Solving the heat equation on the unit sphere via Laplace transforms and radial basis functions
- PDF / 423,717 Bytes
- 23 Pages / 439.642 x 666.49 pts Page_size
- 68 Downloads / 203 Views
Solving the heat equation on the unit sphere via Laplace transforms and radial basis functions Quoc Thong Le Gia · William McLean
Received: 19 July 2012 / Accepted: 24 June 2013 © Springer Science+Business Media New York 2013
Abstract We propose a method to construct numerical solutions of parabolic equations on the unit sphere. The time discretization uses Laplace transforms and quadrature. The spatial approximation of the solution employs radial basis functions restricted to the sphere. The method allows us to construct high accuracy numerical solutions in parallel. We establish L2 error estimates for smooth and nonsmooth initial data, and describe some numerical experiments. Keywords Parabolic equations · Laplace transforms · Unit sphere · Radial basis functions Mathematics Subject Classifications (2010) 35R01 · 65N30
1 Introduction We consider the initial-value problem ∂t u + Au = f (t),
for t > 0, with u(0) = u0 ,
(1.1)
where ∂t = ∂/∂t and A is a linear, self-adjoint, positive-semidefinite, second-order elliptic partial differential operator on the unit sphere. In our standard example, −A is the Laplace–Beltrami operator. The source term f (t) may depend on the spatial
Communicated by: J. Ward. Q. T. Le Gia () · W. McLean School of Mathematics and Statistics, University of New South Wales, Sydney, Australia e-mail: [email protected] W. McLean e-mail: [email protected]
Q.T. Le Gia, W. McLean
variables but we suppress this dependence in our notation, viewing f (t) as an element of a function space on the sphere. Instead of using time stepping for the numerical solution, as was done previously [5], our approach is to represent the solution of Eq. 1.1 as an inverse Laplace transform, which is then approximated by quadrature. Developed first for parabolic problems by Sheen, Sloan and Thom´ee [13], such an approach is also effective for some evolution equations with memory [6]. These and related papers have discussed thoroughly the time discretization, but for the space discretization have considered only piecewise linear finite elements on a bounded domain in Rn . Here, we propose instead a space discretization using spherical radial basis functions (SRBFs), which are convenient for parabolic problems on Riemannian surfaces such as the unit sphere Sn = { x ∈ Rn+1 : |x| = 1 }. Denoting the Laplace transform of u with respect to t by u(z) ˆ = L{u(t)} :=
∞
e−zt u(t)dt,
(1.2)
0
we find that the solution of Eq. 1.1 formally satisfies (zI + A)u(z) ˆ = g(z) := u0 + fˆ(z),
(1.3)
where I denotes the identity operator. The spectrum of A is a subset of the halfline [0, ∞), so if z ∈ / (−∞, 0] and if the Laplace transform fˆ(z) exists, then u(z) ˆ = (zI + A)−1 g(z).
(1.4)
When fˆ(z) is analytic and bounded for z > 0, the solution u(t) can be recovered via the Laplace inversion formula 1 u(t) = 2πi
ezt u(z)dz, ˆ
for t > 0,
(1.5)
0
where 0 is the contour z = ω, for any ω > 0, with z increasing. Section 2 summarizes some technical results and assumptions needed for our subsequent analysis. In
Data Loading...
