arXiv subjects

H. S. Tang

Publications and source records attributed to H. S. Tang.

2 recordsLinked to original sources

arXiv2021

Convergence Analysis for Computation of Coupled Advection-Diffusion-Reaction Problems

A study is presented on the convergence of the computation of coupled advection-diffusion-reaction equations. In the computation, the equations with different coefficients and even types are assigned in two subdomains, and Schwarz iteration is made between the equations when marching from a time level to the next one. The analysis starts with the linear systems resulting from the full discretization of the equations by explicit schemes. Conditions for convergence are derived, and its speedup and the effects of difference in the equations are discussed. Then, it proceeds to an implicit scheme, and a recursive expression for convergence speed is derived. An optimal interface condition for the Schwarz iteration is obtained, and it leads to "perfect convergence", that is, convergence within two times of iteration. Furthermore, the methods and analyses are extended to the coupling of the viscous Burgers equations. Numerical experiments indicate that the conclusions, such as the "perfect convergence, " drawn in the linear situations may remain in the Burgers equations' computation.

math.NA↗

arXiv2020

An exploratory study on machine learning to couple numerical solutions of partial differential equations

As further progress in the accurate and efficient computation of coupled partial differential equations (PDEs) becomes increasingly difficult, it has become highly desired to develop new methods for such computation. In deviation from conventional approaches, this short communication paper explores a computational paradigm that couples numerical solutions of PDEs via machine-learning (ML) based methods, together with a preliminary study on the paradigm. Particularly, it solves PDEs in subdomains as in a conventional approach but develops and trains artificial neural networks (ANN) to couple the PDEs' solutions at their interfaces, leading to solutions to the PDEs in the whole domains. The concepts and algorithms for the ML coupling are discussed using coupled Poisson equations and coupled advection-diffusion equations. Preliminary numerical examples illustrate the feasibility and performance of the ML coupling. Although preliminary, the results of this exploratory study indicate that the ML paradigm is promising and deserves further research.

math.NA↗