Journal of Scientific Computing

, Volume 81, Issue 3, pp 1136–1149 | Cite as

An Iteration-Free Approach to Solving the Navier–Stokes Equations by Implicit Finite Difference Schemes in the Vorticity-Stream Function Formulation

  • V. S. Ryaben’kiiEmail author
  • V. A. Torgashov


The paper introduces a new algorithm for solving the finite difference equations at the upper time level of an implicit scheme that approximates the Navier–Stokes system in the vorticity-stream function formulation. The algorithm requires no iterations and computes the corresponding discrete solution exactly. It is based on the method of difference potentials and allows one to efficiently address the well-known difficulties typical for this type of formulations—two boundary conditions for the stream function and no boundary conditions for vorticity.

Abstract for the translation The paper is translated from the Russian by S. Tsynkov ( The original [12] was published more than 20 years ago (the actual Russian citation is [14]). It has not been previously translated into English and went largely unnoticed by the numerical analysis and scientific computing research community. Yet it presents an important contribution to the discipline as it offers a full answer to the question that has long been outstanding. The incompressible Navier–Stokes equations in the vorticity-stream function formulation require two boundary conditions for the stream function and no boundary conditions for vorticity. A standard approach to addressing numerically the apparent overdetermination in one variable and underdetermination in the other was through the use of iterations. Instead, work [12] shows the unambiguous way of discretizing the Navier–Stokes system implicitly and solving the resulting finite difference equations on the upper time level exactly. It is equivalent to deriving the correct non-local boundary condition for vorticity.

Note from the translator My Ph.D. advisor Prof. Ryaben’kii and my postdoc mentor Prof. Abarbanel were friends. They interacted closely and participated in joint projects at ICASE (NASA Langley Research Center) in the late 1990s and early 2000s. Prof. Ryaben’kii visited Tel Aviv several times, and Prof. Abarbanel traveled to Moscow in 2013 to celebrate Prof. Ryaben’kii’s 90th birthday. This paper, which was written by Prof. Ryaben’kii with his Ph.D. student at the time, V. Torgashov, provides a well-deserved tribute to a friend from a friend.


Method of difference potentials Exact solution Boundary conditions 



This work was supported by the Russian Foundation for Fundamental Research, Project 96-01-01577.


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Authors and Affiliations

  1. 1.Keldysh Institute of Applied MathematicsRussian Academy of SciencesMoscowRussia
  2. 2.Moscow Institute for Physics and TechnologyMoscowRussia

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