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
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. (6) i=1 n The operator IX is well defined for every function f ∈ C(S ) since the matrix (xi,xj ) i,j=1,...,m is positive definite, hence invertible, for every configuration of the set X.
Lemma 2.1. For every f ∈ N,wehave 2 + − 2 = 2 IXf f IXf f .
Proof. Since is the reproducing kernel in the reproducing Hilbert space N,the interpolating condition IXf(xj ) = f(xj ) for all xj ∈ X is equivalent to − · = ∀ ∈ IXf f,(xj , ) 0, xj X.
Since IXf is a linear combination of (xj , ·)’s, we have the orthogonal property
IXf − f,IXf = 0. Hence the desired relation follows from the Pythagorean theorem. 382 Q.T. Le Gia / Approximation of parabolic PDEs on spheres
Based on [4, theorem 4.7], we have the following theorem.
ˆ −σ Theorem 2.1. Let be a positive definite kernel with φ() ∼ (1 + λ) ,andf ∈ 2σ n H (S ). Then there exists a positive constant C independent of hX such that − σ f IXf ChX f 2σ .
Proof. − 2 = − f IXf f, f IXf ∞ N(n,) ˆ ˆ − = fk(fk IXf k) ˆ = = φ() 0 k 1 ∞ 1/2 ∞ 1/2 N(n,) | ˆ |2 N(n,) fk ˆ 2 fk − IXf ˆ 2 k =0 k=1 φ() =0 k=1 f ∗f − IXf L2(Sn). (7) Then by using [4, theorem 4.4] with p = 2, it follows that − σ − f IXf L2(Sn) ChX f IXf . (8) ∼ 2σ n Combining inequalities (7) and (8) and noting that N∗ = H (S ) we have − σ f IXf ChX f 2σ .
3. Semi-discrete problem
The numerical analysis here follows a framework set out in [19], which was used to analyze the approximation of solutions of the heat equation on a bounded domain ⊂ Rn for the finite element method. However, the framework of [19] is modified sig- nificantly with the structure of the reproducing kernel Hilbert space N for a collocation method on Sn.
3.1. The homogeneous problem
By the method of separation of variables, see [14, section 5.7], the exact solution for the homogeneous problem: ∂ u(x, t) = u(x, t), ∂t u(x, 0) = f(x), f ∈ L2 Sn , is given as the infinite series ∞ N(n,) −λt ˆ u(x, t) = e fkYk(x). =0 k=1 Q.T. Le Gia / Approximation of parabolic PDEs on spheres 383
Let the approximate solution be of the following form: m uX(x, t) = ci(t)φi(x), i=1 where φi(x) is the SBF as defined in (5). The homogeneous semi-discrete problem is formulated as the following: we require the equation (1) to be exact on the set X, i.e., ∂ u (x ,t)= u (x ,t), ∀x ∈ X, ∂t X j X j j (9) uX(x, 0) = IXf(x), where IXf is the interpolant of f in VX. Equation (9) can be rewritten as the following: d m m c (t)φ (x ) = c (t)φ (x ), ∀x ∈ X, (10) dt i i j i i j j i=1 i=1 subject to the following initial condition: m ci(0)φi(xj ) = f(xj ), ∀xj ∈ X. i=1
If we set A := [φi(xj )]i,j=1,...,m and B := [φi(xj )]i,j=1,...,m then equation (10) can be written as the following system of ordinary differential equations in time: d c(t) = A−1Bc(t), (11) dt T where c(t) =[c1(t), . . . , cm(t)] . It is known that (see, for example, [11]), in order to solve the system (11), we have to compute the distinct eigenvalues r1,...,rk of the −1 matrix A B with multiplicities n1,...,nk. For each eigenvalue ri,wefindni lin- early independent generalized eigenvectors. Each independent solution of (11) is of the form t2 exp A−1Bt v = ert v + t A−1B − rI v + A−1B − rI 2v +··· , 2 where r is an eigenvalue and v is a corresponding generalized eigenvector. If r has multiplicity ni, then the above series reduces to the first ni terms. The linearly indepen- dent solutions form column vectors of a matrix E(t), and then the fundamental matrix exp(A−1Bt) is given as exp A−1Bt = E(t)E−1(0). The solution of the homogeneous semi-discrete problem is −1 −1 uX(x, t) = φ1(x)...φm(x) exp A Bt c(0), where c(0) = A f |X. (12)
We shall express the solution uX(x, t) in terms of some evolution operator. Let us con- sider the following operator: 384 Q.T. Le Gia / Approximation of parabolic PDEs on spheres n IX : C S → span φi(x): i = 1,...,m ,
f → (IXf).
