2012/05/31 by Edward J. Fuselier, Fuselier, Edward J., Grady B. Wright +1 · 2 citations
Computer Science · Mathematics · #35B36 #35K57 #41A05 #41A25 #41A30 #41A63 #46E22 #58J45 #65D25 #65M20 #65M70 #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1206.0047
openalex publication_date 2012/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we present a high-order kernel method for numerically solving\ndiffusion and reaction-diffusion partial differential equations (PDEs) on\nsmooth, closed surfaces embedded in \ℝd. For two-dimensional\nsurfaces embedded in \ℝ3, these types of problems have received\ngrowing interest in biology, chemistry, and computer graphics to model such\nthings as diffusion of chemicals on biological cells or membranes, pattern\nformations in biology, nonlinear chemical oscillators in excitable media, and\ntexture mappings. Our kernel method is based on radial basis functions (RBFs)\nand uses a semi-discrete approach (or the method-of-lines) in which the surface\nderivative operators that appear in the PDEs are approximated using\ncollocation. The method only requires nodes at "scattered" locations on the\nsurface and the corresponding normal vectors to the surface. Additionally, it\ndoes not rely on any surface-based metrics and avoids any intrinsic coordinate\nsystems, and thus does not suffer from any coordinate distortions or\nsingularities. We provide error estimates for the kernel-based approximate\nsurface derivative operators and numerically study the accuracy and stability\nof the method. Applications to different non-linear systems of PDEs that arise\nin biology and chemistry are also presented.\n