Abstract
We introduce a stable method for solving the incompressible Navier–Stokes equations with variable density and viscosity. Our method is stable in the sense that it does not increase the total energy of dynamics that is the sum of kinetic energy and potential energy. Instead of velocity, a new state variable is taken so that the kinetic energy is formulated by the L2 norm of the new variable. Navier–Stokes equations are rephrased with respect to the new variable, and a stable time discretization for the rephrased equations is presented. Taking into consideration the incompressibility in the Marker-And-Cell (MAC) grid, we present a modified Lax–Friedrich method that is L2 stable. Utilizing the discrete integration-by-parts in MAC grid and the modified Lax–Friedrich method, the time discretization is fully discretized. An explicit CFL condition for the stability of the full discretization is given and mathematically proved.
Original language | English |
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Pages (from-to) | 104-119 |
Number of pages | 16 |
Journal | Journal of Computational Physics |
Volume | 360 |
DOIs | |
State | Published - 1 May 2018 |
Bibliographical note
Funding Information:The research of C. Min was supported by NRF grant 2017R1A2B1006688 and by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827). The research of Byungjoon Lee was supported by NRF grant 2017R1C1B1008626 and POSCO Science Fellowship of POSCO TJ Park Foundation.
Funding Information:
The research of C. Min was supported by NRF grant 2017R1A2B1006688 and by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( 2009-0093827 ). The research of Byungjoon Lee was supported by NRF grant 2017R1C1B1008626 and POSCO Science Fellowship of POSCO TJ Park Foundation .
Publisher Copyright:
© 2018 Elsevier Inc.
Keywords
- Computational fluid dynamics
- Energy stability
- Finite difference method
- Navier–Stokes' equations
- Stability analysis