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AN ENERGY-STABLE AND SECOND-ORDER ACCURATE METHOD FOR SOLVING THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

  • KIM, JEONGHO (DEPARTMENT OF MATHEMATICAL SCIENCES, SEOUL NATIONAL UNIVERSITY) ;
  • JUNG, JINWOOK (DEPARTMENT OF MATHEMATICAL SCIENCES, SEOUL NATIONAL UNIVERSITY) ;
  • PARK, YESOM (DEPARTMENT OF MATHEMATICS, EWHA WOMANS UNIVERSITY) ;
  • MIN, CHOHONG (DEPARTMENT OF MATHEMATICS, EWHA WOMANS UNIVERSITY) ;
  • LEE, BYUNGJOON (DEPARTMENT OF MATHEMATICS, THE CATHOLIC UNIVERSITY OF KOREA)
  • Received : 2019.05.29
  • Accepted : 2019.06.18
  • Published : 2019.06.25

Abstract

In this article, we introduce a finite difference method for solving the Navier-Stokes equations in rectangular domains. The method is proved to be energy stable and shown to be second-order accurate in several benchmark problems. Due to the guaranteed stability and the second order accuracy, the method can be a reliable tool in real-time simulations and physics-based animations with very dynamic fluid motion. We first discuss a simple convection equation, on which many standard explicit methods fail to be energy stable. Our method is an implicit Runge-Kutta method that preserves the energy for inviscid fluid and does not increase the energy for viscous fluid. Integration-by-parts in space is essential to achieve the energy stability, and we could achieve the integration-by-parts in discrete level by using the Marker-And-Cell configuration and central finite differences. The method, which is implicit and second-order accurate, extends our previous method [1] that was explicit and first-order accurate. It satisfies the energy stability and assumes rectangular domains. We acknowledge that the assumption on domains is restrictive, but the method is one of the few methods that are fully stable and second-order accurate.

Acknowledgement

Supported by : National Re-search Foundation of Korea

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