We propose an efficient algorithm for the immersed boundary method on distributedmemory architectures that has the computational complexity of a completely explicit method and also has excellent parallel scaling. The algorithm utilizes the pseudocompressibility method recently proposed by Guermond and Minev that uses a directional splitting strategy to discretize the incompressible Navier-Stokes equations, thereby reducing the linear systems to a series of one-dimensional tridiagonal systems. We perform numerical simulations of several fluid-structure interaction problems in two and three dimensions and study the accuracy and convergence rates of the proposed algorithm. We also compare the proposed algorithm with other second-order projection-based fluid solvers. Lastly, the execution time and scaling properties of the proposed algorithm are investigated and compared to alternate approaches.

We investigate a model for traffic flow based on the Lighthill-Whitham-Richards model that consists of a hyperbolic conservation law with a discontinuous, piecewise-linear flux. A mollifier is used to smooth out the discontinuity in the flux function over a small distance 1 and then the analytical solution to the corresponding Riemann problem is derived in the limit as → 0. For certain initial data, the Riemann problem can give rise to zero waves that propagate with infinite speed but have zero strength. We propose a Godunov-type numerical scheme that avoids the otherwise severely restrictive CFL constraint that would arise from waves with infinite speed by exchanging information between local Riemann problems and thereby incorporating the effects of zero waves directly into the Riemann solver. Numerical simulations are provided to illustrate the behaviour of zero waves and their impact on the solution. The effectiveness of our approach is demonstrated through a careful convergence study and comparisons to computations using a third-order WENO scheme.In the 1950's, Lighthill and Whitham [30] and Richards [36] independently proposed 2 the first macroscopic traffic flow model, now commonly known as the LWR model. Al-3 though this model has proven successful in capturing some aspects of traffic behaviour, 4 its limitations are well-documented and many more sophisticated models have been pro-5 posed to capture the complex dynamics and patterns observed in actual vehicular traffic 6 [22]. Despite this progress, the LWR model remains an important and widely-used model 7 because of its combination of simplicity and explanatory power.The LWR model consists of a single scalar nonlinear conservation law in one dimensionwhere ρ(x, t) is the traffic density (cars/m),is the traffic flow rate or flux (cars/sec), and v(ρ) is the local velocity (m/sec). The most 10 commonly used flux function iswhich was obtained by Greenshields [20] in the 1930's by fitting experimental measure-12 ments of vehicle velocity and traffic density. Here, u max is the maximum free-flow speed, 13 while ρ max is the maximum density corresponding to bumper-to-bumper traffic where 14 speed drops to zero. The LWR model belongs to a more general class of kinematic wave 15 traffic models that couple the conservation law Eq. (1) with a variety of different flux 16 functions. 17 Extensive studies of the empirical correlation between flow rate and density have 18 been performed in the traffic flow literature. This correlation is commonly referred to 19 as the fundamental diagram and is represented graphically by a plot of flux f versus 20density ρ such as that shown in Fig. 1. A striking feature of many experimental results 21is the presence of an apparent discontinuity that separates the free flow (low density) 22 and congested (high density) states, something that has been discussed by many authors, 23 including [7, 14, 15, 23]. In particular, Koshi et al. [24] characterize flux data such as 24 that shown in Fig. 1 as having a reverse lambda shape in w...

Highlights• We propose a new algorithm for simulating suspensions of flexible fibers on distributed-memory clusters. • Our immersed boundary framework captures the full two-way interaction between fluid and flexible fibers.• The algorithm employs a new pseudo-compressible fluid solver recently proposed by Guermond and Minev.• Numerical results are validated against the experimental results of S.G. Mason and co-workers. AbstractWe present an approach for numerically simulating the dynamics of flexible fibers in a three-dimensional shear flow using a scalable immersed boundary (IB) algorithm based on Guermond and Minev's pseudo-compressible fluid solver. The fibers are treated as one-dimensional neutrally-buoyant Kirchhoff rods that resist stretching, bending, and twisting, within the generalized IB framework. We perform a careful numerical comparison against experiments on single fibers performed by S.G. Mason and co-workers, who categorized the fiber dynamics into several distinct orbit classes. We show that the orbit class may be determined using a single dimensionless parameter for low Reynolds flows. Lastly, we simulate dilute suspensions containing up to hundreds of fibers using a distributed-memory computer cluster. These simulations serve as a stepping stone for studying more complex suspension dynamics involving aggregation of fibers (or flocculation) and particle sedimentation due to added mass.