Optimized Schwarz Methods with the Yin-Yang Grid for Shallow Water Equations

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1 Optimize Schwarz Methos with the Yin-Yang Gri for Shallow Water Equations Abessama Qaouri Recherche en prévision numérique, Atmospheric Science an Technology Directorate, Environment Canaa, Dorval, Québec, Canaa. Summary. An efficient implementation of the Schwarz metho is use to solve the 2D linear system of shallow-water equations (SWEs) on the sphere with an overset gri system name Yin-Yang. In this paper the convergence of optimize Schwarz metho for solving an elliptic problem is increase by substructuring the algorithm in terms of interface unknowns. In this work we show, by numerical tests, that the use of the Yin-Yang gri avois the poles problems resulting from the global latitue-longitue mesh convergence in the polar regions. Introuction The Yin-Yang overset gri (see Fig. ) was suggeste in 4 as a quasi-uniform gri free from the polar singularity. This gri system is constructe by overlapping two perpenicularly oriente ientical parts of latitue-longitue gri. In 5 it was shown that the Schwarz omain ecomposition metho can be successfully applie to linear SWEs. In each subomain the same system of equations is iscretize by an implicit time metho using the same time step where at each time step an elliptic problem is solve. In 5 this metho was use to solve the D SWEs on the circle an the elliptic problem was solve by the iterative optimize Schwarz metho (see 2 an references therein). In this paper we apply the same metho to the 2D linear SWEs on the Yin-Yang gri. We show that it is possible to increase the convergence of the optimize Schwarz metho for solving the elliptic problem by using the substructuring formulation. Because the two subgris of the Yin-Yang gri o not match, the variable upate at the subomain interfaces is one by cubic Lagrange interpolation. The complete 2D linear SWE solutions on the Yin-Yang gri are compare to integrations using the spectral moel on a Gaussian gri an a finite-ifference moel on a latitue-longitue gri. The paper is organize as follows: in Section 2, we present the 2D linear SWEs with their iscretize form an solution metho, in Section 3 the 2D positive efinite Helmhotz problem is solve on the Yin-Yang gri, the solver is either an iterative formulation or an substructuring formulation of the optimize Schwarz metho, in Section 4 numerical results are shown an finally concluing remarks are given in Section 5.

2 348 A. Qaouri Fig.. Yin-Yang gri system. The Yang gri is shown on the left, the Yin gri in the mile, an their composition on the right 2 The 2D Linear Shallow Water Equations In this section, we evelop an implicit global linear shallow-water moel on the sphere by using a omain ecomposition metho on Yin-Yang gri. The major goal of this work is the emonstration that this type of gri system removes the poles problems. Each of two component of Yin-Yang overset gri spans the Subregion S l, l =, 2, efine by S = {(λ, θ); θ π 4 + δ λ 3π 4 + δ}, () where (λ, θ) are the longitue an the latitue with respect to the local cartesian referential (see ) an δ is the minimum overlap. The relationship between the Yin coorinate an Yang coorinate is enote in cartesian coorinates by (x, y, z ) = ( x (3 l), z (3 l), y (3 l) ) l =, 2. (2) The governing equations for each subomain l, l =, 2 are the linear SWEs on a non rotating sphere of raius a U + φ =, (3) t a 2 λ V + cos θ φ =, (4) t a2 θ φ + φ D =, (5) t where a is the Earth raius, (U, V ) are the win images win multiplie by cos θ, a φ the perturbation geopotential from the reference geopotential φ an D is the ivergence efine by D = cos 2 θ ( U + cos θ V ). (6) λ θ The φ fiel graient components an ivergence D give rise to high frequency gravity waves. They are always integrate implicitly in time, enabling the use of a long time step t. A time iscretization of equations (3)-(5) is