Lemma 3.1. At the set of points X,wehave −1 n −1 m n m = | = A A B A f(xj ) j=1 (IX) f x xj j=1. [ ]m [ ]T Rm Here and thereafter, the notation aj j=1 stands for a1,...,am which is a vector in .
Proof. For n = 1, we have −1 −1 m −1 m m = =[ | = ] AA BA f(xj ) j=1 BA f(xj ) j=1 IXf x xj j=1. Now assume that for k>1 −1 k −1 m = k m A A B A f(xj ) j=1 (IX) f(xj ) j=1. Then −1 k+1 −1 m = −1 −1 k −1 m A A B A f(xj ) = A A B A B A f(xj ) = j 1 j 1 −1 −1 k −1 m = BA A A B A f(xj ) j=1 = −1 k m = k m BA (IX) f = = (IX)(IX) f = = x xj j 1 x xj j 1 k+1 m = (IX) f . x=xj j=1
Lemma 3.2. For small t>0wehave t2 tn u (x, t) = I f + tI I f + I (I )2f +···+ I (I )nf +···. X X X X 2 X X n! X X
Proof. Using equation (12) and lemma 3.1, we have m m u (x ,t) = A exp A−1Bt A−1 f(x ) X j j=1 j j=1 tn = A I + tA−1B +··· A−1B n +··· A−1 f(x ) m n! j j=1 tn = m + [ | ]m +···+ n m +··· f(xj ) = t IXf x=xj = (IX) f = = . j 1 j 1 n! x xj j 1
Since uX ∈ VX, this implies tn u (x, t) = I f + tI I f +···+ I (I )nf +···. X X X X n! X X Let us define the following evolution operator tn E (t) := I + tI +···+ (I )n +···, X X n! X Q.T. Le Gia / Approximation of parabolic PDEs on spheres 385 then uX(x, t) = EX(t)IXf(x). We can show that EX(t) is a stable operator in VX in · norm by the following lemma.
Lemma 3.3. For every ψ ∈ VX, EX(t)ψ(x) ψ(x) .
Proof. Let θ(x,t) be defined as m θ(x,t) = ci(t)φi(x). i=1 We wish to solve the following PDE by a collocation method ∂ θ(x,t) = θ(x, t), ∂t subject to the initial condition θ(x,0) = ψ(x). In our collocation method, it is required that the PDE is exact on the set of given points X, i.e. ∂ θ(x ,t)= θ(x ,t), ∀x ∈ X, ∂t j j j subject to the initial condition
θ(xj , 0) = ψ(xj ), ∀xj ∈ X.
