1. Introduction
In a limited area model the lateral borders are not physical boundaries to the flow but, instead, artificial constructs imposed by our desire to model what is happening in a subvolume of the global atmosphere. In order to solve the equations describing the evolution of the atmosphere in that subvolume we must supply values for the fields at these artificial boundaries. Doing this correctly is a rather subtle matter. As Oliger and Sundström (1978) have shown, for each inward pointing characteristic a field must be supplied at the lateral boundary. This implies that only a certain subset of all the variables should be imposed. If this subset has been correctly chosen the initial-boundary-value problem is “well posed.”
Constructing a well-posed system for the primitive meteorological equations is a daunting task. For that reason, and because an easy-to-implement alternative solution to the boundary problem exists, numerical weather prediction models with well-posed lateral boundaries have never gotten farther than the “drawing board” stage of existence; see, for example, Elvius and Sundström (1973). The above-mentioned alternative is to overspecify the boundaries and damp the resulting noise with a relaxation scheme; see Davies (1976). This gives stable forecasts and is almost trivial to implement. Evidence of its popularity is its almost universal use in operational limited area numerical weather prediction models. It is also used in many research models.
Despite this there are a number of reasons for taking a fresh look at well-posed boundaries. (i) There is evidence that the flow relaxation scheme can cause mass loss or gain; see McDonald (1998). (ii) At points at which the characteristic velocity is pointing out of the domain of integration we are imposing the external model solution when we do not need to if we use the flow relaxation scheme. Thus we may introduce unnecessary errors when these fields are incorrect (as they will be in an operational environment). (iii) Consider the situation where both the “host model” (the model supplying the boundary fields for the limited area “guest model”) and guest model fields are each in geostrophic balance. Then, as Cats and Åkesson (1983) demonstrate in their appendix the relaxation scheme can destroy this balance throughout the boundary relaxation zone. (iv) In the future 4DVAR (Le Dimet and Talagrand 1986) will be used in the forecast–analysis cycle. This has implications for our boundary treatment. Because of the iterative nature of 4DVAR the influence of the boundary conditions spreads farther into the limited area domain with each iteration. When the adjoint model is integrated backward in time, what were formerly boundary points with incoming characteristic velocities may become points describing error gradient fields with outgoing characteristic velocities. These must now pass out of the limited area without reflection. If our boundary treatment is only partially successful in doing this then one may speculate that this error could build on itself as we continue the forecast–analysis cycle. See Gustafsson et al. (1998). The Davies relaxation scheme may not be sufficiently subtle to accommodate these requirements. (v) In the past the operational limited area philosophy could reasonably be described as follows. The boundaries were placed far enough away from the region of interest that the boundary errors would not have time to corrupt the forecast. The source of these boundary errors was twofold: (a) the boundary formulation, and (b) the inaccuracy of the host model fields (the discretization mesh was coarse, the fields were “old,” they were available only every 6 h, possibly). For the future, with increasing computing power, a new scenario can be envisaged. The host model fields will be sufficiently accurate to describe the broad-scale features of the atmosphere accurately (highs, lows, fronts). The purpose of the operational limited area model will be to forecast the fine details of the meteorological phenomena associated with these broad-scale features (rainfall, surface temperatures and winds, fog, cloud information, pollution dispersion, etc.). In a nesting situation, for example, the finest mesh area could be so small that a front can cross the area during the forecast period. Thus in the later stages the forecast will be dominated by the boundary conditions. Now, if there are errors in our boundary formulation they will corrupt the forecast.
For these reasons the High Resolution Limited Area Model Group has set itself the goal of building a numerical weather prediction model that has well-posed boundaries. This paper takes a first step in that direction by looking at the problems associated with the “multiply upstream well-posedness” problem. That is, how do we invent well-posed boundary conditions when using semi-Lagrangian semi-implicit discretizations? This question is addressed for two one-dimensional systems:the advection equation, and the advection–adjustment equation. Three possible answers are examined: trajectory truncation, time interpolation, and a well-posed buffer zone. Examining the one-dimensional advection equation (see section 2) allows us to ask the following question: how do we maintain stability when the departure point is outside the integration area, in isolation from the complications introduced by the C grid and the semi-implicit scheme? These latter complications are considered in section 3. In sections 2 and 3 we demonstrate stability experimentally. To complement this, in section 4 we make a first attempt at establishing stability theoretically. We do so by applying the energy method to the one-dimensional advection equation for our three postulated solutions for the multiply upstream well-posedness problem. Last, section 5 contains a discussion of the strengths and weaknesses of these different approaches to the boundary value problem.
Naïve extrapolations from the interior to the boundary are risky; see section 6 of Gustafsson et al. (1972). Therefore, as a design strategy, they are introduced only when unavoidable in what follows.