3 V U Optimize Schwarz Methos with the Yin-Yang Gri φ (λ, θ, t) = R U a 2 U λ = + cos θ φ a2 θ φ (λ, θ, t) = R V V = φ + φ D (λ, θ, t) = R φ = φ (λ, θ, t t), (7) a 2 λ a 2 cosθ φ θ (λ, θ, t t),(8) φ D (λ, θ, t t), (9) where = t/2. As for the D SWEs in 5, the secon orer finite ifference spatial iscretization is one on a staggere Arakawa C gri where the variables U, V an φ are carrie at alternate points in space on (U-gri), (v gri) an (φ-gri), respectively. We use (N,M) gri points for the scalar (φ-gri), (N-,M) gri points for (U-gri) an (N,M-) gri points for (V -gri). The iscretize equations on each subomain l are with V U + φ i+,j φ a 2 h λ + a cos φ 2 θj + φ h θ φ + φ D = R U, i =, N ; j =, M () = R V i =, N; j =, M () = R φ i = 2, N ; j = 2, M (2) D = ( U U i,j V + cos θ V cos 2 j ), (3) θ j h λ h θ where h λ, h θ are the gri spacing along the longituinal an latituinal irections respectively. The resulting equations ()-(2) are combine to obtain a single iscretize elliptic equation (4) for φ which is solve by the optimize Schwarz metho iscusse in the following sub-sections, an the win is upate by equations (3)-(4). Then given fiels U, V an φ at the previous time step, the solution metho is summarize as follows: The right-han sies R U, R V an R φ are calculate in parallel on the two subomains. The elliptic problem is solve an the geopotential φ is upate on the two subomains. Interpolation an communication are require to obtain values at subomain interfaces. The win vector fiels (U,V ) are upate in parallel on the two subomains. 2. Iterative Formulation of the Optimize Schwarz Metho Similar to the classical Schwarz metho, we solve the iscretize problems iteratively in each subomain 2 + h θ tan θ j cos 2 θ jh 2 λ 2h 2 θ φ,k φ,k φ,k cos 2 θ jh 2 i,j + (η )φ,k λ cos 2 θ jh 2 λ h 2 θ i+,j 2 h θ tan θ j 2h 2 θ φ,k + = R,k, i =,..., N; j =,..., M, (4)

4 35 A. Qaouri where η = a is a positive an constant parameter an R is the corresponing φ t 2 right-han-sie function. Following the ieas of 2, we use the following iscretizations of the higher orer transmission conitions on each interface Γ ( =, 4) φ,k ν l + β φ,k + α 2 φ,k 2 l = φ(3 l),k ν l + β φ(3 l),k + α 2 φ (3 l),k. (5) The symbol ν l stans for the normal erivative of each subomain an l is the corresponing tangential erivative. The α, β are real parameters introuce to optimize the performance of the metho. We obtain these coefficients numerically, assuming the coefficients α antisymmetric, i.e. an β 2 l are inepenent of the bounary an are α () = α (2), β() = β (2). (6) In Fig. 2 we represent the performance for the solution of the elliptic problem where each subomain consists of 9 3 gri points, the overlap δ is one gri point spacing an the Helmhotz coefficient η is equal to one. For the approximate optimal values of the parameters α an β, the corresponing methos (see Fig. 2) converge in a small number of iterations an the convergence is much better than the convergence of the classical Schwarz metho. This gain more than compensates for the extra cost of computing the neee aitional erivatives. 2.2 Substructuring Formulation of the Optimize Schwarz Metho It is possible to both increase the robustness of the optimize Schwarz metho an its convergence spee by replacing the above fixe point iterative solver by a Krylovtype metho (see 2 an references therein). This is mae possible by substructuring the algorithm in terms of interface unknowns, that we enote here by T for each subomain l an interface. We consier the substructuring formulation of the elliptic problem in equations (4)-(5) where the unknowns T are equal to the right-han sie of equation (5), an we rewrite the problem to be solve on each subomain as A φ = R B φ = B φ(3 l) = T, (7) where B is the transmission operator which is the ientity in the case of Dirichlet conitions. In the previous iterative Schwarz metho we solve iteratively, with iteration number k =,..., k max, in each subomain the system of equations A φ,k = R + B φ(3 l),k = R + T,k, (8) where the matrix A now inclues the corrections from the transmission conitions. The Schwarz metho correspons then to the solution for the interface problem unknowns T by the Jacobi algorithm T,k = B (A(3 l) ) T (3 l),k + B (A(3 l) ) R (3 l). (9) We can improve the convergence by consiering GMRES, or any other Krylov metho, in orer to solve the interface system of equations

5 B (3 l) I (A ) I Optimize Schwarz Methos with the Yin-Yang Gri 35 B (A(3 l) ) B (A(3 l) ) R (3 l) T T (3 l) = B (3 l) (A ) R. (2) The spectral raius of the left-han-sie matrix in equation (2) epens on the choice of the interface conitions B. Once the interfaces functions T are known, the subproblem solutions are upate in parallel. In Table we consier the two interface equation solvers, Jacobi an GMRES, an we give the number of iterations so that the maximum error is smaller than 6. We see that the optimize Robin or secon-orer transmission conitions give a significant improvement when using either solver. 5 Dirichlet Robin Secon orer Maximum Error Iterations Fig. 2. Convergence behavior for the iterative classical Schwarz an the iterative optimize Schwarz methos, overlap = h an η = Table. Number of iterations for 3 interface conitions an 2 solvers. Each panel is 9 3, overlap= h an η =. Bounary Con. ASM GMRES Dirichlet 6 26 Robin 2 2 Secon orer Numerical Results In orer to assess the capability of the Yin-Yang gri to alleviate the pole problem, we consier the solutions for the 2D linear SWEs on both a global latitue-longitue gri an the Yin-Yang gri an we compare them with a spectral moel solution