Since is the reproducing kernel in the Hilbert space N, ∂ · · = · · ∀ ∈ θ( ,t),(,xj ) θ( ,t),(,xj ) , xj X. (13) ∂t
Since VX is spanned by (x, xj ),forj = 1,...,m, equation (13) implies that for every function v ∈ V , X ∂θ ,v =θ, v . ∂t
Since θ ∈ VX, we can take v = θ to obtain 1 ∂ ∂θ 2 = = θ ,θ θ, θ . 2 ∂t ∂t
From the definition of ·, · ,
∞ N(n,) − | ˆ |2 = λ θk ∀ ∈ θ, θ ˆ 0, θ VX. (14) =0 k=1 φ() 386 Q.T. Le Gia / Approximation of parabolic PDEs on spheres
Thus, we obtain the result θ(x,t) θ(x,0) or in other words EX(t)ψ(x) ψ(x) . 3.2. The nonhomogeneous problem
The approximation of the nonhomogeneous equation will be tackled via an elliptic projection from the space of the exact solution u to the finite-dimensional space VX, which is somehow similar to the Ritz projection in the finite element method. To begin, let us define the following operator: 2σ +2 n P : H S → VX
u → uP , where uP (xj ) = u(x j ), ∀xj ∈ X, (15) uP dS = u dS. Sn Sn
It is noted that has zero as an eigenvalue, thus the matrix B =[(xi,xj )]i,j=1,...,m is not invertible. The null space of B has dimension 1. We fix the null space problem by = m := T finding uP j=1 αj φj (x),whereα (α1,...,αm) solves the following system of linear equations = m Bα u(xj ) j=1, m αi φi dS = u dS. n n i=1 S S
We notice that uP is well-defined since the solution α is unique. It is also from the definition that
IXP = IX. (16)
2σ +2 n Lemma 3.4. Let u ∈ H (S ),anduP ∈ VX be constructed from a linear combina- ˆ −σ tionofshiftsofSBF with φ() ∼ (1 + λ) . Then there is a constant C independent of hX so that − σ uP u ChX u 2σ +2.
Proof. Since uP is the interpolation of u, by theorem 2.1, we have − σ σ uP u ChX u 2σ ChX u 2σ +2. (17)
Let ψ = uP − u, then from definition (15) ˆ ψ0 = (uP − u) dS = 0. Sn Q.T. Le Gia / Approximation of parabolic PDEs on spheres 387
Hence, ∞ ∞ N(n,) |ψˆ |2 N(n,) λ2|ψˆ |2 2 = k ˆ 2 + k 2 ψ ˆ ψ0 ˆ ψ . =0 k=1 φ() =1 k=1 φ() Combining with (17), we have − − σ uP u uP u ChX u 2σ +2.
The collocation semi-discrete equation (9) now takes the following form ∂ u (x ,t)− u (x ,t)= F(x ,t), ∀x ∈ X, (18) ∂t X j X j j j subject to the initial condition
uX(x, 0) = IXf(x).
2σ +2 n Theorem 3.1. Let f, ut ∈ H (S ) and u, uX be the solution for (1) and (18), respec- tively. The approximate solution uX is constructed as a linear combination of shifts of a ˆ −σ spherical basis function (x,y) = φ(x · y) which satisfies φ() ∼ (1 + λ) .Then there is a positive constant C, independent of hX, so that the following error estimate holds: T σ − + + + + u(T ) uX(T ) ChX f 2σ f 2σ 2 ut 2σ 2 ds . 0
Proof. Let θ := uX −uP ,andletγ := uP −u. Note that θ ∈ VX. When being restricted on the set X, using the relation uP |X = u|X we have the following equations: ∂ ∂ ∂ θ − θ = uX − uX − uP − uP ∂t ∂t ∂t X X X ∂uP = F |X − − u ∂t X ∂u ∂u ∂uP = F |X − − u + − ∂t X ∂t ∂t X ∂ = (u − uP ) , ∂t X or in terms of a PDE in the finite-dimensional space VX, ∂θ ∂γ − θ =−I . (19) ∂t X ∂t By Duhamel’s principle, see [14, section 3.11], we have T ∂γ θ(T) = EX(T )θ(0) − EX(T − s)IX ds. 0 ∂t 388 Q.T. Le Gia / Approximation of parabolic PDEs on spheres
Since EX(T )v v for all v ∈ VX (by lemma 3.3), we have T + ∂γ θ(T) θ(0) IX (s) ds. 0 ∂t Here, θ(0) =I f − Pf I f − f +Pf − f X X σ Ch f 2σ +f 2σ +2 .
