A PSEUDOSPECTRAL QUADRATURE METHOD FOR NAVIER-STOKES EQUATIONS ON ROTATING SPHERES

Transcription

1 A PSEUDOSPECTRAL QUADRATURE METHOD FOR NAVIER-STOKES EQUATIONS ON ROTATING SPHERES M. GANESH, Q. T. LE GIA, AND I. H. SLOAN Abstract. In this work, we describe, analyze, and implement a pseudospectral quadrature method for a global computer modeling of the incompressible surface Navier-Stokes equations on the rotating unit sphere. Our spectrally accurate numerical error analysis is based on the Gevrey regularity of the solutions of the Navier-Stokes equations on the sphere. The scheme is designed for convenient application of fast evaluation techniques such as the fast Fourier transform (FFT), and the implementation is based on a stable adaptive time discretization. 1. Introduction In this paper we develop a pseudospectral quadrature method for the surface Navier-Stokes partial differential equations (PDEs) on the rotating unit sphere. Whereas the finite element method is best suited for handling non-smooth processes, the spectral global basis computer models are very efficient and perform extremely well for processes with smooth regularity. For example, exponential convergence properties of the global Fourier basis spectral Galerkin methods (without quadrature) for the Ginzburg-Landau and Navier-Stokes PDEs on two dimensional periodic cells are based on the Gevrey regularity of solutions of the PDEs [8, 19]. The complex three dimensional flows in the atmosphere and oceans are considered to be accurately modeled by the Navier-Stokes PDEs of fluid mechanics together with classical thermodynamics [21]. Difficulties in computer modeling in these PDEs resulted in several simplified models for which spectral approximations are well known [2, 14, 21]. A famous open problem is to prove the global regularity for the three dimensional incompressible Navier-Stokes PDEs [24]. However, the precise Gevrey regularity of the unique solution of the (practically relevant) surface Navier-Stokes PDEs on the rotating sphere was proved in [5]. (Because the Earth s surface is an approximate sphere, a standard surface model, to study global atmospheric circulation on large planets, is the sphere.) Consequently, a natural next step is to describe, analyze, and implement an exponentially converging pseudospectral method for the Navier-Stokes PDEs on the rotating sphere. This paper is motivated by the recent work [13], where computer modeling of the Navier-Stokes PDEs on one-dimensional and toroidal domains [9] was extended to the unit sphere. The main result in [13, page 978] establishes only convergence of the semi-discrete Galerkin method without quadrature in L p norms, but does not provide the rate of convergence of the scheme. Date: May 13, Mathematics Subject Classification. Primary 65M12; Secondary 76D05. Key words and phrases. Navier-Stokes equations, unit sphere, vector spherical harmonics. 1

2 2 M. GANESH, Q. T. LE GIA, AND I. H. SLOAN The rate of convergence results, supported by numerical experiments, formed the core part of research on the Navier-Stokes equations on two dimensional domains over the last few decades, see [9, 10, 25] and references therein. There is a vast literature on numerical methods and analysis for the Navier-Stokes PDE on bounded Euclidean domains (see [9, 10, 25] and references therein), but their counterparts on closed manifolds are rarer (see [13] and references therein). The implementation of the scheme in [13, page 978] is based on a fixed time-step explicit Runge-Kutta method that has a small stability region for the systems of ordinary differential equations arising from the spatial discretization. The outline of this paper is as follows. In the next section, we recall various known preliminary results associated with the Navier-Stokes PDEs on the unit sphere, in strong and weak form. In Section 3, we introduce essential computational and numerical analysis tools required for the discretization and analysis of the Navier-Stokes equations. In Section 4, we describe and prove spectral accuracy of a pseudospectral quadrature method and give implementation details required to apply the FFT and adaptive-in-time simulation of the Navier-Stokes equations. In Section 5, we demonstrate computationally the accuracy and applicability of the algorithm for well known benchmark examples. 2. Navier-Stokes equations on the rotating unit sphere The surface Navier-Stokes equations (NSE) describing a tangential, incompressible atmospheric stream on the rotating two-dimensional unit sphere S R 3 can be written as [5, 15, 16, 18, 26] (2.1) t u+ uu ν u+ω u+ 1 ρ Grad p = f, Div u = 0, u t=0 = u 0 on S. Here u = u( x, t) = (u 1 ( x, t), u 2 ( x, t), u 3 ( x, t)) T is the unknown tangential divergencefree velocity field at x S and t [0, T ], p = p( x, t) is the unknown pressure. The known components in (2.1) are the constant viscosity and density of the fluid, respectively denoted by ν, ρ, the normal vector field ω = ω ( x) = ω( x) x for the Coriolis acceleration term, and the external flow driving vector field f = f( x, t). The Coriolis function ω is given by ω( x) = 2Ω cos θ, where Ω is the angular velocity of the rotating sphere, and θ is the angle between x and the north pole. The vorticity of the flow associated with the NSE (2.1), in the curvilinear coordinate system, is a normal vector field, defined, for a fixed t 0, by (2.2) Vort u( x, t) = Curl bx u( x, t) = x Ψ( x, t), x S, for some scalar-valued vorticity stream function Ψ. All spatial derivative operators in (2.1)-(2.2) are surface differential operators, obtained by restricting the corresponding domain operators (defined in a neighborhood of S) to the unit sphere, using standard differential geometry concepts on closed manifolds in R 3 [15, 16]. In particular, with latitude variable θ [0, π] and the longitude variable ϕ [0, 2π), we have (2.3) x = p(θ, φ) = (sin x 1 cos x 2, sin x 1 sin x 2, cos x 1 ) T, x 1 = θ, x 2 = φ, x S. All variables and equations in this paper are based on the coordinate system in (2.3). To obtain computable representations of the spatial operators in (2.1), we

3 A PSEUDOSPECTRAL METHOD FOR NSEs ON ROTATING SPHERES 3 define (2.4) e 1 = p θ, e 2 = p φ, g ij = e i e j = 0 if i j 1 if i = j = 1 sin 2 θ if i = j = 2 The set of orthogonal vectors {e 1, e 2 } forms a basis for the tangent vector space of S, denoted throughout the paper by T S; the 2 2 matrix g ij is the Riemannian metric on S. Let g ij be the inverse of the matrix g ij. For a scalar valued function v defined on S, the surface gradient Grad v is defined by (2.5) Grad v = 2 i,j=1 ij v g e j. x i For a tangential vector field v on S, with the representation v = v 1 e 1 + v 2 e 2, using the fact that the determinant of the Riemannian metric on S is sin 2 θ, the surface divergence Div v is defined by (2.6) Div v = 1 sin(x 1 ) 2 i=1 [sin(x 1 )v i ] = 1 x i sin(θ) [ ] θ sin(θ)v 1 + φ v 2. The covariant derivative w v is defined as follows: for any two tangential vectors v and w, with v = v 1 e 1 + v 2 e 2 and w = w 1 e 1 + w 2 e 2, (2.7) w v = w i v j Γ k v k ij + w i e k, Γ k ij = 1 2 ( g kl gli + g jl g ) ji. x i 2 x j x i x l k=1 i=1 j=1 The Laplace-Beltrami operator for a scalar valued function v on S is defined by (2.8) v = Div Grad v. Using the fact that the outward unit normal at x S is x, the Curl of a scalar function v, of a normal vector field w = w x, and of a tangential vector field v on S are respectively defined by (2.9) Curl v = x Grad v, Curl w = x Grad w, Curl bx v = x Div ( x v). The surface diffusion operator acting on tangential vector fields on S is denoted by (known as the Laplace-Beltrami or Laplace-de Rham operator) and is defined as (2.10) v = Grad Div v Curl Curl bx v. The following relations connecting the above operators will be used throughout the paper: (2.11) Div Curl v = 0, Curl bx Curl v = x v, Curl v = Curl v, (2.12) 2 w v = Curl (w v)+grad (w v) vdiv w+wdiv v v Curl bx w w Curl bx v. In particular, for tangential divergence-free vector fields, such as the solution u of the NSE, using (2.12), the nonlinear term in (2.1) can be written as (2.13) u u = Grad u 2 2 u Curl bxu. l=1.

