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Approximation of Parabolic Pdes on Spheres Using Spherical Basis Functions ∗ Advances in Computational Mathematics (2005) 22: 377–397 Springer 2005 Approximation of parabolic PDEs on spheres using spherical basis functions ∗ Q.T. Le Gia Department of Mathematics, Texas A&M University, TX 77843-3368, USA E-mail: [email protected] Received 17 June 2003; accepted 29 September 2003 Communicated by C.A. Micchelli In this paper we investigate the approximation of a class of parabolic partial differential + equations on the unit spheres Sn ⊂ Rn 1 using spherical basis functions. Error estimates in the Sobolev norm are derived. Keywords: heat equation, radial basis functions, collocation method, spheres AMS subject classification: 35K05, 65M70, 46E22 1. Introduction Approximation of partial differential equations on spheres has many applications in physical geodesy, potential theory, oceanography, and meteorology [2,17,18]. Evolu- tion equations on spherical geometry such as shallow water equations have been studied in weather forecasting services [3,23]. The geometry of the sphere is a major obsta- cle in constructing the approximation space for the solution of the PDEs. One way to overcome the obstacle is to construct basis functions which depend only on the geodesic distance between two points on the sphere, which are called spherical basis functions in literature [2,5,13]. Error estimates of pseudo-differential operator (which are time- independent) were studied in [2,9] but error estimates for the evolution equations remain unexplored. In this paper we consider the following parabolic partial differential equation de- fined on the unit sphere Sn ⊂ Rn+1: ∂ u(x, t) − u(x, t) = F(x,t) ∂t (1) u(x, 0) = f(x), f ∈ H 2σ Sn , ∗ The results presented in this paper are taken from the author’s Ph.D. dissertation under supervision of Professor J.D. Ward and Professor F.J. Narcowich at Texas A&M University. 378 Q.T. Le Gia / Approximation of parabolic PDEs on spheres where is the Laplace–Beltrami operator on Sn and H 2σ (Sn) is the Sobolev space defined on Sn (see section 2). It is known that equation (1) describes the heat diffusion process on the surface of the sphere with external heat source F(x,t). In many applications in geophysics and global weather forecast, it is common that the function f is not known analytically everywhere but only at a finite set of scattered points. We propose a collocation method in which the spherical basis functions are used to construct the approximate solution. The approximate solution of the partial differential equation will be of the form m uX(x, t) = ci(t)φi(x), (2) i=1 subject to the initial condition uX(x, 0) = IXf(x), where φi(x) = φ(x·xi) = (xi,x)’s are the shifts of a spherical basis function (SBF) φ and IXf is the SBF interpolant of the function f . In case the basis function φ satisfies certain regularity conditions, we are able to obtain error estimates in certain Sobolev norms. The paper is organized as follows: section 2 gives the necessary background on spherical harmonics and the Laplace–Beltrami operator together with the problem of interpolation on spheres using spherical basis functions. In section 3, we present the semi-discrete problem, in which the exact solution u(x, t) is approximated by uX of the form (2) and uX is a solution of a system of ordinary differential equations. In section 4, we discretize (1) also in time variable so as to produce a completely discrete scheme for the approximate solution of our problem. Finally, some numerical experiments are presented in section 5. 2. Preliminaries 2.1. Spherical harmonics and Sobolev spaces Spherical harmonics are polynomials which satisfy xY(x) = 0(wherex is the Laplacian operator in Rn+1) and are restricted to the surface of the Euclidean sphere Sn. A more detailed discussion of spherical harmonics can be found in [10]. It is well known that Laplace–Beltrami operator is linear, self-adjoint and negative definite in the spa- tial variables. The eigenvalues for − are λ = ( + n − 1) for = 0, 1, 2, 3 ...and the respective eigenfunctions are the spherical harmonics Y(x) of order , i.e., Y(x) =−λY(x). n The space of all spherical harmonics of degree on S , denoted by V, has an orthonor- mal basis Yk(x): k = 1,...,N(n,) , Q.T. Le Gia / Approximation of parabolic PDEs on spheres 379 where (2 + n − 1)( + n − 1) N(n,0) = 1andN(n,) = for 1. ( + 1)(n) Every function f in L2(Sn) can be expanded in terms of spherical harmonics ∞ N(n,) ˆ ˆ f = fkYk, fk = f Yk dS, n =0 k=1 S where dS is the surface measure of the unit sphere. The Sobolev space H σ (Sn) with real parameter σ consists of all distributions f such that ∞ N(n,) 2 = + σ ˆ 2 ∞ f σ (1 λ) fk < . =0 k=1 For more details we refer to [7, section 1.7]. 