We can use lemma 3.4 to obtain ∂ = − σ γt (u uP ) ChX ut 2σ +2. ∂t Using lemma 2.1, we obtain σ IXγt γt ChX ut 2σ +2. We know from lemma 3.4 that σ γ(T) = u(T ) − uP (T ) Ch u(T ) X 2σ +2 T σ + ChX f ut (s) ds 0 2σ +2 T σ + ChX f 2σ +2 ut 2σ +2 ds . 0 Therefore, after adjusting the constant C, we obtain u − uX θ(T) + γ(T) T σ + + ChX f 2σ f 2σ +2 ut 2σ +2 ds . 0
4. Time discretization
4.1. Backward Euler method
Let us discretize the time derivative using backward Euler method as u(x, t) − u(x, t − τ) + o(1) − u(x, t) = F(x,t). τ
The collocation equation for uX is
uX(xj ,t)− uX(xj ,t − τ)− τuX(xj ,t)= τF(xj ,t), ∀xj ∈ X. (20)
Let us define tN := Nτ, UN (x) := uX(x, tN ) and introduce the notation U − U − ∂ U := N N 1 . t N τ Q.T. Le Gia / Approximation of parabolic PDEs on spheres 389
The collocation equation (20) can be rewritten as
∂t UN (xj ) − UN (xj ) = F(xj ,tN ), ∀xj ∈ X, (21) subject to the initial condition
U0 = IXf. = m If we write UN i=1 cN,iφi(x) then in terms of matrices A and B,definedinsection3, we have m − = − + (A τB)cN AcN 1 τ F(xj ,Nτ) j=1, (22) with the initial condition = m Ac0 f(xj ) j=1.
We now estimate the difference between UN and the exact solution u at the time tN .
σ n 2σ +2 n Theorem 4.1. Let us assume that utt ∈ H (S ) and ut ,f ∈ H (S ) and let UN be the solution of (21). The approximate solution UN is constructed from shifts of a ˆ −σ spherical basis function with φ() ∼ (1 + λ) . Then there are positive constants C1 and C2 so that we have the following error estimate: tN − σ + UN u(tN ) C1hX(f,ut ) C2τ utt σ ds, 0 where tN N := + + + + (f,ut ) f 2σ f 2σ 2 ut (s) 2σ +2 ds τ ut (tj ) 2σ . 0 j=1
Proof.
UN − u(tN ) = UN − Pu(tN ) + Pu(tN ) − u(tN ) =: θN + γN . We already know Pu(t ) − u(t ) =γ Chσ u(t ) N N N X N 2σ +2 tN σ + + ChX f 2σ 2 ut (s) 2σ +2 ds . 0 Similar to (19), we have
∂t θN (xj ) − θN (xj ) =−ωN (xj ), ∀xj ∈ X, (23) where
ωN = ∂t Pu(tN ) − IXut (tN ). 390 Q.T. Le Gia / Approximation of parabolic PDEs on spheres
We can rewrite equation (23) as
(1 − τ)θN (xj ) = θN−1(xj ) − τωN (xj ), ∀xj ∈ X. (24)
In terms of the inner product ·, · in the reproducing kernel Hilbert space N, − · = − − · ∀ ∈ θN τθN ,(xj , ) θN 1 τωN ,(xj , ) , xj X. (25)
Since VX is spanned by (xj , ·)’s, j = 1,...,m, this means for every v ∈ VX,
θN − τθN ,v =θN−1 − τωN ,v . (26)
By taking v = θN ,wehave
θN − τθN ,θN =θN−1 − τωN ,θN , 2 − = − θN τ θN ,θN θN−1,θN τ ωN ,θN .
Since θN ,θN 0 (cf. inequality (14)), we can conclude 2 + θN θN−1,θN τ ωN ,θN θN−1θN + τωN θN .
