Efficient Variable-Coefficient Finite-Volume Stokes Solvers

We investigate several robust preconditioners for solving the saddle-point linear systems that arise from spatial discretization of unsteady and steady variablecoefficient Stokes equations on a uniform staggered grid.Building on the success of using the classical projection method as a preconditioner for the coupled velocitypressure system[B.E.Griffith,J.Comp.Phys.,228(2009),pp.7565–7595],as well as established techniques for steady and unsteady Stokes flow in the finite-element literature,we construct preconditioners that employ independent generalized Helmholtz and Poisson solvers for the velocity and pressure subproblems.We demonstrate that only a single cycle of a standard geometric multigrid algorithm serves as an effective inexact solver for each of these subproblems.Contrary to traditional wisdom,we find that the Stokes problem can be solved nearly as efficiently as the independent pressure and velocity subproblems,making the overall cost of solving the Stokes system comparable to the cost of classical projection or fractional step methods for incompressible flow,even for steady flow and in the presence of large density and viscosity contrasts.Two of the five preconditioners considered here are found to be robust to GMRES restarts and to increasing problem size,making them suitable for large-scale problems.Our work opens many possibilities for constructing novel unsplit temporal integrators for finite-volume spatial discretizations of the equations of low Mach and incompressible flow dynamics.

On the Volume Conservation of the Immersed Boundary Method

The immersed boundary(IB)method is an approach to problems of fluid-structure interaction in which an elastic structure is immersed in a viscous incompressible fluid.The IB formulation of such problems uses a Lagrangian description of the structure and an Eulerian description of the fluid.It is well known that some versions of the IB method can suffer from poor volume conservation.Methods have been introduced to improve the volume-conservation properties of the IB method,but they either have been fairly specialized,or have used complex,nonstandard Eulerian finite-difference discretizations.In this paper,we use quasi-static and dynamic benchmark problems to investigate the effect of the choice of Eulerian discretization on the volume-conservation properties of a formally second-order accurate IB method.We consider both collocated and staggered-grid discretization methods.For the tests considered herein,the staggered-grid IB scheme generally yields at least a modest improvement in volume conservation when compared to cell-centered methods,and in many cases considered in this work,the spurious volume changes exhibited by the staggered-grid IB method are more than an order of magnitude smaller than those of the collocated schemes.We also compare the performance of cell-centered schemes that use either exact or approximate projection methods.We find that the volumeconservation properties of approximate projection IB methods depend strongly on the formulation of the projection method.When used with the IB method,we find that pressure-free approximate projection methods can yield extremely poor volume conservation,whereas pressure-increment approximate projection methods yield volume conservation that is nearly identical to that of a cell-centered exact projection method.

Simulating an Elastic Ring with Bend and Twist by an Adaptive Generalized Immersed Boundary Method

Many problems involving the interaction of an elastic structure and a viscous fluid can be solved by the immersed boundary(IB)method.In the IB approach to such problems,the elastic forces generated by the immersed structure are applied to the surrounding fluid,and the motion of the immersed structure is determined by the local motion of the fluid.Recently,the IB method has been extended to treatmore general elasticity models that include both positional and rotational degrees of freedom.For such models,force and torque must both be applied to the fluid.The positional degrees of freedomof the immersed structuremove according to the local linear velocity of the fluid,whereas the rotational degrees of freedom move according to the local angular velocity.This paper introduces a spatially adaptive,formally second-order accurate version of this generalized immersed boundary method.We use this adaptive scheme to simulate the dynamics of an elastic ring immersed in fluid.To describe the elasticity of the ring,we use an unconstrained version of Kirchhoff rod theory.We demonstrate empirically that our numerical scheme yields essentially second-order convergence rates when applied to such problems.We also study dynamical instabilities of such fluid-structure systems,and we compare numerical results produced by our method to classical analytic results from elastic rod theory.