinear Terms of the Shallow-Water Model

We predict h for time level (n + 1) by the forward scheme using
i1520-0493-135-11-3862-ea1
where the components of vχ are obtained by
i1520-0493-135-11-3862-ea2a
and
i1520-0493-135-11-3862-ea2b
at the half-integer grid points shown in Fig. 2. The velocity potential χ in (A2a) and (A2b) is obtained by solving
i1520-0493-135-11-3862-ea3
in the domain with proper boundary conditions. A second-order-centered finite differencing is used in the discretization of the divergence ∇̃ and Laplacian ∇̃2. The kinetic energy at the point (i, j) is defined by
i1520-0493-135-11-3862-ea4
Using (A4), we predict the divergence by
i1520-0493-135-11-3862-ea5
Then, from [∇̃2χ(*)]i,j = δi,j(*) and following the same procedure given by (A2a), (A2b), and (A4), we find vχ(*) and K(*). To find h(*), we use
i1520-0493-135-11-3862-ea6
where h(†)is obtained from (A1). Here vχ(*), K(*), and h(*) are used in (36a) and (36b).

Fig. 1.
Fig. 1.

A space–time grid used in the discretization of (1). The dashed line shows a characteristic line along which the initial value at the open circle is carried.

Citation: Monthly Weather Review 135, 11; 10.1175/2007MWR1923.1

Fig. 2.
Fig. 2.

The portion of the grid used in the discretizations.

Citation: Monthly Weather Review 135, 11; 10.1175/2007MWR1923.1

Fig. 3.
Fig. 3.

Coefficients ai′,0 and ci′,0 as a function of i′ for selected Courant numbers with a sufficiently large J. Only coefficients for −5 ≤ i′ ≤ 5 are shown.

Citation: Monthly Weather Review 135, 11; 10.1175/2007MWR1923.1

Fig. 4.
Fig. 4.

The maximum of (8μ*T)2 in the interval of −πξπ and −πηπ as a function of J.

Citation: Monthly Weather Review 135, 11; 10.1175/2007MWR1923.1

Fig. 5.
Fig. 5.

Selected cells of an arbitrary grid.

Citation: Monthly Weather Review 135, 11; 10.1175/2007MWR1923.1

Fig. 6.
Fig. 6.

A 1D section of surface height perturbation of a two-dimensionally propagating wave at t = 320 min obtained by explicit, TZ, and the TETZ schemes for different Courant numbers. Figure shows half of the domain from the domain center indicated by grid point 0 to the boundary of the domain. Simulation starts from a cone-shaped initial perturbation at the center of the domain. Arrows indicate the propagation direction.

Citation: Monthly Weather Review 135, 11; 10.1175/2007MWR1923.1

Fig. 7.
Fig. 7.

RMS errors for different spatial and temporal resolutions for selected Courant numbers. Squares and circles represent solutions of the TZ and TETZ schemes, respectively. The slope of the dashed line represents second-order convergence.

Citation: Monthly Weather Review 135, 11; 10.1175/2007MWR1923.1

Fig. 8.
Fig. 8.

Initial free-surface height perturbation placed at the center of the domain used in the simulations with the cyclic boundaries.

Citation: Monthly Weather Review 135, 11; 10.1175/2007MWR1923.1

Fig. 9.
Fig. 9.

Time evolution of the free-surface height simulated with the cyclic boundaries by using the Courant numbers μ = 1, 3, 5, and 10 and corresponding J = 3, 7, 12, and 24, respectively.

Citation: Monthly Weather Review 135, 11; 10.1175/2007MWR1923.1

Fig. 10.
Fig. 10.

Time evolution of the free-surface height simulated with the rigid boundaries by using the Courant numbers μ = 3 and 5 and corresponding J = 7 and 12, respectively.

Citation: Monthly Weather Review 135, 11; 10.1175/2007MWR1923.1

Fig. 11.
Fig. 11.

Initial free-surface height perturbation used in the simulations with an irregular rigid boundary. Green patches indicate land.

Citation: Monthly Weather Review 135, 11; 10.1175/2007MWR1923.1

Fig. 12.
Fig. 12.

Same as in Fig. 8, but for a domain with an arbitrarily shaped rigid boundary. Green patches indicate land.

Citation: Monthly Weather Review 135, 11; 10.1175/2007MWR1923.1

Table 1.

Minimum Js that maintain (8μ*T)2max less than (top row) 1.05 and (bottom row) 1.01.

Table 1.
Table 2.

Truncated coefficients ãi′,j for selected Courant numbers and corresponding Js.

Table 2.
Table 3.

Same as in Table 2, but for i′,j.

Table 3.
Save
  • Bourke, W., 1974: A multi-level spectral model. I. Formulation and hemispheric integrations. Mon. Wea. Rev., 102 , 687701.

  • Brandt, A., 1977: Multi-level adaptive solutions to boundary value problems. Math. Comput., 31 , 333390.

  • Briggs, W. L., V. E. Henson, and F. McCormick, 2000: A Multigrid Tutorial. 2d ed. SIAM, 193 pp.

  • Burridge, D. M., 1975: A split semi-implicit reformulation of the Bushby–Timpson 10-level model. Quart. J. Roy. Meteor. Soc., 101 , 777792.

    • Search Google Scholar
    • Export Citation
  • Cullen, J. P., 1990: A test of a semi-implicit integration technique for a fully compressible non-hydrostatic model. Quart. J. Roy. Meteor. Soc., 116 , 12531258.