Simplifying θN on both sides, we obtain
θN θN−1 + τωN . By repeated application, N θN θ0 + τ ωj . j=1 Here, as before, = − σ + θ0 IXf Pf ChX f 2σ f 2σ +2 . Now for every 1 j N, ωj = ∂t Pu(tj ) − ∂t u(tj ) + ∂t u(tj ) − IXut (tj )
=: ωj,1 + ωj,2. We note that tj tj −1 −1 ωj,1 = (P − I)τ ut ds = τ (P − I)ut ds, tj−1 tj−1 whence N N tj tN σ = σ τ ωj,1 ChX ut (s) 2σ +2 ds ChX ut (s) 2σ +2 ds. j=1 j=1 tj−1 0 Q.T. Le Gia / Approximation of parabolic PDEs on spheres 391
Further,
u(tj ) − u(tj−1) ωj,2 = − ut (tj ) + ut (tj ) − IXut (tj ) τ 1 tj =− (s − tj−1)utt(s) ds + ut (tj ) − IXut (tj ), τ tj−1 so that N N tj N − − + − τ ωj,2 (s tj 1)utt(s) ds τ ut (tj ) IXut (tj ) j=1 j=1 tj−1 j=1 tN N + σ τ utt ds CτhX ut (tj ) 2σ . 0 j=1
Therefore, by setting C1 := C and noting · ∼·σ , we obtain a constant C2 so that N N N τ ωj τ ωj,1 + τ ωj,2 j=1 j=1 j=1 tN N σ + C1hX ut (s) 2σ +2 ds τ ut (tj ) 2σ 0 = j 1 tN + C2τ utt(s) σ ds. 0 Thus N − + + + u(T ) UN (T ) γN θN γN θ0 τ ωj = j 1 tN σ + C1hX(f,ut ) C2τ utt(s) σ ds, 0 where tN N := + + + + (f,ut ) f 2σ f 2σ 2 ut (s) 2σ +2 ds τ ut (tj ) 2σ . 0 j=1 4.2. CrankÐNicolson method We now turn to the Crank–Nicolson method in which the semi-discrete equation is discretized in a symmetric fashion around the point tN−1/2 := (N − 1/2)τ,which will produce a second order in time accurate method. More precisely, UN in VX can be defined recursively by
(UN (xj ) + UN−1(xj )) ∂ U (x ) − = F(x ,t − ), ∀x ∈ X, (27) t N j 2 j N 1/2 j 392 Q.T. Le Gia / Approximation of parabolic PDEs on spheres given that
U0 = IXf. In matrix form 1 1 m A − τB c = A + τB c − + τ F(x ,t − ) , 2 N 2 N 1 j N 1/2 j=1 given that = m Ac0 f(xj ) j=1.
Theorem 4.2. Let UN and u be the solutions of (27) and (1), respectively. We assume 2σ +2 n σ n that f, ut ∈ H (S ) and uttt,utt ∈ H (S ). The approximate solution UN is ˆ −σ constructed from shifts of a spherical basis function with φ() ∼ (1 + λ) .Then there are positive constants C1 and C2, independent of hX,sothat tN − σ + 2 + UN u(tN ) C1hX(f,ut ) C2τ uttt σ utt σ ds , 0 where tN N := + + + + − (f,ut ) f 2σ f 2σ 2 ut (s) 2σ +2 ds τ ut (tj 1/2) 2σ . 0 j=1
Let
UN − u(tN ) = UN − Pu(tN ) + Pu(tN ) − u(tN ) =: θN + γN . With the above notation we have
(θ (x ) + θ − (x )) ∂ θ (x ) − N j N 1 j =−η (x ), ∀x ∈ X, t N j 2 N j j where now u(tN ) + u(tN−1) ηN = ∂t Pu(tN ) − ∂t IXu(tN−1/2) + IX u(tN−1/2) − 2 = (P − I)∂ u(t ) + ∂ u(t ) − I u (t − ) t N t N X t N 1/2 u(tN ) + u(tN−1) + I u(t − ) − X N 1/2 2 =: ηN,1 + ηN,2 + ηN,3. Applying arguments similar to (25) and (26) we arriving at τ θN − θN−1 − (θN + θN−1), χ =−τηN ,χ , ∀χ ∈ VX. 2 Q.T. Le Gia / Approximation of parabolic PDEs on spheres 393
By taking χ = θN + θN−1 and note that (θN + θN−1), θN + θN−1 0 (cf. inequal- ity (17)), we have 2 2 − − − + − + − θN θN 1 τ ηN ,(θN θN 1) τ ηN θN θN 1 .