4 4 M. GANESH, Q. T. LE GIA, AND I. H. SLOAN 2.1. A weak formulation. A standard technique for removing the scalar pressure field from the Navier-Stokes equations is to multiply the first equation in (2.1) by test functions v from a space with elements having the main properties of the unknown velocity field u (in particular, Div v = 0) and then integrate to obtain a weak formulation. (The unknown p, if required, can then be computed by solving a pressure Poisson equation, obtained by applying the surface divergence operator in (2.1).) To this end, we introduce the standard inner products on the space of all square integrable (i) scalar functions on S, denoted by L 2 (S); and (ii) tangential vector fields on S, denoted by L 2 (T S): (2.14) (v 1, v 2 ) = (v 1, v 2 ) L 2 (S) = v 1 v 2 ds, v 2, v 2 L 2 (S), S (2.15) (v 1, v 2 ) = (v 1, v 2 ) L 2 (T S) = v 1 v 2 ds, u, v L 2 (T S), where ds = sin θdθdφ. Throughout the paper, the induced norm on L 2 (T S) is denoted by and for other inner product spaces, say X with inner product (, ) X, the associated norm is denoted by X. For example, for s > 0, standard norms in the scalar and vector valued functions Sobolev spaces H s (S) and H s (T S) are denoted by H s (S) and H s (T S), respectively. Since H 0 (T S) = L 2 (T S), H 0 (T S) =. We have the following identities for appropriate scalar and vector fields [15, (2.4)-(2.6)]: S (2.16) (2.17) (2.18) (Grad ψ, v) = (ψ, Div v), (Curl ψ, v) = (ψ, Curl bx v), (Curl Curl bx w, z) = (Curl bx w, Curl bx z). In (2.16),(2.17), the L 2 (T S) inner product is used on the left hand side and the L 2 (S) inner product is used on the right hand side. Throughout the paper, we identify a normal vector field w with a scalar field and hence (2.19) (ψ, w) := (ψ, w) L 2 (S), w = xw, ψ, w L 2 (S). Using (2.11), smooth (C ) tangential fields on S can be decomposed into two components, one in the space of all divergence-free fields and the other complement, through the Hodge decomposition theorem [1]: (2.20) C (T S) = C (T S; Grad ) C (T S; Curl ), where (2.21) C (T S; Grad ) = {Grad ψ : ψ C (S)}, C (T S; Curl ) = {Curl ψ : ψ C (S)}. For s 0, let C,s (T S; Curl ) denote the closure of C (T S; Curl ) in the H s (T S) norm. In particular, following [5] we introduce a simpler notation H = closure of C (T S; Curl ) in L 2 (T S) = C,0 (T S; Curl ), V = closure of C (T S; Curl ) in H 1 (T S) = C,1 (T S; Curl ).

5 A PSEUDOSPECTRAL METHOD FOR NSEs ON ROTATING SPHERES 5 Using the Gauss surface divergence theorem, for any scalar valued function v on S with Grad v L 2 (T S), using (2.17), we have (2.22) (Grad v, w) = Grad v w ds = v Div w ds = 0, w V, S and hence the unknown pressure can be eliminated from the first equation in (2.1) through the weak formulation. Following [15, Page 567], for the diffusion part of the NSE, we consider the Stokes operator (2.23) A = Curl Curl bx. Using (2.10) and (2.11), it is easy to see that the Stokes operator is the restriction of the vector Laplace-de Rham operator on V ; A = P Curl, where P Curl : L 2 (T S) H is the orthogonal projection onto the divergence-free tangent space. For each positive integer L = 1, 2,..., the eigenvalue λ L and the corresponding eigenvectors of the Stokes operator A are given by (2.24) λ L = L(L + 1), Z L,m (θ, ϕ) = λ 1/2 L Curl Y L,m (θ, ϕ), m = L,..., L, where Y L,m are the scalar orthonormal spherical harmonics of degree L, defined by [ ] 1/2 (2L + 1) (L m )! (2.25) Y L,m (θ, ϕ) = PL m (cos θ)e imϕ, m = L,..., L, 4π (L + m )! with P m L being the associated Legendre polynomials { (2.26) P m L (t) = (1 t 2 ) m/2 d L+m 2 L L! (t 2 1) L, for m 0, dt L+m ( 1) m P m L for m < 0, so that Y L,m = ( 1) m Y L, m. The spectral property AZ L,m = λ L Z L,m follows from the fact that Y L,m are eigenfunctions of the scalar Laplace-Beltrami operator in (2.8) with eigenvalues λ L, the definition of the Stokes operator A in (2.23), Z L,m in (2.24), (2.11), and (2.9). Since {Y L,m : L = 0, 1,... ; m = L,..., L} is an orthonormal basis for L 2 (S), it is easy to see that {Z L,m : L = 1,... ; m = L,..., L} is an orthonormal basis for H. Thus an arbitrary v H can be written as L (2.27) v = v L,m Z L,m, v L,m = v Z L,m ds = (v, Z L,m ). L=1 m= L We consider a subset of H, (2.28) { D(A s/2 ) = v H : v = L L=1 m= L v L,m Z L,m, S S L L=1 m= L λ s L v L,m 2 < which is the divergence-free subset of the Sobolev space H s (T S). For every v D(A s/2 ), we set [ ] 1/2 L (2.29) v H s (T S) = λ s L v L,m 2, L=1 m= L },

6 6 M. GANESH, Q. T. LE GIA, AND I. H. SLOAN and for v D(A s/2 ), we define (2.30) A s/2 v := L L=1 m= L λ s/2 L v L,mZ L,m H. For a tangential vector field v on S, using (2.3), we define the Coriolis operator C, (2.31) (Cv)( x) = ω ( x) v( x) = ω( x)( x v), ω( x) = 2Ω cos θ. To treat the nonlinear term in (2.1), we consider the trilinear form b on V V V, defined as (2.32) b(v, w, z) = ( v w, z) = v w z ds, v, w, z V. S Using (2.12) and (2.17), for divergence free fields v, w, z, the trilinear form can be written as (2.33) b(v, w, z) = 1 [ v w Curl bx z + Curl bx v w z v Curl bx w z] ds. 2 S Lemma 2.1. [15, Lemma 2.1] Let V be equipped with 2 V = (A, ). The trilinear form b is continuous on V V V, i.e. Moreover, b(v, w, z) C v V w V z V, v, w, z V. (2.34) b(v, w, w) = 0, b(v, z, w) = b(v, w, z) v, w, z V. Thus, using (2.10), (2.17), (2.23), and (2.33), a weak solution of the Navier-Stokes equations (2.1) is a vector field u L 2 ([0, T ]; V ) with u(0) = u 0 that satisfies the weak form (2.35) (u t, v) + b(u, u, v) + ν(curl bx u, Curl bx v) + (Cu, v) = (f, v), v V. This weak formulation can be written in operator equation form on V, the adjoint of V : Let f L 2 ([0, T ]; V ) and u 0 H. Find a vector field u L 2 ([0, T ]; V ), with u t L 2 ([0, T ]; V ) such that (2.36) u t + νau + B(u, u) + Cu = f, u(0) = u 0, where the bilinear form B(u, v) V is defined by (2.37) (B(u, v), w) = b(u, v, w) w V. In the subsequent error analysis, we need the following estimate for the nonlinear term (see Lemma 6.1 in Appendix): (2.38) A δ B(u, v) { C A 1 δ u v C A 1/2 u v, C u A 1 δ v C u A 1/2 v, δ (1/2, 1) u, v V. In (2.38), as throughout the paper, C is a generic constant independent of u and v, (and the discretization parameter N introduced in Section 3). From (2.38) we deduce the weak Lipschitz continuity property (2.39) A δ (B(v, v) B(w, w)) C v w, δ (1/2, 1), if A 1/2 v, A 1/2 w < C. The existence and uniqueness of the solution u L 2 ([0, T ]; V ) of the weak formulation (2.35) are discussed in [15, 16, 18]. A regular solution of the Navier-Stokes