2.2. Positive definite kernels on spheres Bizonal functions on Sn are functions that can be represented as φ(x · y) for all x,y ∈ Sn,wherex ·y is the usual dot product in Rn+1 and φ(t)is a continuous functions on [−1, 1]. We shall be concerned exclusively with bizonal kernels of the type ∞ ∞ (x,y) = φ(x · y) = aP(n + 1; x · y), a 0, a < ∞, =0 =0 { + ; }∞ + where P(n 1 t) =0 is the sequence of (n 1)-dimensional Legendre polynomials. Recall from [10] that 1 2 (n−2)/2 P(n + 1; t)Pk(n + 1; t) 1 − t dt = 0, for = k, −1 and 1 n (n− )/ |S | P (n + 1; t) 2 1 − t2 2 2 dt = , n−1 −1 |S |N(n,) where |Sn| is the surface area of Sn, |Sn−1| is the surface area of Sn−1. Thanks to seminal work of Schoenberg [16], we know that such a is positive def- n := [ ]m inite on S ,thatis,thematrixA (xi,xj ) i,j=1 is positive semidefinite for every set n of distinct points {x1,...,xm} on S for any positive integer m. When the coefficients a are positive for every , we say that is strictly positive definite, hence invertible, for n every set of distinct points {x1,...,xm} on S and every m (see [24]). Using the addition theorem (see [10]), we can express (x,y) as the following: ∞ N(n,) |Sn| (x,y) = φ()ˆ Y (x)Y (y), φ()ˆ = a > 0, ∀ 0. (3) k k N(n,) =0 k=1 380 Q.T. Le Gia / Approximation of parabolic PDEs on spheres Upon completion, the kernel defines a reproducing kernel Hilbert space N with respect to the following inner product ∞ N(n,) ˆ ˆ = ukvk u, v ˆ . =0 k=1 φ() More precisely, we define the native space N to be the completion of the following set: ∞ N(n,) |fˆ |2 := ∈ D n 2 = k ∞ N f S : f ˆ < , =0 k=1 φ() where D (Sn) denotes the set of all tempered distributions defined on Sn. Note that is the reproducing kernel in N in the sense that for every f ∈ N and for any fixed x ∈ Sn, ∞ N(n,) Y (x)fˆ (·,x),f = φ()ˆ k k = f(x). ˆ =0 k=1 φ() ˆ −σ Throughout the paper we make further assumption that φ() ∼ (1 + λ) , i.e., there are positive constants c and C and σ>n/2 such that −σ ˆ −σ c(1 + λ) φ() C(1 + λ) . (4) We define the convolution kernel of by ∗ (x,y) := (x, z)(z, y) dS(z), x,y ∈ Sn. Sn In terms of Fourier expansions we have ∞ N(n,) ˆ 2 ∗ (x,y) = φ() Yk(x)Yk(y). =0 k=1 This observation allows us to define a convolution native space to be the completion of the following set ∞ N(n,) | ˆ |2 n 2 fk N∗ = f ∈ D S : f ∗ = < ∞ . ˆ 2 =0 k=1 (φ()) If the kernel satisfies condition (4) then ∼ 2σ n σ n ∼ N∗ = H S ⊂ H S = N. 2.3. Interpolation of scattered data on Sn n Now let X ={xi: i = 1,...,m} be a set of m distinct scattered points on S . Schoenberg, in [16], establishes that if the coefficients φ()ˆ 0forall 0 then the Q.T. Le Gia / Approximation of parabolic PDEs on spheres 381 matrix [(xi,xj )] is symmetric positive semi-definite for any configuration of X.Xu and Cheney [24] have shown that if φ()ˆ > 0forall 0 then the matrix is symmetric positive definite, hence invertible. The distribution of the set X is measured by its mesh norm hX := sup inf d(x,y), ∈ x∈Sn y X −1 n where d(x,y) = cos (x · y) is the geodesic distance on S . The separation distance qX is defined as 1 qX := min d(xi,xj ). 2 i=j We always have ρX := hX/qX 1. The set X is called quasi-uniform if 1 ρX <c, where c is a constant independent of the set X. The finite-dimensional vector space is defined as VX = span φi(x): i = 1,...,m , where the spherical basis functions φi are the shifts of a strictly positive definite func- tion φ on the set X,i.e. φi(x) := (xi,x), i = 1,...,m. (5) n Let IX be the interpolation operator IX : C(S ) → VX such that IXf(xj ) = f(xj ) for all xj ∈ X. In practice, we have to solve the following linear system in order to find coefficients ci’s such that m ci(xi,xj ) = f(xj ) for all j = 1,...,m.
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