Simplifying the common factor (θN +θN−1) on both sides of the inequality, we obtain
θN θN−1 + τηN . After repeated application this yields N θN θ0 + τ ηj,1 +ηj,2 +ηj,3 . j=1
The term θ0 can be estimated as before. For the latter sum, we have tj −1 σ = − + ηj,1 (P I)∂t u(tj ) Cτ hX ut 2σ 2 ds. tj−1 Further, η = ∂ u(t ) − I u (t − ) j,2 t j X t j 1/ 2 ∂ u(t ) − u (t − ) + u (t − ) − I u (t − ) t j t j 1/2 t j 1 /2 X t j 1/2 tj−1/2 tj 1 2 2 = (s − tj−1) uttt(s) ds + (s − tj ) uttt(s) ds 2τ tj−1 tj−1/2 + u (t − ) − I u (t − ) t j 1/2 X t j 1/2 tj σ + − τ uttt ds ChX ut (tj 1/2) 2σ . tj−1 Let
u(tj ) + u(tj−1) ψ := u(tj−1/2) − 2 1 tj−1/2 1 tj = (tj−1 − s)utt(s) ds + (s − tj )utt(s) ds. 2 tj−1 2 tj−1/2 Therefore, we have 1 ηj,3 = IX u(tj−1/2) − u(tj ) + u(tj−1) 2 = IXψ ψ (see lemma 2.1) tj C2τ uttσ ds, since · ∼·σ . tj−1 394 Q.T. Le Gia / Approximation of parabolic PDEs on spheres
Altogether, with C1 := C,wehave N τ ηj,1 +ηj,2 +ηj,3 j=1 tN N σ + + − C1hX ut 2σ 2 ds τ ut (tj 1/2) 2σ 0 = j 1 tN + 2 + C2τ uttt(s) σ utt(s) σ ds. 0
Thus tN + σ + 2 + θN γN C1hX(f,ut ) C2τ uttt σ utt σ ds , 0 where tN N := + + + + − (f,ut ) f 2σ f 2σ 2 ut (s) 2σ +2 ds τ ut (tj 1/2) 2σ . 0 j=1
5. Numerical experiments on S2
Let us consider the function 1 − z G(z) = 1 − 2ln 1 + . 2 We can expand G(z) as a series of Legendre polynomials (cf. [8]): ∞ 1 G(z) = P (z). ( + 1) =1 The following PDE describes the heat diffusion process from the north pole onto the surface of the unit sphere: ∂ u(x, t) = u(x, t), x ∈ S2, ∂t (28) u(x, 0) = G(x · p), where p = (0, 0, 1)T. Since the initial condition u(x, 0) is a zonal function which depends only on the geodesic distance from any given point on the sphere to the north pole, the solution u(x, t) also depends only on the geodesic distance to the north pole. The problem (28) is reduced to ∂u ∂ ∂u = 1 − z2 , ∂t ∂z ∂z Q.T. Le Gia / Approximation of parabolic PDEs on spheres 395 subject to the following initial condition: u(z, 0) = G(z), z ∈[−1, 1]. We know that the Legendre polynomials are eigenfunctions of the operator ∂ ∂ 1 − z2 . ∂z ∂z Thus, by the method of separation of variables, the exact solution of (28) is given as ∞ e−(+1)t u(z, t) = P (z). ( + 1) =1 We can approximate u(z, t) by the truncated series:
L e−(+1)t u (z, t) = P (z). L ( + 1) =1
The error is estimated by using the fact that P (z)∞ = 1 (see [10]) ∞ e−(+1)t u − u ∞ = P (z) L ( + 1) =L+1 ∞ ∞ dx 1 e−L(L+1)t e−L(L+1)τ ln 1 + . L x(x + 1) L For time-step τ = 0.00125, in order to obtain the accuracy of order 10−16 it is required that L 160. The spherical basis functions used to construct the approximate solution are de- rived from a class of locally supported radial basis function proposed by Wendland [21]. These functions ψ(x) are rotation invariant and