Communications in Applied Mathematics and Computational Science

A higher-order upwind method for viscoelastic flow

Andrew Nonaka, David Trebotich, Gregory Miller, Daniel Graves, and Phillip Colella

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We present a conservative finite difference method designed to capture elastic wave propagation in viscoelastic fluids in two dimensions. We model the incompressible Navier–Stokes equations with an extra viscoelastic stress described by the Oldroyd-B constitutive equations. The equations are cast into a hybrid conservation form which is amenable to the use of a second-order Godunov method for the hyperbolic part of the equations, including a new exact Riemann solver. A numerical stress splitting technique provides a well-posed discretization for the entire range of Newtonian and elastic fluids. Incompressibility is enforced through a projection method and a partitioning of variables that suppresses compressive waves. Irregular geometry is treated with an embedded boundary/volume-of-fluid approach. The method is stable for time steps governed by the advective Courant–Friedrichs–Lewy (CFL) condition. We present second-order convergence results in L1 for a range of Oldroyd-B fluids.

Article information

Commun. Appl. Math. Comput. Sci., Volume 4, Number 1 (2009), 57-83.

Received: 7 August 2008
Revised: 22 May 2009
Accepted: 25 May 2009
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 65N06: Finite difference methods 76D05: Navier-Stokes equations [See also 35Q30]

viscoelasticity Oldroyd-B fluid Godunov method Riemann solver projection method embedded boundaries


Nonaka, Andrew; Trebotich, David; Miller, Gregory; Graves, Daniel; Colella, Phillip. A higher-order upwind method for viscoelastic flow. Commun. Appl. Math. Comput. Sci. 4 (2009), no. 1, 57--83. doi:10.2140/camcos.2009.4.57.

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