Numerical analysis of a fractional step theta-method for fluid flow problems

The accurate numerical approximation of viscoelastic fluid flow poses two difficulties: the large number of unknowns in the approximating algebraic system (corresponding to velocity, pressure, and stress), and the different mathematical types of the modeling equations. Specifically, the viscoelastic modeling equations have a hyperbolic constitutive equation coupled to a parabolic conservation of momentum equation. An appealing approximation approach is to use a fractional step theta-method. The theta-method is an operator splitting technique that may be used to decouple mathematical equations of different types as well as separate the updates of distinct modeling equation variables when modeling mixed systems of partial differential equations.;In this work a fractional step theta-method is described and analyzed for the numerical computation of both the time dependent convection-diffusion equation and the time dependent equations of viscoelastic fluid flow using the Johnson-Segalman constitutive model. For convection-diffusion the theta-method presented allows for a decoupling within time steps of the parabolic diffusion operator from the hyperbolic convection operator. The hyperbolic convection update is stabilized using a Streamline Upwinded Petrov-Galerkin (SUPG)-method. The analysis given for the convection-diffusion equation serves as a template for the analysis of the more complicated viscoelastic fluid flow modeling equations.;The theta-method implementation analyzed for the viscoelastic modeling equations allows the velocity and pressure approximations within time steps to be decoupled from the stress, reducing the number of unknowns resolved at each step of the method. Additionally the theta-method decoupling results in the approximation of the nonlinear viscoelastic modeling system using only the solution of linear systems of equations. Similar to the scheme implemented for convection-diffusion, the hyperbolic constitutive equation is stabilized using a SUPG-method. For both the convection-diffusion and the viscoelastic modeling equations a priori error estimates are established for their theta-method approximations. Numerical computations supporting the theoretical results and demonstrating the theta-methods are also included.