are thus function of |x| only. So the n corresponding√ convolution kernel ψ(x − y),x,y ∈ S , is a function of |x − y|= 2 − 2x · y. We may therefore define a function (x,y) = φ(x · y) := ψ(x − y), x,y ∈ Sn. Note that (x,y) inherits the property of positive definiteness from ψ,andφ()ˆ ∼ −σ (1 + λ) for some σ>0 (see [13, section 4]). For our numerical study, we use the 4 function ψ(r) = (1 − r)+(4r + 1). The set of points which are used in constructing the SBFs is generated according to an algorithm in [15]. These points are generated uniformly, in the sense that each point is a center of a cell on the unit sphere of area 4π/m. The iterative equation (22) becomes −1 I − τA B cN = cN−1, with the initial equation −1 c0 = A f |X. 396 Q.T. Le Gia / Approximation of parabolic PDEs on spheres
Table 1 Backward Euler method with different sets of points and time-steps.
mhX qX E∞(τ = 0.01)E∞(τ = 0.005)E∞(τ = 0.0025) 200 0.1942 0.1130 0.0224 0.0225 0.0225 400 0.1288 0.0731 0.0137 0.0138 0.0139 600 0.1122 0.0675 0.0088 0.0089 0.0090 800 0.0950 0.0577 0.0060 0.0061 0.0062 1000 0.0849 0.0516 0.0044 0.0045 0.0046 1200 0.0789 0.0476 0.0034 0.0036 0.0036
Since A is positive definite and B has nonpositive eigenvalues, it can be shown that all the eigenvalues of the matrix (I − τA−1B) are in the interval (0, 1] (see the appendix). Hence the numerical algorithm is stable. The following table show the numerical errors between the iterated solution UN obtained by backward Euler method and u160. Here, N = 1.5/τ and
E∞(τ) := max |UN − uL|. x∈S2
Appendix
Lemma A.1 (cf. [22, chapter 1, section 31]). Let A be a symmetric positive definite matrix and B be a symmetric positive semi-definite (negative semi-definite). Then all of the eigenvalues of AB are non-negative (nonpositive).
Proof. Since A is symmetric positive definite, there is an invertible matrix P such that A = P TP .LetC = PBPT,thenC and AB have the same set of eigenvalues since (P T)−1ABP T = PBPT = C. The matrix C is symmetric since CT = PBTP T = PBPT = C since B is symmetric. Now since B is positive semi-definite, P Tx TB P Tx 0forallx ∈ Rm.
Hence C is symmetric semi-positive definite. Hence all of the eigenvalues of C are non-negative, so are the eigenvalues of the matrix AB.
Lemma A.2. Let A be a symmetric positive definite matrix and B be a symmetric neg- ative semi-definite. Then for any ε>0, all of the eigenvalues of (I − εA−1B)−1 are in the interval (0, 1].
Proof. Let µ be an eigenvalue of I − εA−1B,then1/µ is an eigenvalue of (I − εA−1B)−1. It is observed that µ = 1 − εδ where δ is an eigenvalue of A−1B.By lemma A.1, δ 0, and therefore, µ ∈[1, ∞). Thus, 1/µ ∈ (0, 1]. Q.T. Le Gia / Approximation of parabolic PDEs on spheres 397
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