7 A PSEUDOSPECTRAL METHOD FOR NSEs ON ROTATING SPHERES 7 equations (2.1) on [0, T ] is a tangential divergence-free velocity field u that satisfies the equation obtained by integrating in time the weak form (2.35), from t 0 to t, for almost every t 0, t [0, T ]. In order to recall the existence, uniqueness, and Gevrey regularity of the regular solution, we need a few more additional details from [5]. These are also needed as tools for analysing our pseudospectral quadrature method Gevrey regularity of regular solution. The Gevrey class of functions of order s > 0 and index σ > 0, associated with the Stokes operator defined in (2.23), is denoted by G s/2 σ and is defined as (2.40) G s/2 σ := D(A s/2 e σa1/2 ) D(A s/2 ). Using (2.28), the Gevrey space (2.41) { G s/2 σ = v D(A s/2 ) : v = L L=1 m= L v L,m Z L,m, L L=1 m= L λ s Le 2σλ1/2 L vl,m 2 < is a Hilbert space with respect to the inner product L (2.42) v, w s/2 G = λ s Le 2σλ1/2 L vl,m ŵ L,m, v, w G s/2 σ. σ L=1 m= L First we recall the following result from [5, 6]. Theorem 2.1. If u 0 D(A s+1/2 ) and f L ((0, ); D(A s e σ1a1/2 )), for some s, σ 1 > 0, then for all t > 0 there exists a T > 0, depending only on ν, f, and A s+1/2 u 0 L 2 (T S), such that the NSE (2.1) on S have a unique regular solution u(, t) and u(, t) G s+1/2 σ(t), where σ(t) = min{t, T, σ 1 }. In addition, from the assumption and proof of Theorem 2.1 [5, page 355], we also have (2.43) A s+1/2 e σ(t)a1/2 u(t) 2 M 0, t > 0, where M 0 depends on A s+1/2 u(0), sup t 0 A s f(, t) and ν but not on t. The bound in (2.43) is useful for establishing the quality of approximation of the Stokes projection of u in the next section (see Theorem 3.1). It is also convenient to have a similar bound for the time derivative of the Stokes projections with t in (2.43) replaced with certain complex times ζ C, to prove the power of approximation of the time derivative of the Stokes projection (see Theorem 3.2). To this end, we consider the NSE extended to complex times ζ, (2.44) du dζ + νau + B(u, u) + Cu = f, Div u = 0, u(0) = u 0, ζ C, on S, with standard complexification (see [10]) of all the spaces and operators introduced earlier. In the next theorem, we extend arguments used in [10] for the solution of the NSE on the plane to the case of the sphere. The arguments differ in an essential way only for the nonlinear term. Theorem 2.2. Let u 0 D(A s+1/2 ) and f C([0, T ]; D(A s e σ1a1/2 )), with s 1/4. Let the domain T be defined by T := {ζ = re iθ : 0 r T ; θ π/4}. }

8 8 M. GANESH, Q. T. LE GIA, AND I. H. SLOAN We assume further that f(, ζ) is analytic for ζ T and that (2.45) K := sup{ A s e ψ(r cos θ)a1/2 f(, ζ) 2 : ζ = re iθ T} <. Then there exists T > 0 such that (2.46) A s+1/2 e ψ(r cos θ)a1/2 u(ζ) 2 M 1, ζ T and ζ T, where M 1 depends on A s+1/2 u(0) and hence u(, ζ) G s+1/2 ψ(r cos θ), where ψ(x) := min{x, T, σ 1 }. Proof. Let ζ = re iθ with r > 0 and θ π/4, and let u A (ζ) := A s+1/2 e ψ(r cos θ)a1/2 u(ζ). The definition of ψ gives d dx ψ := ψ (x) 1, and ψ σ 1, thus for fixed θ we find (2.47) Using (2.47) in d dr u A(ζ) = ψ (r cos θ) cos θa s+1 1/2 ψ(r cos θ)a e u(ζ) +e iθ A s+1/2 ψ(r cos du θ)a1/2 e dζ (ζ). ( ) 1 d 2 dr u A(ζ) 2 d = R dr u A(ζ), u A (ζ), where R( ) denotes the real-part function, we get (2.48) 1 d 2 dr u A(ζ) 2 = ψ (r cos θ) cos θ R (A 1/2 u A (ζ), u A (ζ)) +R e iθ (A s+1/2 ψ(r cos du θ)a1/2 e dζ (ζ), u A(ζ)). Using (2.44) for the last term in (2.48) together with the fact that (see [5, Lemma 1]), we find (A s e ψ(r cos θ)a1/2 Cu, A s+1 e ψ(r cos θ)a1/2 u) = 0 (2.49) 1 d 2 dr u A(ζ) 2 + ν cos θ A 1/2 u A (ζ) 2 = ψ (r cos θ) cos θ(a 1/2 u A, u A ) R e iθ (A s+1/2 e ψa1/2 B(u, u), u A ) +R e iθ (A s e ψa1/2 f, A 1/2 u A ), where in (2.49) and below we write u A = u A (ζ), u = u(ζ), f = f(ζ), ψ = ψ(r cos θ). From [5, Lemma 2], we have (with p = max{2 s, 7/4} = 7/4, since s 1/4), (2.50) (A s+1/2 e ψa1/2 B(u, u), u A ) C u A 3 p A 1/2 u A p. Applying ψ 1, the Cauchy-Schwarz inequality, (2.50) and Young s inequality (ab a q /q + b q /q with 1/q + 1/q = 1) with q = 2 and q = 2/(2 p) in (2.49), we

9 A PSEUDOSPECTRAL METHOD FOR NSEs ON ROTATING SPHERES 9 get 1 d 2 dr u A 2 + ν cos θ A 1/2 u A 2 cos θ A 1/2 u A u A + C u A 3 p A 1/2 u A p + A s e ψa1/2 f A 1/2 u A ν cos θ A 1/2 u A 2 + cos θ 4 ν u A 2 ( p +C 2 (2 p) ) p 2 p 2 p u A 2(3 p) 2 p p ν cos θ + 1 ν cos θ As e ψa 1/2f 2 + ν cos θ A 1/2 u A 2. 4 Therefore, d dr u A 2 2 cos θ ν (2.51) ( 1 u A 2 + C ν cos θ 2 cos θ (1 + u A 2 ) + C ν 2 ν cos θ As e ψa 1/2 f 2. + ν cos θ A 1/2 u A 2 2 ) p 2 p ua 2(3 p) 2 p + 2 ( 1 ν cos θ ν cos θ As e ψa ) p 2 p (1 + ua 2 ) 3 p 2 p + Using θ π/4, and 3 p 2 p = 5, in (2.51), we get d (2.52) dr u A 2 C(1 + u A 2 ) ν K. With θ π/4 fixed, let y(r) = 1 + u A 2. Then on using (2.52) and y 1 we obtain d dr y Cy5, and hence on integrating the inequality we find that provided that By setting 0 r 15 64C y(r) 2y(0) 1 (y(0)) 4 = 15 64C 1 (1 + A s+1/2 u(0) 2 ) 4. T := C (1 + A s+1/2 u(0) 2 ), M 4 1 = A s+1/2 u(0) 2, we deduce that (2.46) holds for 0 r T. 1/2f 2 We can extend the bound (2.46) to a larger domain which contains the interval [0, T ] using the following property of the NSE solution on the sphere [6]: (2.53) A s+1/2 u(, t) M 2 for all t [0, T ], where the constant M 2 depends only on A s+1/2 u 0, sup 0 t T A s f(, t) and ν but not on T.

