A Meshfree RBF-FD Constant along Normal Method for Solving PDEs on Surfaces Articles uri icon

publication date

  • December 2024

start page

  • A3897

end page

  • A392

issue

  • 6

volume

  • 46

International Standard Serial Number (ISSN)

  • 1064-8275

Electronic International Standard Serial Number (EISSN)

  • 1095-7197

abstract

  • This paper introduces a novel meshfree methodology based on radial basis function-finite difference (RBF-FD) approximations for the numerical solution of partial differential equations (PDEs) on surfaces of codimension 1 embedded in R3. The method is built upon the principles of the closest point method, without the use of a grid or a closest point mapping. We show that the combination of local embedded stencils with these principles can be employed to approximate surface derivatives using polyharmonic spline kernels and polynomials (PHS + Poly) RBF-FD. Specifically, we show that it is enough to consider a constant extension along the normal direction only at a single node to overcome the rank deficiency of the polynomial basis. An extensive parameter analysis is presented to test the dependence of the approach. We demonstrate high-order convergence rates on problems involving surface advection and surface diffusion, and solve Turing pattern formations on surfaces defined either implicitly or by point clouds. Moreover, a simple coupling approach with a particle tracking method demonstrates the potential of the proposed method in solving PDEs on evolving surfaces in the normal direction. Our numerical results confirm the stability, flexibility, and high-order algebraic convergence of the approach.

subjects

  • Mathematics

keywords

  • pdes on surfaces; closest point method; meshfree; rbf-fd; high-order methods