Kernel-based collocation methods for heat transport on evolving surfaces

Abstract

We propose algorithms for solving convective-diffusion partial differential equations (PDEs), which model surfactant concentration and heat transport on evolving surfaces, based on intrinsic kernel-based meshless collocation methods. The algorithms can be classified into two categories: one collocates PDEs directly and analytically, and the other approximates surface differential operators by meshless pseudospectral approaches. The former is specifically designed to handle PDEs on evolving surfaces defined by parametric equations, and the latter works on surface evolutions based on point clouds. After some convergence studies and comparisons, we demonstrate that the proposed method can solve challenging PDEs posed on surfaces with high curvatures with discontinuous initial conditions with correct physics.

A Time-Continuous Embedding Method for Scalar Hyperbolic Conservation Laws on Manifolds

“…The solution of the embedded PDEs (7), when restricted to the surface M, coincides with the solution of the SPDEs (1) with…”

“…Numerically, the spatial operator (in this case, the gradient of the flux) in the embedded PDEs (7) can be discretized by any finite volume scheme. The resulting semi-discrete temporal ODEs can be integrated to advance the solution of the SPDEs (1) near the surface of the manifold for one time step (or one stage of a multi-stage Runge-Kutta scheme).…”

“…Note that there is no temporal treatment in this formulation. More precisely, once (7) is advanced in time, the numerical solution Q cp ( • , t 0 + t) will violate the CAN-property for any t > 0. An interpolation process of all grid points is needed to reconstruct the CAN-property of the numerical solution, which is mathematically equivalent to constructing a new CAN-extension based on Q M ( • , t 0 + t).…”

“…Instead of studying linear and quasi-linear SPDEs with smooth solutions ( [12,14] for level set approach, [31] for a semi-Lagrangian method, [37] for meshfree finite difference method, [3,7,10,22,[28][29][30] for closest point (cp-)embedding methods) on Cartesian grids and orthogonal gradient grids, as a proof of concept, this study explores the application of the embedding methods and develops a numerical scheme, based on our extensive knowledge of essentially non-oscillatory shock-capturing schemes, to solve the (nonlinear scalar) hyperbolic conservation laws (Burgers' equation) with a discontinuous solution on one-dimensional manifolds. For the rest of the paper, the manifold is implicitly referred to as the one-dimensional, connected, smooth, closed, and complex (or simple) manifold unless specified otherwise.…”

Physics-informed Neural Networks for Elliptic Partial Differential Equations on 3D Manifolds

“…On the other side, the embedding techniques extend the PDE on manifolds into one additional embedding space and then solve it. In recent years, various methods including either mesh-based methods like the Finite Element Method (FEM) [8,9], Finite Volume Method (FVM) [10] or mesh-free methods like RBF-type methods [11][12][13], meshless finite difference methods [14] have been proposed to solve PDEs on manifolds. Different from these typical numerical methods, machine learning methods [15] provide another option for solving PDEs on manifolds.…”