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Annals of Mathematical Sciences and Applications
Volume 5 (2020)
Number 1
Mathematical sciences related to theoretical physics, engineering, biology and economics
Guest Editor: Tony Wen-Hann Sheu, National Taiwan University
High-accuracy compact difference schemes for differential equations in mathematical sciences
Pages: 101 – 138
DOI: https://dx.doi.org/10.4310/AMSA.2020.v5.n1.a5
Author
Abstract
Our work on high-order compact difference schemes was initiated about 35 years ago when we first presented new 4th- and 6th‑order discretizations for convection-diffusion equations in $2$ dimensions. This work is now routinely applied to complex fluid flow problems, and has also been developed for $3$-dimensional differential equations. In our quest to apply these ideas to the biharmonic equation, we discovered that it is beneficial to carry the unknowns and their derivatives as computational parameters. This allowed us to propose the streamfunction-velocity formulation for the Navier–Stokes equations.
In this paper, I describe the historical developments of the high-order compact difference schemes and their evolution into the powerful computational techniques that are now available to solve fluid flow problems of important physical interest. Some theoretical analysis on stability and convergence of these schemes will also be presented.
This paper is based, in part, on the presentation I gave in March 2019 at the Taiwan-India joint conference “Recent Progress on Flow Simulation and Stability Analysis: 2019 Spring Progress in Mathematical and Computational Studies on Science and Engineering Problems,” at National Taiwan University, Taipei, Taiwan.
Keywords
high-order compact, Navier–Stokes, streamfunction-velocity formulation, high-accuracy, fluid flow
2010 Mathematics Subject Classification
Primary 65M06, 65N06. Secondary 76M20.
Received 27 August 2019
Accepted 28 December 2019
Published 27 February 2020