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



Murli M. Gupta (Department of Mathematics, George Washington University, Washington, District of Columbia, U.S.A.)


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.


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