## 1. Introduction

Global weather forecasting and climate models still have grids too coarse to resolve thunderstorms. A common remedy is to embed a high-resolution window—a limited-area model (LAM)—within a global model. A common LAM task is to solve the Poisson equation to calculate the streamfunction from the vorticity and the velocity potential from the divergence.

*inhomogeneous*differential equation with

*homogeneous*boundary conditions (i.e., Poisson equation) plus (ii) a solution to a

*homogeneous*PDE with

*inhomogeneous*boundary conditions (i.e., Laplace equation):From a Fourier perspective, the Poisson subproblem is more challenging then the Laplace subproblem, so we shall mostly concentrate upon the former before returning to discuss the Laplace equation near the end. Such a decomposition into subproblems is the heart of the harmonic-sine method of Chen and Kuo (1992a,b) and its successors (Chen et al. 1997).

*f*(

*x*,

*y*) in a bivariate sine series:where the coefficients can be calculated by the usual Fourier integrals:Assuming a similar expansion for

*u*(

*x*,

*y*) with coefficients

*u*and matching terms giveswhereThe simplicity of bivariate Fourier series has inspired a host of Fourier LAMs and meteorological data analysis schemes including Bubnová et al. (1995), Cocke (1998), Cocke and LaRow (2000), Denis et al. (2002), Errico (1985), Haugen and Machenhauer (1993), Hong et al. (1999), Hong and Leetmaa (1999), Juang (1992), Juang and Kanamitsu (1994), Juang et al. (1997), Juang and Hong (2001), Juang et al. (2003), Kuo and Williams (1992, 1998), Park et al. (2011), Radnoti (1995), Sasaki et al. (1995), Segami et al. (1989), Tatsumi (1986), Boyd (2005), Termonia et al. (2012), and Degrauwe et al. (2012).

_{mn}*f*(

*x*,

*y*) has generally only a first-order rate of convergence while the coefficients

*u*have only a third-order rate of convergence:Worse still, the partial sums converge nonuniformly in space to a function that is discontinuous. If the series is truncated to include all coefficients

_{mn}*u*withfor some integer

_{mn}*N*, then Sköllermo (1975) has proved that maximum pointwise error is

*O*(1/

*N*

^{2}) as

*N*→ ∞.

In spite of this rather modest predicted rate of convergence, Fourier LAM models and data analysis have been successful, spawning the large number of applications already cited. As noted in the abstract, part of the reason for the success is that accuracy in the *meteorologically interesting* region is an order of magnitude higher than predicted for the maximum error over the entire Fourier domain (i.e., the square).

The reasons that meteorologists discard a strip of width *D* around the boundary of the unit square as shown schematically in Fig. 1 are fully described in Boyd (2005) and the references he cites. In this article, we shall concentrate on the mathematics. However, when a storm straddles the boundary between the high-resolution LAM domain (here nondimensionalized to the unit square), and the coarser grid of the global model that fills the rest of the sphere, the dynamics of the storm will be unreliable. Only data that is a finite distance away from the discontinuity in resolution can be trusted. Furthermore, there are several issues about inflow and outflow boundary conditions and blending the limited-area and global approximations smoothly through “Davies relaxation” and so on. It is then silly to worry about numerical accuracy in such regions where the meteorology is already corrupted.

The reason that the exclusion of this boundary strip is important is illustrated in Fig. 2, which is a cross section of the error from the center of the domain to the boundary for a typical truncated, unaccelerated Poisson Fourier solution. The error is large near the boundary but diminishes very rapidly away from the boundary at *x* = 1. This extreme nonuniformity in error is the point of this article.

In the rest of this article, we shall prove the higher rate of convergence in the interior in one dimension, demonstrate this in two dimensions, and describe how the order of convergence can be raised by simple modifications to the standard Fourier procedure. We concentrate primarily upon solving the Poisson equation with homogeneous boundary conditions. We shall show in the next-to-last section that when a boundary strip is deleted, the analogous Fourier series that solves the Laplace equation converges *exponentially* fast.