10 10 M. GANESH, Q. T. LE GIA, AND I. H. SLOAN Theorem 2.3. Suppose u 0 and f satisfy all the conditions in the domain T as in Theorem 2.2. Then (2.54) A s+1/2 e ψ(r cos θ)a1/2 u(ζ) 2 M 3, ζ T and Im ζ T / 2, where M 3 := 1 + 2M2 2, T depends on M 2, and hence u(, ζ) G s+1/2 ψ(r cos θ) for all ζ T with Im ζ T / 2. Proof. We proceed as in the proof of Theorem 2.2 to obtain the ordinary differential equation d dr y Cy5, where y(r) = 1 + A s+1/2 e ψ(r cos θ)a1/2 u(re iθ ) 2. On integrating the inequality we find that y(r) 2y(0) provided that We define 0 r C (1 + A s+1/2 u(0) 2 ). 4 T 1 (ρ) := C (1 + ρ 2 ) 4, ρ 0. If T 1 ( A s+1/2 u(0) ) T we have finished the proof. Otherwise, we let T = T 1 (M 2 ), where M 2 is given in (2.53). For ζ = re iθ T, 0 r T, (2.55) A s+1/2 e ψ(r cos θ)a1/2 u(re iθ ) A s+1/2 u(0) 2. Consequently, (2.54) holds for 0 r T with M 3 = A s+1/2 u(0) 2. Next we consider the case ζ = T + re iθ, with r [0, T ]. We define, for θ π/4, v(re iθ ) := u(t + re iθ ), r [0, T ]. Using A s+1/2 v(0) M 2 we can apply the previous arguments to obtain (2.54) (with u replaced with v) for 0 r T. We complete the proof and obtain the bound (2.54) by repeating the last argument n times, where n = T/T. 3. Finite dimensional spaces and Stokes projections Throughout the remaining of the paper, with s and σ 1 as in Theorem 2.1 and Theorem 2.2, we assume that (3.1) u 0 D(A s+1/2 ), f C([0, T ]; D(A s+1/2 e σ1a1/2 )), s 1/4. Natural finite dimensional spaces (depending on a parameter N > 0) in which to seek approximations to u(t) are (3.2) V N := span {Z L,m : L = 1,..., N; m = L,..., L}. The dimension of V N is N 2 + 2N. Let Π N : H V N be the orthogonal projection with respect to the L 2 (T S) inner product defined by N L (3.3) Π N (v) = v L,m Z L,m. L=1 m= L

11 A PSEUDOSPECTRAL METHOD FOR NSEs ON ROTATING SPHERES 11 Lemma 3.1. Let α > 0 be given. If v D(A α/2 ) then (3.4) v Π N v N α v H α (T S). Proof. Using (2.24), (2.27), and (2.29) we get L v Π N v 2 = v L,m 2 N 2α L=N+1 m= L L=N+1 m= L N 2α v 2 H α (T S). L λ α L v L,m 2 In particular, using (3.1), we get (3.5) f Π N f N (2s+1) f H 2s+1 (T S), t [0, T ]. For computer implementation, the Fourier coefficients in (3.3) and all Galerkin type integrals in computational schemes for the NSE need to be approximated by cubature rules on the sphere, leading to a pseudospectral method. To this end, for a continuous scalar field ψ on S, we consider a Gauss-rectangle quadrature sum Q M (ψ) with quadrature points { ξ p,q = p(θ p, φ q )} and positive weights w p of the form (3.6) Q M (ψ) := 2π M M/2 M q=1 p=1 w p ψ( ξ p,q ) = 2π M M/2 M w p ψ(θ p, φ q ), q=1 p=1 where M 2 is an even integer, w p and cos θ p for p = 1,..., M/2 are the Gauss- Legendre weights and nodes on [ 1, 1] and φ q = 2qπ/M, q = 1,..., M. The rule (3.6) is exact when ψ is a polynomial of degree M 1 on S, that is, Q M ψ = ψ ds, ψ P M 1. S Hence corresponding to (2.14) and (2.15), we define discrete inner products for scalar and vector fields on the unit sphere as (3.7) (v 1, v 2 ) M = Q M (v 1 v 2 ), (v 1, v 2 ) M = Q M (v 1 v 2 ). The choice of M is very important; we choose M such that all Galerkin integrals with polynomial terms in our scheme are evaluated exactly. In particular, with the unknown tangential divergence-free velocity field sought in the polynomial space V N, and knowing that the NSE nonlinearity is quadratic, we choose M such that (3.8) (B(v, w), z) = (B(v, w), z) M, v, w, z V N. This holds, for example, if M = 3N + 2. We define a computable counterpart of (3.3), using L N : H C(T S) V N, a discrete orthogonal projection with respect to the M 2 /2 point discrete inner product, as (3.9) N L L N (v) = v L,m,M Z L,m, v L,m,M = Q M (v Z L,m ) = (v, Z L,m ) M. L=1 m= L With M chosen to satisfy (3.8), it is easy to see that (3.10) L N (v) = Π N (v) = v, v V N,

12 12 M. GANESH, Q. T. LE GIA, AND I. H. SLOAN and for v H C(T S) and w H C k (T S), (3.11) L N (v) C v, w L N (w) CN k w C k (T S), where the last two inequalities follow from simple arguments used in Theorem 13 and Lemma 14 of [23]. In particular, since D(A s+1/2 ) H C 2s (T S), for an integer 2s, using (3.1) (3.12) f L N (f) CN 2s f C 2s (T S), t [0, T ]. Next we consider the Stokes projection in V N of the exact unique regular solution u(t) := u(., t) of (2.1). For each fixed t, the Stokes projection ũ N V N of u is defined by (3.13) (Aũ N, v) = (Au, v), v V N. Since Π N A = AΠ N, it follows that ũ N = Π N (u). Following standard techniques in finite element analysis, the Stokes projection of u plays an important role as a comparison function in the main analysis in the next section. Theorem 3.1. Let u 0 and f satisfy (3.1). Then (3.14) u ũ N Cλ s 1/2 N+1 e σ(t)λ1/2 N+1 CN 2s 1 e σ(t)n, t [0, T ], where σ(t) is as in Theorem 2.1. Proof. Using the fact that ũ N = Π N u, we have u ũ N 2 = û L,m 2 L>N m L λ 2s 1 N+1 e 2σ(t)λ1/2 N+1 L>N m L λ 2s+1 L e 2σ(t)λ1/2 L ûl,m 2 λ 2s 1 N+1 e 2σ(t)λ1/2 N+1 A s+1/2 e σ(t)a1/2 u(t) 2 Cλ 2s 1 N+1 e 2σ(t)λ1/2 N+1 where in the last step we used (2.43). The last inequality in (3.14) follows from the fact that N 2 λ N+1 = (N + 1)(N + 2). Theorem 3.2. Let u 0, f satisfy (3.1). We assume further that f is analytic in T and (2.45) holds. Then for t (0, T ), (3.15) d dt (u ũ N) Cλ s 1/2 N+1 e ψ1(t)λ1/2 N+1 CN 2s 1 e ψ1(t)n, where ψ 1 (t) = min{(1 1/ 2)t, T, σ 1 }, and T is as in Theorem 2.3. Proof. Let t (0, T ) be fixed. Let p N (ζ) = (u ũ N )(ζ) be the standard complexification of u ũ N at ζ = re iθ. Using Theorem 2.3 and the Cauchy integral formula, dp N (t) dt = 1 2πi Γ p N (ζ) (t ζ) 2 dζ, where for t > 0, Γ is a circle in the ζ plane with center (t, 0) and radius min{t/ 2, T / 2, T t}, a condition that ensures that ζ = re iθ Γ lies in the