6 352 A. Qaouri which is free of the pole problem. We start from an initial state at rest (U an V = ) with the initial geopotential perturbation given by: φ(λ, θ, t = ) = 2 Ωau sin 3 (θ) cos(θ) sin(λ) (2) where Ω =.7292s, an u = 2m/s. If we expan φ in terms of truncate series of spherical harmonics, its spectrum shows that the non-zero contributions are only from total wavenumbers 2 an 4. We take a global lat-lon gri with the resolution 8 4, Yin-Yang gri with resolution 6 2 in each panel, an we compare with an equivalent spectral moel with a truncation at wavenumber 39. There must be no interaction between the time tenencies of ifferent spectral coefficients. No new spectral moe can appear uring the evolution from the initial state. The spectrum (not plotte here) of the perturbation of the geopotential given by the 3 moels after 4 hours shows that for the spectral moel the only non zeros are the contributions from wavenumbers 2 an 4 an the initial energy, initially potential, is now ivie into kinetic an potential parts. In the spectrum of the perturbation of the geopotential given by using the lat-lon gri an the Yin-Yang gri, there are non zero contributions from wavenumbers other than 2 an 4, however the energy at these wavenumbers is much smaller when using the Yin-Yang gri. We show in Fig. 3 the ifference between the perturbation of the geopotential after 4 hours given by using the lat-lon an Yin-Yang gripoint moels an by the spectral moel. For latlon we can see that the biggest ifference is near the poles. The ifference between the solution on the Yin-Yang gri an the spectral moel is the same everywhere on the globe. The maximum ifference after 24 hours (not shown here) between the Yin-Yang gripoint moel an the spectral moel solutions correspons to only m height ifference. We can conclue that the use of the quasi-uniform Yin-Yang gri eliminates the pole problem an gives much more accurate solutions than with the stanar lat-lon gri Fig. 3. Geopotential ifferences between spectral an Lat/lon (left), Yin-Yang an spectral (right). The two subfigures have not the same scale 4 Conclusion In this paper we have shown that the Schwarz omain ecomposition methos are practical for obtaining solutions of the linear SWEs on the sphere with the overset

7 Optimize Schwarz Methos with the Yin-Yang Gri 353 gri system Yin-Yang. In a previous stuy 5 the convergence analysis of the D iterative solution of the elliptic problem on the circle yiele an analytical formula for the optimize coefficients of the transmission conitions. In this paper the optimize coefficients for the 2D case on the Yin-Yang gri were foun numerically an it was possible to both increase the robustness of the optimize Schwarz metho an its convergence spee by using the substructuring formulation. We have emonstrate that the use of the quasi-uniform Yin-Yang gri effectively eliminates the pole problem an gives more accurate solutions than when using the latitue-longitue gri. We have not yet valiate the full 2D SWEs moel on the Yin-Yang gri, but 2D passive semi-lagrangian avection has been thoroughly teste for this gri, see 5, 6, an the results were comparable to the value in 3. In future work we will complete the valiation of the SWEs with the Yin-Yang gri using the test set of Williamson et al. 7, an finally we will consier the Yin-Yang gri system with more than one regular subomain in each panel. Acknowlegement. This research was partly supporte by the Canaian Founation for Climate an Atmospheric Sciences through a grant to the QPF network an by the Office of Science (BER), U.S. Department of Energy, Grant No. DE-FG2- ER6399. I woul like to thank Jean Côté an Martin Ganer for reviewing the manuscript. References A. Arakawa an V. Lamb. Computational esign of the basic ynamical processes of the ucla general circulation moel. In Methos in Computational Physics, volume 7, pages Acaemic Press, M. Ganer. Optimize Schwarz methos. SIAM J. Numer. Anal., 44(2):699 73, R. Jacob-Chien, J.J. Hack, an D.L. Williamson. Spectral transform solutions to shallow water test set. J. Comput. Phys., 9:64 87, A. Kageyama an T. Sato. Yin-yang gri: An overset gri in spherical geometry. Geochem. Geophys. Geosyst., 5(9), A. Qaouri, J. Côté, M. Ganer, an S. Loisel. Optimize Schwarz methos with an overset gri system for the shallow-water equations: Preliminary results. Appl. Numer. Math., 27. In press. 6 A. Qaouri, L. Laayouni, J. Côté, an M. Ganer. Optimize Schwarz methos with an overset gri system for the shallow-water equations. Research Activities in Atmospheric an Oceanic Moelling WMO/TD-276, 35(3):2 22, D.L. Williamson, J.B. Drake, J.J. Hack, R. Jacob-Chien, an P.N. Swarztrauber. A stanar test set for numerical approximations to the shallow water equations in spherical geometry. J. Comput. Phys., 2:2 224, 992.

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