A high-order meshless Galerkin method for semilinear parabolic equations on spheres

A kernel-based least-squares collocation method for surface diffusion

“…Example 1: Comparing with a meshless Galerkin method. Our first aim is to compare the proposed method with a weak formulation in [3]. We consider an example there: u − 0.1∆ S + 3u = f on the unit sphere with u * (y, t) = exp(x…”

“…In this numerical experiment, we consider the Allen-Cahn equation u = ∆ S u+ 1 ε 2 u(1−u 2 ), which is a reaction diffusion equation that models phase separation of two fluids [25] with ε > 0 being the width of the diffusion interface between two fluids. We set up the test problem on the unit sphere as in [3,26]. For some R 0 > 0, we define initial condition u(0) to be +1 within radius R 0 to the northpole and −1 otherwise.…”

“…Using R 0 = 0.717, ε = 0.05, and |Z| = 3721 as in [3] for another round of comparison with the Galerkin method, we solve the semi-discrete equation We end this example with an Allen-Cahn phase separation simulation on a torus 2 . Using the generator in [27] with h Z1 = 0.1333 and h Z2 = 0.1, and h X = 0.066, we obtain point terface.…”

“…(1.1) were solved by the method of Rothe, in which the PDE is discretized first in time, and then by kernel-based collocation method in space. In this paper, we focus on the method of lines that is theoretically more completed for parabolic PDEs on spheres [3] using Galerkin formulations and also in bulk domains, see [4,5]. An advantage of using collocation method is that we can completely remove surface integrations from our algorithms (but not theories).…”

“…and,(3) all components a ij (y, t) := [A(y, t)] ij , 1 ≤ i, j ≤ d,of A are sufficient smooth with bounded derivatives. i=1 ⊂ R d of the diffusion tensor A(y, t) that spans T y…”