2. The one-dimensional advection equation
a. Discretization of the equations

Let us consider the outflow boundary first. Recall that for well posedness we must not impose a value of ϕ at this boundary; we must extrapolate from the interior. The semi-Lagrangian scheme provides us with an ideal way of doing this; see Robert and Yakimiw (1986). When |α| < 1, then we use Eq. (2.4). It yields an O(Δx3/Δt) accurate solution. If |α| ≥ 1, then we use Eq. (2.3), which is O(Δx4/Δt) accurate.
It is interesting to note that if, for example, we were using the leapfrog Eulerian scheme to integrate Eq. (2.1) we would have to switch to either Eq. (2.4) or some other upwind type of discretization in order to compute ϕ at the outflow boundary point. See McDonald (1999a), for example.
Next consider the inflow boundary and points in its immediate vicinity. On the boundary line itself we must impose a value on the field. If |α| > 1, then updating ϕ at the grid points adjacent to the inflow boundary causes difficulties, which we examine in sections 2d–2f. If |α| ⩽ 1, then p = 0, and we proceed as in section 2c.
b. Setup for the numerical tests
The following parameters will be used in the tests of our discretization: Δx = 10 km,
Test 1: start with the bell curve centered at x = L/2 and choose the time such that at the end of the integration it will be centered at 3L/2 in order to see whether it passes through the boundary at x = L without false reflections.
Test 2: start with the bell curve centered at x = −L/2 and choose the time such that at the end of the integration it will be centered at L/2 in order to see whether it passes through the boundary at x = 0 without distortion.
c. Testing with |α| < 1
Assuming
With α = 0.5 it passes both tests. Figure 2 shows the results of test 1. The bell curve has exited through the boundary accurately. There are no false reflections (the dots coincide exactly with the diamonds). Figure 3 shows the results of test 2. The bell curve has entered the area accurately (the dots coincide almost exactly with the diamonds).
Let us return to the problem near the inflow boundary when |α| > 1. We now have |p| > 0, and when, for example,
d. Option 1, when |α| > 1: Trajectory truncation
e. Option 2, when |α| > 1: Time interpolation
This is stable from a von Neumann point of view, anyway, because 0 < μ/α < 1. When |α| < μ, we do not use this approach to finding the departure point value of the field; we use Eq. (2.3) or (2.4). Equation (2.9) is only used when i < [α], where [α] is the integer part of α. Thus, in Eq. (2.10), where we have chosen i = μ, μ/α ⩽ 1.
Repeating test 2 with α = 2.5, using quadratic interpolation in time at the inflow boundary, we see from Fig. 5 that full accuracy is restored.
f. Option 3, when |α| > 1: Well-posed buffer zone
From a semi-Lagrangian point of view it would be ideal if we had a buffer zone external to the boundary. Then if the departure point were located there, we could interpolate in the same way as if it were in the interior. In this section we generate such a buffer zone, which gives stable integrations by using only information from the inflow boundary line. Again, for the purpose of demonstration consider the situation when
Naïve extrapolation from the interior to the inflow buffer zone will cause instability. Thus we cannot, for example, use Eq. (2.3) to extrapolate, abandoning the restriction 0 <
Repeating test 2 with α = 2.5, using this procedure to compute the necessary fields in the boundary zone restores full accuracy. The figure (not shown) looks identical to Fig. 5.
3. The one-dimensional advection-adjustment equation
We saw in section 2 that we could integrate the one-dimensional advection equation stably for large |α| by using time interpolation, a well-posed buffer zone, or, less accurately, trajectory truncation. In the real atmosphere adjustment and rotation also play significant roles. Thus in this section we add adjustment and rotation terms to the advection equation in order to see whether these schemes can still be implemented in this more realistic environment.
a. Discretization of the equations


The indices for which Eqs. (3.14)–(3.17) are valid depend on the grid points at which the boundaries are imposed. Examples are given in Table 1 for four different boundary choices: u1/2,
b. Trajectory truncation
To clarify the discussion let us impose our boundary values on ϕ0 and
In fact, as is shown in McDonald (2000) by experiment, the integration is inaccurate unless we make sure to use
Extrapolations from the interior are also necessary if we impose boundary conditions on u, p, or q. See McDonald (2000) for details.
c. Time interpolation
Here b represents a boundary line, and μ is the integer number of points the ϕ point of interest is from the boundary. Also, ν is a half odd-integer, which measures the distance of the u point of interest from the boundary.
Assume
From a computing point of view these drawbacks may not be too serious in a single-processor scalar environment. We would simply pay a possibly moderate computational price for getting the boundaries right. However, with vector processors in a multiprocessor environment these complications seem very serious. For these reasons it is interesting to consider an alternative that avoids these pitfalls, that of iterating.