## 2. Rates of convergence

### Why is the convergence so slow?

*f*(

*x*) is sufficiently smooth, we may integrate by parts again. However, unless

*f*(0) =

*f*(1) = 0, we are stuck with the boundary terms which decay only as slowly as

*O*(1/

*n*). The sum of the sine series must be zero at

*x*= 0 and

*x*= 1 because every term in the series is individually zero at these points. If

*f*(0) and

*f*(1) are nonzero, then the Fourier series must approximate a function that jumps discontinuously at these endpoints, giving rise to the Gibbs Phenomenon. Classical textbook analysis shows that in thin layers near these endpoints, the error is

*O*(1)

*independent*of the series truncation

*N*—note that the maximum pointwise error of the

*sum*is larger by a factor of

*N*than the

*N*th coefficient, which is

*O*(1/

*N*). The partial sum overshoots near the jump by about 17% of the jump, and this does not diminish with increasing series truncation

*N*(Boyd 2001; Jerri 2011).

*m*and

*n*in front of the transformed integral. The boundedness of the integral then implies that

*f*must decay asymptotically as fast as all the negative powers in front of the integral. This is a standard argument for asymptotically estimating Fourier coefficients (Sköllermo 1975; Boyd 2009). However, unless

_{mn}*f*(

*x*,

*y*) satisfies compatibility conditions on the boundaries such that all the boundary terms in (13) sum to zero, the leading term (which is the sum of the boundary contributions) will be nonzero, implying that

*f*will decay only as

_{mn}*O*[1/(

*mn*)]. A concrete example is

*f*(

*x*,

*y*) ≡ 1 on the unit square, which has the following series:Because the sines are antisymmetric with respect to both

*x*= 0 and

*x*=

*π*, that is, sin(−

*mπx*) = −sin(

*mπx*) for all

*x*, and similarly in

*y*, the sine series above converges to ±1. (This is called the “checkerboard function” because if one visualizes a western style checkerboard/chessboard with squares of unit area, the function is equal to one on the white squares and minus one on the black.) Because the checkerboard function is discontinuous everywhere on the boundary of the unit square, its series exhibits a two-dimensional Gibbs Phenomenon (Boyd 2009; Jerri 2011). The discontinuities create a global pollution in the sense that the error is

*O*(1/

*N*) everywhere, but rises to

*O*(1) in a layer of thickness

*O*(1/

*N*) at the boundary. The solution tohas coefficients converging faster than

*f*

^{checkerboard}by a factor of (

*m*

^{2}+

*n*

^{2}), but the error in

*u*

^{checkerboard}shows the same concentration near the boundaries.

*N*wherewhere

*D*> 0 is a user-choosable parameter. The discarded buffer strip error ignores all points that lie within a distance

*D*of the boundary of the unit square. The second-order convergence for the

*square*versus third-order for the

*square with discarded strip*is clear.

Figure 4 shows the errors for *N* = 40. The dashed square shows the boundary of the region used to compute *E*^{deleted,strip}. Third-order convergence is obviously better than second order, but can one do better?

## 3. Raising the order of convergence by boundary subtractions

*f*(

*x*,

*y*) in two, choosing a function

*B*(

*x*,

*y*) which matches

*f*(

*x*,

*y*) everywhere on the boundary of the square, and a second part,

*O*(

*N*

^{2}) faster than the Fourier series of

*f*(

*x*,

*y*), which lacks the vanish-at-the-boundary property. She constructs the function

*B*(

*x*,

*y*) using what today is called transfinite interpolation (Gordon and Hall 1973):Sköllermo then analytically solved the Poisson equation with forcing equal to

*B*(

*x*,

*y*). She decided to impose the homogeneous Dirichlet boundary conditions satisfied by the

*whole solution*on each individual part of the response to

*B*(

*x*,

*y*). Unfortunately,

*u*+

_{xx}*u*=

_{yy}*xy*with

*u*= 0 on the boundary can be analytically solved only as a slowly convergent Fourier series—precisely what one was trying to avoid by splitting

*f*(

*x*,

*y*) into parts.

*particular*solution to

*u*+

_{xx}*u*=

_{yy}*B*without imposition of homogeneous boundary conditions. Define

*B*

_{1}(

*x*,

*y*) as the sum of the first four terms of

*B*(

*x*,

*y*):whereA solution to

*u*

_{1,xx}+

*u*

_{1,yy}=

*B*

_{1}(

*x*,

*y*) is

*υ*

_{5,yy}=

*f*(0,

*y*) −

*w*

_{5,yy,}accurate

*O*(1/

*N*

^{5}) on the interior of [0, 1]. Similarlywherewherewhere

*O*(1/

*N*

^{5}) accuracy on the interior of the square iswhereand

*u*

_{Lap}(

*x*,

*y*) is a solution to the Laplace equation,with the boundary conditionsthe boundary of the unit square.