13 A PSEUDOSPECTRAL METHOD FOR NSEs ON ROTATING SPHERES 13 region T with Im ζ T / 2. Using the fact that ũ N = Π N u, for ζ = re iθ Γ we have, from Theorem 2.3, (3.16) p N (ζ) = u(ζ) ũ N (ζ) 2 = û L,m 2 L>N m L λ 2s 1 cos θ)λ1/2 N+1 e 2ψ(r N+1 L>N m L λ 2s+1 2ψ(r cos θ)λ1/2 L e L ûl,m 2 λ 2s 1 cos θ)λ1/2 N+1 e 2ψ(r N+1 A s+1/2 e ψ(r cos θ)a1/2 u(ζ) 2 Cλ 2s 1 cos θ)λ1/2 N+1 e 2ψ(r N+1 CN 2(2s+1) e 2ψ(r cos θ)n. For ζ = re iθ Γ it is easily seen that r cos θ (1 1/ 2)t, and hence that (3.17) ψ(r cos θ) min{(1 1/ 2)t, T, σ 1 } =: ψ 1 (t). On using (3.16) in (3.14) we get d dt (u ũ N) (t) 1 2π Γ p N (ζ) t ζ 2 dζ CN 2s 1 e ψ1(t)n. 4. A fully discrete pseudospectral quadrature method We are now ready to describe, analyze, and implement a spectrally accurate scheme to compute approximate solutions of the NSE (2.1) in V N, through its weak formulation (2.35). The task is then to compute u N (, t) V N for t [0, T ] with u N (0, x) = L N u 0 ( x), x S, satisfying the (spatially) fully discrete system of ordinary differential equations (4.1) d dt (u N, v) M + b(u N, u N, v) M + ν(curl bx u N, Curl bx v) M + (Cu N, v) M = (f, v) M, for all v V N and prove that the scheme is spectrally accurate with respect to the parameter N (that is, converges with rate determined by the smoothness of the given data), and demonstrate the theory with numerical experiments. In order to the make the above system fully discrete in space and time, and hence compute the N 2 + 2N unknown time-dependent coefficients in the representation of the tangential divergence-free approximate real velocity vector field, for x S and t 0, (4.2) N u N ( x, t) := α L,m (t)z L,m ( x), α L,m = α L, m, α L,m (0) = (u 0, Z L,m ) M, L=1 m L one may use the standard finite element fixed-time-step Crank-Nicolson-Galerkin approach in (4.1), leading to a second-order in time non-adaptive scheme. However, due to the complicated flow behavior of the NSE solutions, especially with random initial states, it is efficient instead to integrate (4.1) using multi-order backward differentiation stiff solvers with adaptive evolution in time, for a chosen accuracy. Accordingly, in this paper we follow the latter approach.

14 14 M. GANESH, Q. T. LE GIA, AND I. H. SLOAN Using (3.2), the exactness properties of the discrete inner product, (3.8), (2.17), (2.23), (2.31), and (2.37), the system (4.1) can be written as (4.3) ( ) d dt u N + νau N + B(u N, u N ) + Cu N, Z L,m L = 1,, N; m = L,, L. = (f, Z L,m ) M, We now substitute (4.2) in (4.3), and write the resulting system of ordinary differential equations as (4.4) d α(t) = F(t, α(t)). dt For the adaptive solver, it is important to have an efficient method for the evaluation of the nonlinear function F. To this end, we follow the fast evaluation procedure described in [4]. Using the spectral properties of the Stokes operator A given by (2.24), the linear second term in (4.3) is trivial to evaluate using the diagonal matrix consisting of the eigenvalues of A. The Coriolis term can be evaluated similarly using the identity [5, Equation (24)] (4.5) (C Curl Y J,k, Z L,m ) = (2Ω cos θ x Curl Y J,k, Z L,m ) = 2Ωi m λ 1/2 L δ L,J δ k,m. For the first component in the nonlinear third term in (4.3), we use (2.13) to write (4.6) B(Z R,s, Z J,k ) = 1 λr λ J P Curl ( Y R,s Grad Y J,k ), (4.7) (B(Z R,s, Z J,k ), Z L,m ) = λr λ J λ L (Y R,s Grad Y J,k, x Grad Y L,m ). It is convenient to write Grad Y J,k and x Grad Y L,m in terms of expressions similar to those in (2.25). Such explicit representations are also useful for the efficient evaluation of the N 2 Fourier coefficients (f, Z L,m ) M of the source term in (4.3), and eventually for the computation of the vorticity field. In order to express the tangential (and normal, needed for computing the approximate vorticity from u N ) vector harmonics as a linear combination of the scalar harmonics (2.25), we first recall, from the classical quantum mechanics literature (see, for example, [27]), the covariant spherical basis vectors (4.8) e +1 = 1 ([1, 0, 0] T +i[0, 1, 0] T ), e 0 = [0, 0, 1] T, e 1 = 1 ([1, 0, 0] T i[0, 1, 0] T ), 2 2 and the Clebsch-Gordan coefficients (4.9) C j,m j 1,m 1,j 2,m 2 := ( 1) (m+j1 j2) ( j1 j 2j j m 1 m 2 m ),

15 A PSEUDOSPECTRAL METHOD FOR NSEs ON ROTATING SPHERES 15 ( ) a b c where are the Wigner 3j-symbols given, for example, by the Racah α β γ formula, ( a b c ) α β γ = ( 1) (a b γ) T (abc) (a + α)!(a α)!(b + β)!(b β)!(c + γ)!(c γ)! ( 1) t t!(c b + t + α)!(c a + t β)!(a + b c t)!(a t α)!(b t + β)!, t where the sum is over all integers t for which the factorials in the denominator all have nonnegative arguments. In particular, the number of terms in the sum is 1 + min{a ± α, b ± β, c ± γ, a + b c, b + c a, c + a b}. The triangle coefficient T (abc) is defined by [ ] (a + b c)!(a b + c)!( a + b + c)! T (abc) =. (a + b + c + 1)! Below, we require C j,m j 1,m 1,j 2,m 2 only for some j 2, m 2 { 1, 0, 1}, and using various symmetry and other known properties (such as C j,m j 1,m 1,j 2,m 2 = 0 unless the conditions j 1 j 2 j j 1 + j 2 and m 1 + m 2 = m hold) of Wigner 3j-symbols, these coefficients can be efficiently pre-computed and stored. In our computation, we used the following basis functions for the tangential vector fields: (i) Grad Y L,m, (ii) x Grad Y J,k. For the vorticity components of u N, in addition we used (iii) Vort Z J,m = Curl bx Z J,m = λ 1/2 J xy J,m = λ 1/2 J x Y J,m. In particular, using (4.2) our approximation to the vorticity in (2.2), for a fixed t 0 and x S, is (4.10) Vort u N ( x, t) = Curl bx u N ( x) = x Ψ N ( x), where Ψ N ( x, t) = N L=1 m L λ 1/2 L α L,m (t)y L,m ( x). To facilitate easy application of fast transforms to evaluate these functions at the M = O(N 2 ) quadrature points { ξ p,q = p(θ p, φ q )}, we represent these three types of fields first as a linear combination of the covariant vectors in (4.8): (4.11) Grad Y L,m = B +1,L,m e +1 + B 0,L,m e 0 + B 1,L,m e 1, (4.12) ( x Grad Y J,k ) = D +1,J,k e +1 + D 0,J,k e 0 + D 1,J,k e 1, L With c L = (L + 1) 2L+1, d L+1 L = L 2L+1, these coefficients are explicitly given by B +1,L,m = { } c L C L,m L 1,m 1,1,1 P m 1 L 1 (cos θ) + d LC L,m L+1,m 1,1,1 P m 1 L+1 (cos θ) e i(m 1)ϕ { } B 0,L,m = c L C L,m L 1,m,1,0 P L 1(cos m θ) + d L C L,m L+1,m,1,0 P L+1(cos m θ) e imϕ { } B 1,L,m = c L C L,m L 1,m+1,1, 1 P m+1 L 1 (cos θ) + d LC L,m L+1,m+1,1, 1 P m+1 L+1 (cos θ) e i(m+1)ϕ.