When discretized, these equations give rise to a set that is formally the same as Eqs. (3.14)–(3.16) and is thus very easy to program. (The tridiagonal system now has variable coefficients and there are a few extra terms on the right side to be evaluated during each iteration.)
d. Well-posed buffer zone
How can we fill a well-posed buffer zone when some of the fields needed for the extrapolation cannot be imposed externally? For discussion purposes assume
Gustafsson et al. (1972) showed that, for the Lax–Wendroff scheme, many extrapolations cause instability on 0 ⩽ x ⩽ L; t > 0. We can infer that such extrapolations will also be unstable for the semi-Lagrangian schemes since the Lax–Wendroff scheme corresponds to the latter with quadratic interpolation when |α| < 1. A scheme that they proved to be stable uses Eqs. (3.3)–(3.5) to extrapolate the field corresponding to the outgoing characteristic to the boundary. Let us put a semi-Lagrangian twist to this idea.
e. Numerical testing (1): Advection
Let us perform numerical tests using a slow solution for which u = 0, and
With this solution we mirror what we would be attempting with the full three-dimensional system of equations in an operational context. There, the analysis and initialization produces a well-balanced initial state; the slow initial state mirrors this. On the boundary we want the meteorological fields to enter the domain without corruption from the gravity waves; here we model this by imposing the slow solution on the boundary.
We use L = 1000 km and Δx = 10 km. In addition,
Let us impose the boundary conditions on ϕ(0, t) and ϕ(L, t) and υ(0, t) given by Eq. (3.39). The initial state, which can be seen in Fig. 7 is given by Eq. (3.39) with t = 0.
First, the forecast was run using trajectory truncation. That is, Eqs. (3.14)–(3.17) were solved with Ru, Rυ, and Rϕ computed as described in section 3b when the departure point was adjacent to the boundary. The result is shown in Fig. 7. The positive aspects of this forecast are (a) that the front has entered and exited the area reasonably accurately, (b) that there are no false reflections at the boundary, and (c) that there is no short-wave noise generated. The negative aspects are, of course, (a) the “tail,” which the ϕ solution has attained, and (b) the slowing down of the ϕ and υ solutions, just as in section 2.
Compare this with the forecast obtained using time interpolation on the boundary. That is, Eqs. (3.14)–(3.17) were solved when the departure point was within the boundary, and Eqs. (3.31)–(3.33) when the departure point was outside the boundary. The result is shown in Fig. 8. The positive aspects of the solution are retained and the negative ones eliminated. The solution is almost as good as the analytical one.
The well-posed buffer zone also gives stable forecasts, better than trajectory truncation. In the region x < 800 km the forecast is not quite as good as the time interpolation forecast; for x > 800 km it is marginally better; see Fig. 9.
f. Numerical testing (2): Adjustment
The analysis of Oliger and Sundström (1978) does not derive a unique set of fields required for well posedness. There are many choices. Obviously, “transparent” boundaries, that is, those that allow waves to pass through them without reflection, are the most desirable. In this section we demonstrate that imposing p/q on the boundary allows the gravity waves to pass through with little or no reflection, making it a more attractive option than imposing ϕ, which causes the gravity waves to be reflected.


The trajectories are truncated as follows: Ru =
4. Stability
a. Trajectory truncation
b. Time interpolation
c. Well-posed buffer zone
5. Discussion
As we have seen, inventing stable boundaries even for simple one-dimensional systems is a nontrivial task. When we move to two (or three) dimensions the complications multiply. As is discussed in Durran (1999, section 8.2), there are fundamental difficulties encountered when we try to extend to two dimensions the idea of controlling the gravity waves as we did in section 3f. Nevertheless, Engquist and Majda (1977) have established a method constructing boundary conditions that are well posed and theoretically perfectly absorbing at least for waves striking at angles sufficiently close to normal incidence. For waves impinging at other angles of incidence (except tangential) reflections at the boundary can be reduced to a minimum by their method. The plan is to use their ideas and those discussed in this paper to construct a two-dimensional semi-implicit semi-Lagrangian shallow water model with boundaries that are “as transparent as possible.”
Extrapolations from the interior were found to be unavoidable. However, they did not lead to instabilities in our tests. This is consistent with the findings of Gustafsson et al. (1972), who proved that first-order extrapolations are stable for the Lax–Wendroff scheme (which is the same as a semi-Lagrangian scheme with quadratic interpolation and |α| < 1).