*inhomogeneous*boundary conditions by the splitting described in the introduction,

*u*solves the Poisson equation and

*u*

_{Lap}is a solution to the Laplace equation. If the Poisson solution does not satisfy homogeneous boundary conditions, then it is merely necessary to alter the inhomogeneous boundary conditions of the Laplace solution toso that the sum of

*u*and

*u*

_{Lap}is the specified

*g*(

*x*,

*y*) on the boundary of the unit square. The fact that

*u*(

*x*,

*y*), the Poisson solution, does not satisfy homogeneous boundary conditions is of no significance.

## 4. Fourier solution of the Laplace equation

*g*into its components on each of the four sides of the square, the boundary conditions arewhere we have used “

*N*” and “

*S*” to denote the top and bottom boundaries of the square and “

*W*” and “

*E*” to denote the sides. The solution splits naturally into four pieces, each piece satisfying homogeneous Dirichlet boundary conditions on three of the four sides of the square. For example, the “northern” piece is derived by expanding the boundary condition along

*y*= 1 as a sine series in

*x*:Substituting a sine series in

*x*with

*y*-dependent coefficients yields a set of uncoupled, constant coefficient ordinary differential equations for the coefficients; imposing the boundary conditions

*y*= 1, the Laplace solution has the poor convergence rate of the expansion of the (generally nonperiodic) function

*g*(

_{N}*x*); the error is

*O*(1) in the Gibbs layers near

*x*= 0 and

*x*= 1. Away from the northern boundary, the fact that sinh(

*z*) ~ (1/2) exp(

*z*) for large

*z*impliesThis means that the Laplace solution has an

*exponential*rate of convergence away from the boundaries. We can state this as a general theorem incorporating all four pieces of the solution. The Fourier–sinh solution to Laplace's equation is classical [48–51 in Jackson (1962) and online at http://eqworld.ipmnet.ru/en/solutions/lpde/lpde301.pdf], but the rate-of-convergence theorem is new.

*x*,

*y*) ∈ [

*D*, 1 −

*D*] ⊗ [

*D*, 1 −

*D*],for some positive constant

*D.*In other words, the Laplace solution series converges exponentially fast with

*n.*

*N*

^{2}by writing

*ℸ*(

*x*,

*y*) is a Laplace solution that interpolates the boundary values at the four corners of the unit square:The Fourier method is then applied to compute

*g*=

*ℸ*. The series for

*O*(1/

*N*

^{2}) error within

*O*(1/

*N*) of the corners and with

*O*(1/

*N*

^{3}) error on the interior of the domain.

All this Fourier–Laplace mechanics are well described in appendix B of Chen and Kuo (1992a). The only novelty in this section is the proof of exponential convergence in the inner square of meteorological interest, given in the appendix.

## 5. Summary

We have shown that for applications to limited-area modeling (LAM), the domain of needed accuracy is not the whole LAM window, here normalized to the unit square, but is rather a smaller square embedded within the larger. In the smaller domain, we show that the error of the bivariate Poisson Fourier solution is *O*(1/*N*^{3}). The previously published estimate *O*(1/*N*^{2}), a factor of *N* more pessimistic, is correct only for the *entire* square including the meteorologically irrelevant boundary buffer.

We also explicitly constructed an improved Fourier algorithm that solves the Poisson equation with fifth-order accuracy. Because we seek only particular solutions, our order-raising procedure is significantly simpler than previously proposed methods.

The Fourier solution to the Laplace equation, also needed for the harmonic-sine LAM analysis of Chen and Kuo (1992a,b), is shown to converge *exponentially* fast on the square of meteorological interest.

An *exponential* rate of convergence can be recovered for the Poisson solution, too, in the domain with the deleted strip by applying a series acceleration method (Boyd 2011). The rate of convergence diminishes to nothing as the discontinuities of the boundary of the domain are approached, but again, because a boundary strip is of no interest, the exponential convergence rate of the accelerated series is always nonzero in the region of meteorological interest. [Another oceanic–atmospheric example where series acceleration on only *part* of the domain is needed—the tropics for Hermite series on the latitudinally infinite equatorial beta plane—is described in Boyd and Moore (1986).] Other periodize-with-windowing procedures appear promising (Boyd 2005; Termonia et al. 2012; Degrauwe et al. 2012). “Boundary subtraction” can be continued to arbitrary order (Averbuch et al. 1997, 1998; Huybrechs 2010; Huybrechs et al. 2011; Adcock and Hansen 2012).