16 16 M. GANESH, Q. T. LE GIA, AND I. H. SLOAN D +1,J,k = i λ J C J,k J,k 1,1,1 P k 1 J (cos θ)e i(k 1)ϕ, D 0,J,k = i λ J C J,k J,k,1,0 P k J (cos θ)e ikϕ, D 1,J,k = i λ J C J,k J,k+1,1, 1 P k+1 J (cos θ)e i(k+1)ϕ. Noting (i) the complex azimuthal exponential terms e ikϕ, e imϕ in (2.25) and (4.11)-(4.12) (via the above representations for B and D) for k J, m L; 1 L, J N, and (ii) the need to evaluate several O(N 2 ) sums, of the form in (4.2) and (4.6)-(4.6), at the equally spaced O(N) azimuthal quadrature points (see (3.6)), we reduce the complexity by O(N) in each of these sums, at each adaptive-time step, by using the FFT for setting up F(t, α(t)) in (4.4). In our numerical experiments (see Section 5) for adaptive-time simulation of a flow induced by random initial states, we observed that such an efficient FFT based implementation reduced the (non-fft code) computing time by several weeks for the case N = 100, to simulate from t = 0 to t = 60 (by saving a few days for each adaptive time step). In addition, by using the fast Legendre/spherical transforms along the latitudinal direction (obtained, for example, by modifying the NFFT algorithm in [20] for evaluation of the Legendre functions in the above terms at O(N) non-uniform latitudinal quadrature points), we could reduce the complexity by O(N 2 ). We did not use the fast Legendre/spherical transforms in our implementation due to the spectral convergence of the scheme and the fact that N 100 in our simulations. (In these complexity counts, we ignored O(log N) and O(log 2 N) terms.) 4.1. Stability and convergence analysis. First we establish the stability of the approximate solution u N of (4.2). That is, similar to [7, Proposition 9.1], we prove that max t [0,T ] u N V is uniformly bounded with the bound depending only on the initial data, forcing function, and the viscosity term in (4.2). Theorem 4.1. Let u 0 and f satisfy (3.1). Let N 1 be an integer. Let u N be the solution of the pseudospectral quadrature Galerkin system (4.1). Then there exists a constant C depending on ν, u 0 V and f := max t [0,T ] f(t) C(T S) so that max u N V C. t [0,T ] Proof. The proof follows by repeating the arguments described in the first four pages of [7, Section 9] (proving [7, Proposition 9.1]), provided that we establish [7, Inequalities (9.3) and (9.13)] for our system (4.1) on the spherical surface with additional Coriolis term and quadrature approximations. By Lemma 2.1, (2.37) and the exactness of the quadrature rule (given by (3.7)- (3.8)), we have (B(u N, u N ), u N ) M = 0. Using (4.5), the symmetry of the coefficients of u N in (4.2) and the exactness of the quadrature, we get (4.13) (Cu N, u N ) M = (Cu N, u N ) = ( 2Ωi) N L=1 λ 1 L m L m α L,m 2 = 0. By taking v to be u N in (4.1) and using u N 2 = (u N, u N ) M, u N 2 V = (Au N, u N ) M, (4.13), Young s inequality and the fact that all the eigenvalues λ J of A (corresponding to eigenvectors in u N ) satisfy λ J λ 1 = 2, J = 1,, N, we obtain 1 d 2 dt u N 2 +ν u N 2 V = (f, u N ) M f u N f 2 4ν +ν u N 2 f 2 4ν +ν 2 u N 2 V,

17 A PSEUDOSPECTRAL METHOD FOR NSEs ON ROTATING SPHERES 17 Hence, for our discrete system (4.1), we obtain [7, Inequality (9.3)]: (4.14) d dt u N 2 + ν u N 2 V f 2 νλ 1. Again using (4.5), the exactness of the quadrature rule, eigenfunction properties of A, and the symmetry of the coefficients of u N in (4.2), we get (4.15) (Cu N, Au N ) = ( 2Ωi) N L=1 m L m α L,m 2 = 0, By taking v to be Au N in equation (4.1), and using Au N 2 = (Au N, Au N ) M, (3.8), and (4.13), we obtain (4.16) 1 d 2 dt u N 2 V + ν Au N 2 + b(u N, u N, Au N ) = (f, Au N ) M. Using [15, Lemma 3.1], b(u N, u N, Au N ) = 0. The term (f, Au N ) M can be estimated by using the exactness of the quadrature and Young s inequality: (f, Au N ) M f Au N f 2 2ν + ν 2 Au N 2 Hence we obtain a stronger version of [7, Inequality (9.13)] for our quadrature discrete scheme (4.1): d dt u N 2 V + ν Au N 2 f 2 νλ 1. Thus, the result follows from arguments in [7, Page 74-77]. Next, using Theorem 3.1, 3.2, and 4.1, we prove the spectral convergence of the solution u N of (4.3) to the solution of u of (2.35). Theorem 4.2. Let u 0, f satisfy (2.45), (3.1). Then there exists a T # > 0, depending only on ν, f, u 0 and the uniform bound in (2.53) (and hence there exists 0 < µ(t) < min{t, T #, σ 1 }) such that for all t (0, T ), [ ] (4.17) u u N C N 2s 1 e µ(t)n + Π N f L N f. In particular, with 2s being an integer (4.18) u u N CN 2s. Proof. Let w N = ũ N u N, where the comparison function ũ N is the solution of (3.13). Since u u N = p N + w N, where p N = u ũ N, in view of Theorem 3.1 and 3.2, existence of T # and µ(t) follows and it is sufficient to show that w N C [ N 2s 1 e µ(t)n + Π N f L N f ]. For any v V N, using (2.36), (3.13), and (4.3), ((w N ) t, v) + ν(aw N, v) + (Cw N, v) = ((ũ N ) t, v) + ν(aũ N, v) + (Cũ N, v) ((u N ) t, v) ν(au N, v) (Cu N, v) = ((ũ N ) t, v) + ν(aũ N, v) + (Cũ N, v) (f, v) M + (B(u N, u N ), v) = ((ũ N ) t, v) + ν(au, v) + (Cũ N, v) (f, v) M + (B(u N, u N ), v) = ((ũ N u) t, v) + (f, v) (f, v) M + (Cũ N Cu, v) + (B(u N, u N ) B(u, u), v).