    • Search Google Scholar
    • Export Citation
  • Eriksson, C., C. Johnson, and A. Logg, 2003: Explicit time stepping for stiff ODE’s. SIAM J. Sci. Comput., 25 , 11421157.

  • Frank, J., S. Reich, A. Staniforth, A. White, and N. Wood, 2005: Analysis of a regularized, time-staggered method and its link to the semi-implicit method. Atmos. Sci. Lett., 6 , 97104.

    • Search Google Scholar
    • Export Citation
  • Fulton, S. R., P. E. Ciesielski, and W. Schubert, 1986: Multi-grid methods for elliptic problems: A review. Mon. Wea. Rev., 114 , 943959.

    • Search Google Scholar
    • Export Citation
  • Holton, J. R., 1967: A stable finite difference scheme for the linearized vorticity and divergence equation system. J. Appl. Meteor., 6 , 519522.

    • Search Google Scholar
    • Export Citation
  • Israeli, M., N. H. Naik, and M. A. Cane, 2000: An unconditionally stable scheme for the shallow water equations. Mon. Wea. Rev., 128 , 810823.

    • Search Google Scholar
    • Export Citation
  • Konor, C. S., and A. Arakawa, 1997: Design of an atmospheric model based on a generalized vertical coordinate. Mon. Wea. Rev., 125 , 16491673.

    • Search Google Scholar
    • Export Citation
  • Kurihara, Y., 1965: On the use of implicit and iterative methods for the time integration of the wave equation. Mon. Wea. Rev., 93 , 3346.

    • Search Google Scholar
    • Export Citation
  • Kwizak, M., and A. J. Robert, 1971: A semi-implicit scheme for grid point atmospheric models of the primitive equations. Mon. Wea. Rev., 99 , 3236.

    • Search Google Scholar
    • Export Citation
  • Purser, R. J., and L. M. Leslie, 1994: An efficient semi-Lagrangian scheme using third-order semi-implicit time integration and forward trajectories. Mon. Wea. Rev., 122 , 745756.

    • Search Google Scholar
    • Export Citation
  • Robert, A., J. Henderson, and C. Turnbull, 1972: An implicit time integration scheme for baroclinic models of the atmosphere. Mon. Wea. Rev., 100 , 329334.

    • Search Google Scholar
    • Export Citation
  • Simmons, A. J., B. J. Hoskins, and D. M. Burridge, 1978: Stability of the semi-implicit method of time integration. Mon. Wea. Rev., 106 , 405412.

    • Search Google Scholar
    • Export Citation
  • Tanguay, M., A. Robert, and R. Laprise, 1990: A semi-implicit semi-Lagrangian fully compressible regional forecast model. Mon. Wea. Rev., 118 , 19701980.

    • Search Google Scholar
    • Export Citation
  • Tapp, M., and P. W. White, 1976: A non-hydrostatic mesoscale model. Quart. J. Roy. Meteor. Soc., 102 , 277296.

  • Verwer, J. G., 1996: Explicit Runge–Kutta method for parabolic partial differential equations. Appl. Num. Math, 22 , 359379.

  • Yeh, K-S., J. Côté, S. Gravel, A. Méthot, A. Patoine, M. Roch, and A. Staniforth, 2002: The CMC–MRB Global Environmental Multiscale (GEM) Model. Part III: Nonhydrostatic formulation. Mon. Wea. Rev., 130 , 339356.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    A space–time grid used in the discretization of (1). The dashed line shows a characteristic line along which the initial value at the open circle is carried.

  • Fig. 2.

    The portion of the grid used in the discretizations.

  • Fig. 3.

    Coefficients ai′,0 and ci′,0 as a function of i′ for selected Courant numbers with a sufficiently large J. Only coefficients for −5 ≤ i′ ≤ 5 are shown.

  • Fig. 4.

    The maximum of (8μ*T)2 in the interval of −πξπ and −πηπ as a function of J.

  • Fig. 5.

    Selected cells of an arbitrary grid.

  • Fig. 6.

    A 1D section of surface height perturbation of a two-dimensionally propagating wave at t = 320 min obtained by explicit, TZ, and the TETZ schemes for different Courant numbers. Figure shows half of the domain from the domain center indicated by grid point 0 to the boundary of the domain. Simulation starts from a cone-shaped initial perturbation at the center of the domain. Arrows indicate the propagation direction.

  • Fig. 7.

    RMS errors for different spatial and temporal resolutions for selected Courant numbers. Squares and circles represent solutions of the TZ and TETZ schemes, respectively. The slope of the dashed line represents second-order convergence.

  • Fig. 8.

    Initial free-surface height perturbation placed at the center of the domain used in the simulations with the cyclic boundaries.

  • Fig. 9.

    Time evolution of the free-surface height simulated with the cyclic boundaries by using the Courant numbers μ = 1, 3, 5, and 10 and corresponding J = 3, 7, 12, and 24, respectively.

  • Fig. 10.

    Time evolution of the free-surface height simulated with the rigid boundaries by using the Courant numbers μ = 3 and 5 and corresponding J = 7 and 12, respectively.

  • Fig. 11.

    Initial free-surface height perturbation used in the simulations with an irregular rigid boundary. Green patches indicate land.

  • Fig. 12.

    Same as in Fig. 8, but for a domain with an arbitrarily shaped rigid boundary. Green patches indicate land.

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