### A NEW BOUNDARY CONDITION FOR RATE-TYPE NON-NEWTONIAN DIFFUSIVE MODELS AND THE STABLE MAC SCHEME

Kun Li1, Youngju Lee2, Christina Starkey2

1. 1. BGP R & D Center, CNPC, Beijing 100871, China;
2. Department of Mathematics, Texas State University, San Marcos, TX 78666 USA
• Received:2015-08-17 Revised:2016-02-18 Online:2018-07-15 Published:2018-07-15
• Supported by:

The research of the second author was supported in part by NSF-DMS 1358953 and American Chemical Society PRF# 57552-ND9.
The authors thank Professors Michael Graham, Chun Liu, and Yong-Lak Joo for helpful discussions and encouragement. Lee has been supported in part by NSF-DMS 1358953 and American Chemical Society PRF# 57552-ND9.

Kun Li, Youngju Lee, Christina Starkey. A NEW BOUNDARY CONDITION FOR RATE-TYPE NON-NEWTONIAN DIFFUSIVE MODELS AND THE STABLE MAC SCHEME[J]. Journal of Computational Mathematics, 2018, 36(4): 605-626.

We present a new Dirichlet boundary condition for the rate-type non-Newtonian diffusive constitutive models. The newly proposed boundary condition is compared with two such well-known and popularly used boundary conditions as the pure Neumann condition[1] and the Dirichlet condition by Sureshkumar and Beris[2]. Our condition is demonstrated to be more stable and robust in a number of numerical test cases. A new Dirichlet boundary condition is implemented in the framework of the finite difference Marker and Cell (MAC) method. In this paper, we also present an energy-stable finite difference MAC scheme that preserves the positivity for the conformation tensor and show how the addition of the diffusion helps the energy-stability in a finite difference MAC scheme-setting.

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