18 18 M. GANESH, Q. T. LE GIA, AND I. H. SLOAN Using the orthogonal projection Π N in (3.2), we can write this relation in functional form as dw N = ( νa C)w N Π N [(p N ) t + Cp N ]+Π N f L N f+π N [B(u N, u N ) B(u, u)]. dt Integrating with respect to t and using w N (0) = 0, we have (4.19) Let w N (t) = t 0 t + e (t s)(νa+c) [ Π N ( d ds p N + Cp N ) + Π N f L N f 0 e (t s)(νa+c) Π N [B(u N, u N ) B(u, u)] (s) ds. (4.20) R N (ɛ, t s) = ν ɛ Π N A ɛ e (t s)(νa+c). ] (s) ds On using (2.39), with δ (1/2, 1), and the uniform boundedness of the orthogonal projection Π N, we get (4.21) e (t s)(νa+c) Π N [B(u N, u N ) B(u, u)] R N(δ, t s) A δ Π N (B(u N, u N ) B(u, u)) CR N (δ, t s) u N u. ν δ Taking norms and using u N u w N + p N together with (4.20), and (4.22) in (4.19), we obtain [ w N (t) d ] t dt p N + Cp N + Π N f L N f R N (0, t s) ds +C t 0 R N (δ, t s)( p N (s) + w N (s) )ds. Using Gronwall s inequality, we obtain for each t [0, T ], {[ w N (t) C d ] t dt p N + Cp N + Π N f L N f R N (0, t s) ds (4.22) + p N For ɛ [0, 1], we have to bound (4.23) t Using also (4.20) and (4.5), 0 t 0 } R N (δ, t s) ds. R N (ɛ, t s) ds = t 0 R N (ɛ, r) dr. R N (ɛ, r) max (νλ L) ɛ e νλlr e 2Ωirmλ L max 1 L N; m L z [νλ 1,νλ N ] zɛ e rz. 1/2 Thus (νλ N ) ɛ e νλ N r if r ɛ/(νλ N ), (4.24) R N (r) ɛ ɛ e ɛ r ɛ if ɛ/(νλ N ) r ɛ/(νλ 1 ), (νλ 1 ) ɛ e νλ1r if r ɛ/(νλ 1 ). With (4.25) I 1 = [0, ɛ/(νλ N )] [0, t], I 2 = [ɛ/(νλ N ), ɛ/(νλ 1 )] [0, t], I 3 = [ɛ/(νλ 1 ), t] [0, t], 0 0

19 A PSEUDOSPECTRAL METHOD FOR NSEs ON ROTATING SPHERES 19 the interval of integration in (4.23) can be subdivided into these three intervals. In particular, using (4.24) and (4.25), we get (4.26) R N (r)dr (νλ N ) ɛ e νλ N r dr (1 e ɛ ) C, I 1 I 1 (νλ N ) 1 ɛ (4.27) and (4.28) I 2 R N (r)dr [ ( ) (1 ɛ) ( ) ] (1 ɛ) ɛ I2 ɛ e ɛ r ɛ dr ɛe ɛ 1 1 C, (1 ɛ) νλ 1 νλ N R N (r)dr (νλ 1 ) ɛ e νλ1r dr (e ɛ e νλ1t ) I 3 I 3 (νλ 1 ) 1 ɛ C. Using (4.26)- (4.28) in (4.23), we get (4.29) t 0 R N (ɛ, t s) ds C, ɛ [0, 1]. Substituting this in (4.22), we get [ (4.30) w N (t) C d ] dt p N + Cp N + p N + Π N f L N f. The term Cp N in (4.30) can be simplified using (2.31) and the fact that p N is tangential (and hence x p N ( x) = 0), [ ] [ ] [ x p N ( x)] ( x p N ( x) = [ x x] p N ( x) p N ( x) = p N 2, and hence (4.31) w N (t) C [ d ] dt p N + p N + Π N f L N f. Hence from Theorem 3.1 and 3.2, for all t (0, T ) we have [ ] (4.32) u u N C N 2s 1 e µ(t)n + Π N f L N f. In particular, using (3.5) and (3.12) with 2s being an integer, we get (4.33) Π N f L N f Π N f f + f L N f CN 2s. Now the result (4.18) follows from (4.32) and (4.33). 5. Numerical Experiments We demonstrate the fully discrete pseudospectral quadrature algorithm by simulating (i) a known solution test case with low to high frequency modes and (ii) a benchmark example [9, page 305] in which the unknown velocity and vorticity fields are generated by a random initial state. The first test example is useful to demonstrate that the pseudospectral quadrature algorithm reproduces any number of high frequency modes in the solution (provided V N contains all these modes), with computational error dominated only by the chosen accuracy for the adaptive time evolution for the ordinary differential system (4.4).

20 20 M. GANESH, Q. T. LE GIA, AND I. H. SLOAN 5.1. Example 1. Our test case first example is (2.1) with (5.1) u t=0 ( x) = u 0 ( x) = g(0)[w 1 ( x) W 2 ( x)], (5.2) g(t) = νe t [sin(5t)+cos(10t)], W 1 = Z 1,0 +2R(Z 1,1 ), W 2 = Z 2,0 +2R(Z 2,1 +Z 2,2 ), where Z L,m is given by (2.24), R( ) denotes the real-part function, and the external force f( x, t) in (2.1) is chosen so that (5.3) [ ] N 0 L u( x, t) = tg(t) Z L,0 + 2 R(Z L,m ) ( x) + g(t)w 1 ( x) + (t 1)g(t)W 2 ( x), L=1 m=1 is the exact tangential divergence-free velocity field, solving the NSE (2.1). The exact test field (5.2)-(5.3) has high oscillations both in space and time, and exponentially decays in time. Note the dependence on a parameter N 0, the maximum order of the spherical harmonics in the exact solution. In our calculation of the approximate solution u N, we chose N = N 0, so that all frequencies of the exact solution can be recovered. The solution (5.3) is then used to validate our algorithm and code by numerical adaptive time-integration of (4.1), for various values of N = N 0. In particular, for a fixed integration tolerance error, we demonstrate in Figure 1 that all N modes in (5.3) can indeed be recovered by the approximate solution u N, within the chosen error tolerance, for all N = N 0 = 70, 80, 90, 100. Figure 1. u u N for Example 1 with a fixed time-integration error, N = N 0 = 70, 80, 90, 100.

21 A PSEUDOSPECTRAL METHOD FOR NSEs ON ROTATING SPHERES Example 2. Having established the validity of our algorithm for a simple known solution, we use the same code to simulate unknown velocity and vorticity fields generated by (2.1) with random initial velocity field as in [9, 13] with angular velocity of the rotation Ω = 1 and ν 1 = 10, 000 in (2.1). The initial flow and external force in this benchmark example satisfy the main assumption (3.1) for any s, σ 1 > 0 and hence, as proved in Theorem 4.2, the approximate solution u N is spectrally accurate and converges super-algebraically with order given by (4.18) for any s > 0. As mentioned in Section 1, this is the main advantage of the present paper over the recent paper [13], where such spectrally accurate convergence results are neither discussed nor proved. On the other hand, convergence results for two-dimensional problems on a Euclidean plane, supported by numerical experiments, have formed a core part of research on the NSE over the last few decades, see [9, 10, 25] and references therein. The random initial tangential divergence-free velocity field, having properties similar to those considered in [9, page 305] and [13, page 988] (but not exactly same as in [9, 13], due to randomness), is a smooth function u 0 = v G s/2 σ, a Gevrey class of order s and index σ, see (2.41), for any s, σ > 0, with Fourier coefficients v L,m, L = 1, 2,, m L, defined by a L exp(iφ m ) L = 1,..., 20; m = 0,..., L, (5.4) v L,m = a L ( 1) m exp( iφ m ) L = 1,..., 20; m = L,, 1, 0, L > 20; m = L,..., L, where φ 0 = 0, and φ m (0, 2π) are random numbers for m > 0, and a L = b L / b, with b R 20 having components b L = 2/ [ L + (νl) 2.5], L = 1,..., 20. The vorticity stream function Ψ (see (2.2)) of Vort u(, 0) in Figure 2 demonstrates the randomness of the field at time t = 0. The external force field f = f( x, t) in (2.1) for our simulation is motivated by that considered in [9, page 305] and is exactly same as that in [13, page 988]. The source term f is a decaying tangential divergence-free field which, for any s, σ > 0, belongs to C([0, T ]; D(A s+1/2 e σa1/2 )) (for any T > 0) with the only nonzero Fourier coefficient being f(t) 3,0. The Fourier coefficient f(t) 3,0 is a continuous function in time and is defined by { 1, 0 t 10, f(t) 3,0 = cos(πt/5) exp( (t 10)/5) t > 10. The impact of the external force on the numerical velocity and its complement (generating the approximate inertial manifold) is demonstrated in Figure 3 with the time evolution of the velocity field matching that of the external force, leading to little change in evolution process of the velocity as the external force gets smaller and smaller. We chose a fixed relative error tolerance to be of accuracy at least O(10 3 ), for adaptive time-integration solving the N 2 + 2N-dimensional system ordinary differential equations (4.1) in time, using backward differential formulas with variable order (one to five) and variable adaptive time step sizes that meet the fixed error tolerance. As discussed in the next subsection, the high-frequency components of the solution turn out to be unreliable as t increases, [9], presumably because of the time