Thus, we do not claim to have spoken the last word or offered the most accurate algorithm. We only claim that our third- and fifth-order Poisson solvers are simple, and sufficiently accurate for most practical LAM applications.

## Acknowledgments

The first author was supported by the NSF through Grant OCE 1059703 and by the Distinguished International Visiting Scholar Program of the Chinese Academy of Sciences. The remaining authors are supported by the project from Research Plan 973 (2012CB417201) and the project from the Natural Science Foundation of China (40930950, 41075043).

## APPENDIX

### A Proof of the Interior Rate of Convergence for the Fourier Series of a Discontinuous Function

Gibbs Phenomenon and the ensuing *O*(1) error in the Fourier approximation of a general, nonperiodic function are discussed in standard textbooks including the first author's book (Boyd 2001) and especially in the collection of articles entirely on this topic (Jerri 2011). However, it is difficult to find a careful treatment that proves that the error improves to *O*(1/*N*) everywhere outside the “Gibbs layers,” which shrink in width inversely proportional to *N*, so we provide one. Note that in this appendix, the domain is [0, *π*] instead of [0, 1].

Our error analysis requires a strengthened form of a classic lemma.

#### a. Theorem 2 [Riemann–Lebesgue (strong form)]

Suppose *f(x)* is sufficiently nice as to allow one integration by parts in the integrals below. It is sufficient for *f(x)* to belong to *x* ∈ [*a*, *b*] and both *f(x)* and *df*/*dx*(*x*) are integrable on the same interval.

*n*> 0,Note that the limits of integration are arbitrary and not restricted to [−π, π].

*g*(

*x*), the integral is less than or equal to the area under a rectangle whose height is equal to the maximum absolute value of the integrand and whose width is equal to the interval of integration,

*f*(

*x*) with a Fourier series, the partial sum of degree

*N*can be represented without approximation using the Dirichlet integral asSince the partial sum for the trivial function

*f*(

*x*) = 1 is identical to

*f*(

*x*), the identityfollows. We can use this to write the

*error*in

*f*(

*x*), again without approximation, as

As noted in the main body of this article, a function can always be written on *x* ∈ [0, 1] as the sum of a linear polynomial *B*(*x*) that interpolates *f*(0) and *f*(1) plus the function *O*(1/*n*^{3}). We can then invoke the following lemma.

*N*th partial sum of a Fourier sine series asSuppose thatThen

*nx*)| ≤ 1 for all real

*x*and

*n*, so each term in the Fourier sum is bounded by |

*b*|. The specified inequality obeyed by the Fourier coefficients then implies

_{n}*x*−1)

^{k}>

*ξ*(

*x*;

*k*), it follows that the integral of the 1/(

*x*− 1)

^{k}, which can be evaluated analytically, must be larger than the integral of

*ξ*, giving

The lemma shows that the error in the *N*th partial sum of *O*(1/*N*^{2}), so the error in the Fourier series of *f*(*x*) is dominated by the error in the Fourier series of *B*(*x*).

*σ*(

*x*+

*y*) jumps so that

The limits of integration do not include the points *y* = 0, *π* where the denominator sin(*y*/2) = 0. It follows that for any fixed *x* on the interior of the interval [0, *π*], we can invoke the strong form of the Riemann–Lebesgue theorem to prove the following.

#### b. Theorem 3 (interior rate of convergence)

Suppose *f*(*x*) is sufficiently “nice” as to allow one integration by parts in the usual Fourier coefficient integrals below. It is sufficient for *f*(*x*) to belong to *x* ∈ [*a*, *b*] and both *f*(*x*) and *df*/*dx*(*x*) are integrable on the same interval.

*x*on the interior of [0, π],

There is no contradiction with the classical analysis of Gibbs Phenomenon. At fixed *x* near a discontinuity, the error is indeed large and *O*(1) for small *N*, but as the Gibbs layers shrink with *N*, the error at fixed *x* asymptotes to its predicted *O*(1/*N*) decay as illustrated in Fig. A2.

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