We have seen that both the time interpolation and well-posed buffer zone schemes overcome the accuracy limitation of trajectory truncation for large |α|. Looking ahead, it is interesting to ask what are the limitations of the former schemes in the context of an operational numerical weather prediction model? There is a limitation that applies to both; that is, their accuracy depends on how often the boundary is updated. For instance, if this is done every 3 h, neither scheme will be much more accurate than trajectory truncation. Each scheme has a particular limitation, at least as we have implemented them here. Convergence may be slow for an iterative implementation of the time interpolation scheme. Also, for our particular implementation of the well-posed buffer zone, looking at Eqs. (3.36) and (3.37) we see that it cannot be used when |
Although these are early days it is still nice to have a “fallback” position. One is to use trajectory truncation and accept the associated inaccuracy when |α| is large on the boundary. The second is based on an idea of J. E. Haugen (1999, personal communication). If uΔt/Δx < 0.5 on the western boundary, say, we need never invoke the time interpolation nor well-posed buffer zone schemes. To accomplish this we surround the uniformly discretized area with a buffer zone in which we use stretched coordinates, each Δx being, say, 15% bigger than its neighbor until this criterion is fulfilled for the maximum advecting velocity we expect perpendicular to the boundary. We choose this as our boundary and apply trajectory truncation there as an insurance against instability for unanticipated large inflow velocities.
Finally, although we have concentrated on looking at well-posed boundaries for semi-Lagrangian schemes in this paper, Eulerian systems are equally problematic. Consider an Eulerian approach using fourth-order spatial differencing. This will, in principle, require knowledge of fields outside the boundaries. Thus, stable extrapolations will be required to update the fields in the vicinity of the boundary. A possible solution is to adopt the upstream ideas inherent in the semi-Lagrangian approach. See also, however, Elvius and Sundström (1973) for an alternative approach.
Acknowledgments
Suggestions for improving the original manuscript by an anonymous reviewer and Dale Durran are gratefully acknowledged. In particular, the latter’s comment that the Rossby adjustment problem would provide an interesting challenge inspired section 3f.
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Durran, D. R., 1999: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer-Verlag, 465 pp.
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Distribution of variables on the x axis
Citation: Monthly Weather Review 128, 12; 10.1175/1520-0493(2000)129<4084:BCFSLS>2.0.CO;2
The solution for test 1. At time zero the bell curve, centered at 500 km, is shown by the ×’s. The analytical solution, for which the bell curve is centered at 1500 km, is shown by the diamonds. The result of integrating with α = 0.5 is shown by the dots
Citation: Monthly Weather Review 128, 12; 10.1175/1520-0493(2000)129<4084:BCFSLS>2.0.CO;2
The solution for test 2. At time zero the bell curve, centered at −500 km, is shown by the ×’s. The analytical solution, for which the bell curve is centered at 500 km, is shown by the diamonds. The result of integrating with α = 0.5 is shown by the dots
Citation: Monthly Weather Review 128, 12; 10.1175/1520-0493(2000)129<4084:BCFSLS>2.0.CO;2
Same as Fig. 3 but using the trajectory truncation scheme (option 1) and α = 2.5
Citation: Monthly Weather Review 128, 12; 10.1175/1520-0493(2000)129<4084:BCFSLS>2.0.CO;2
Same as Fig. 3 but using time interpolation (option 2) and α = 2.5
Citation: Monthly Weather Review 128, 12; 10.1175/1520-0493(2000)129<4084:BCFSLS>2.0.CO;2
Distribution of variables on the x axis using the C grid
Citation: Monthly Weather Review 128, 12; 10.1175/1520-0493(2000)129<4084:BCFSLS>2.0.CO;2
The solution of Eqs. (3.14)–(3.17) with initial and boundary conditions given by Eq. (3.39) and ϕ imposed at both boundaries and υ at inflow. We use the trajectory truncation scheme. At time zero ϕ is shown by the diamonds, u by the plus signs, and υ by the dots. The analytical solutions at time T for all three fields are shown by symbol-free lines. The result of integrating with Δt = 416.667 s,
Citation: Monthly Weather Review 128, 12; 10.1175/1520-0493(2000)129<4084:BCFSLS>2.0.CO;2
Same as Fig. 7 but now using time interpolation on the boundary instead of trajectory truncation
Citation: Monthly Weather Review 128, 12; 10.1175/1520-0493(2000)129<4084:BCFSLS>2.0.CO;2
Same as Fig. 7 but now using a well-posed buffer zone instead of trajectory truncation
Citation: Monthly Weather Review 128, 12; 10.1175/1520-0493(2000)129<4084:BCFSLS>2.0.CO;2
The solution of Eqs. (3.14)–(3.17) after 10 days. The dashed line represents the ϕ solution; the solid line the υ solution multiplied by
Citation: Monthly Weather Review 128, 12; 10.1175/1520-0493(2000)129<4084:BCFSLS>2.0.CO;2
Same as Fig. 10 except that on the boundaries, while still imposing υ at inflow, we now hold p and q constant, as described in Eq. (3.45)
Citation: Monthly Weather Review 128, 12; 10.1175/1520-0493(2000)129<4084:BCFSLS>2.0.CO;2
Range of validity the indices in Eqs. (3.14)–(3.17) for different boundary choices