22 22 M. GANESH, Q. T. LE GIA, AND I. H. SLOAN discretization error. We therefore retained only the frequency components up to some order N 1 N, correspondingly, we define an additional approximation of u N, defined in (4.1)-(4.2), by (5.5) u N1;N(, t) := Π N1 u N (, t), N 1 N. As in Theorem 4.2, for the semi-discrete pseudospectral quadrature method, assuming (3.1) and exact time integration of (4.1), and hence using (2.43), (3.4), and (4.18), we get spectral convergence: [ u u N1;N u Π N1 u + Π N1 (u u N ) C N 2(s+1) 1 + N 2s] CN1 2s. Our simulated approximate velocity fields U 75 (t) := u 75;100 (t), for t = 10, 20, 30, 40, 50, 60, are in Figure 4 9 and the associated vorticity stream function Ψ 75 (t) of Vort U 75 (t) (computed using (4.10)) are in Figure These figures demonstrate that the initial random flow with several smaller structures evolve into regular flow with larger structures, similar to those observed in [9, page 307]. The choice of N 1 will be discussed in the next subsection Energy spectrum of the solution. If u (and hence the number of modes in u) is unknown, as it is in the case in the benchmark test Example 2, and if u N does not contain all of the modes in u, the higher modes of u N are usually less accurate than the chosen practical error tolerance and hence can even violate important physical properties of u (because of the error time-integration tolerance being not very small). In such cases, it is important to choose N 1 < N, depending on certain known physical properties of u. For a fixed time t, the L-th mode energy spectrum of a tangential divergence-free flow u on the sphere is defined by (5.6) E(L) = E(u, L) = β L,m (t) 2, u( x, t) := β L,m (t)z L,m ( x). m L L=1 m L Although the analytical form of the flow in Example 2 is not known, several investigations have been carried out for such fields with initial spectrum of the L-th mode decaying with order L 1 or L 2. In particular, it is well known (see [9]), for this benchmark test case (on periodic two dimensional geometries), that the energy spectrum of the velocity has a power-law inertial range and an exponential decay (dissipation range) for wave numbers larger than the Kraichnan s dissipation wave number. Further, several random smaller structures built in the initial random vorticity evolve into regular flow with larger structures. With u being the unique solution of the NSE (2.1), let us decompose u = ũ N1 +w N1, where ũ N1 = Π N1 (u) contains all modes lower or equal N 1 and w N1 := u Π N1 (u) contains all higher modes. The existence of a relation between w N1 and ũ N1 of a form w N1 = Φ(ũ N1 ) was established in [26]. The graph of Φ is known as the inertial manifold of (2.1). For computational purposes, the higher modes can be computed efficiently using an approximate inertial manifold: (5.7) Φ ( ) fn1 (ũ N1 ) = (νa + C) 1 Π fn1 Π N1 [f B (ũ N1, ũ N1 )], Ñ 1 > N 1.

23 A PSEUDOSPECTRAL METHOD FOR NSEs ON ROTATING SPHERES 23 This well known approximation (without the Coriolis term) was introduced in [11, 12] for nonlinear dissipative systems, including the NSE on domains and we choose Ñ 1 = 2N 1. For the benchmark test case, in the dissipation range, even L 4 E( Φ 2N1 (ũ N1 ), L) decays exponentially to zero, further justifying restriction of the infinite dimensional range after certain values of N 1. As discussed in [9, page 280, 307] (and repeated in [13, page 991]), such a faster decay (one order higher than the L 3 decay known is Kraichnan theory of turbulence) is expected due to the Reynolds number considered in [9, 13] being much small than 25, 000. (Extensive study in [3, 22] shows that the turbulence theory decay can occur only when the Reynolds number is of the order 25, 000.) Briefly (without repeating technical details in [9]), using the viscosity term ν in (2.1), the Reynolds number is O(ν 1 ) with the order constant r given by the product of the mean fluid velocity and the characteristic length-scale. With ν 1 = 10, 000 and the constant rotation rate Ω = 1, for the Coriolis parameter in (2.31), the decay of energy spectrum of a numerical velocity and exponential decay of its approximate complement in Figure is well supported by extensive simulations in [3, 22, 9], highlighting further the benchmark applicability of our algorithm, extending periodic domain results in [3, 22, 9] to a practically relevant rotating sphere case with Coriolis effect. Further, the simulated results are substantiated by the well known exponential decay of E( Φ 2N1 (ũ N1 ), L), observed in Figures Finally, the exponential decay of the energy spectrum show that N = 100, N 1 = 3N/4 is sufficient to understand the flow behavior for this benchmark example with our algorithm using adaptive variable order and variable time-step highly stable backward differentiation formulas with a practically useful relative error tolerance O(10 3 ).

24 24 M. GANESH, Q. T. LE GIA, AND I. H. SLOAN Figure 2. Initial random vorticity stream function Ψ of Vort u at t = 0. Time evolution of U 75 (t) and f(t). Time evolution of Φ 150 (U 75 (t)) and f(t). Figure 3. Impact of the external force on a numerical velocity and its complement.

25 A PSEUDOSPECTRAL METHOD FOR NSEs ON ROTATING SPHERES Figure 4. Numerical velocity U75 (t), at t = 10. Figure 5. Numerical velocity U75 (t), at t = 20. Figure 6. Numerical velocity U75 (t), at t =

26 26 M. GANESH, Q. T. LE GIA, AND I. H. SLOAN Figure 7. Numerical velocity U75 (t), at t = 40. Figure 8. Numerical velocity U75 (t), at t = 50. Figure 9. Numerical velocity U75 (t), at t = 60.

27 A PSEUDOSPECTRAL METHOD FOR NSEs ON ROTATING SPHERES 27 Figure 10. Vorticity stream function Ψ 75 of Vort U 75 at t = 10. Figure 11. Vorticity stream function Ψ 75 of Vort U 75 at t = 20. Figure 12. Vorticity stream function Ψ 75 of Vort U 75 at t = 30.

28 28 M. GANESH, Q. T. LE GIA, AND I. H. SLOAN Figure 13. Vorticity stream function Ψ 75 of Vort U 75 at t = 40. Figure 14. Vorticity stream function Ψ 75 of Vort U 75 at t = 50. Figure 15. Vorticity stream function Ψ 75 of Vort U 75 at t = 60.