The interactions between incompressible fluid flows and immersed structures are nonlinear multi-physics phenomena that have applications to a wide range of scientific and engineering disciplines. In this article, we review representative numerical methods based on conforming and non-conforming meshes that are currently available for computing fluid-structure interaction problems, with an emphasis on some of the recent developments in the field. A goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study in fluid-structure interactions.

In this paper, a novel approach for quantifying the parametric uncertainty associated with a stochastic problem output is presented. As with Monte-Carlo and stochastic collocation methods, only point-wise evaluations of the stochastic output response surface are required allowing the use of legacy deterministic codes and precluding the need for any dedicated stochastic code to solve the uncertain problem of interest. The new approach differs from these standard methods in that it is based on ideas directly linked to the recently developed compressed sensing theory. The technique allows the retrieval of the modes that contribute most significantly to the approximation of the solution using a minimal amount of information. The generation of this information, via many solver calls, is almost always the bottle-neck of an uncertainty quantification procedure. If the stochastic model output has a reasonably compressible representation in the retained approximation basis, the proposed method makes the best use of the available information and retrieves the dominant modes. Uncertainty quantification of the solution of both a 2-D and 8-D stochastic Shallow Water problem is used to demonstrate the significant performance improvement of the new method, requiring up to several orders of magnitude fewer solver calls than the usual sparse grid-based Polynomial Chaos (Smolyak scheme) to achieve comparable approximation accuracy.

In this article, we detail the methodology developed to construct arbitrarily high order schemes — linear and WENO — on 3D mixed-element unstructured meshes made up of general convex polyhedral elements. The approach is tailored specifically for the solution of scalar level set equations for application to incompressible two-phase flow problems. The construction of WENO schemes on 3D unstructured meshes is notoriously difficult, as it involves a much higher level of complexity than 2D approaches. This due to the multiplicity of geometrical considerations introduced by the extra dimension, especially on mixed-element meshes. Therefore, we have specifically developed a number of algorithms to handle mixed-element meshes composed of convex polyhedra with convex polygonal faces. The contribution of this work concerns several areas of interest: the formulation of an improved methodology in 3D, the minimisation of computational runtime in the implementation through the maximum use of pre-processing operations, the generation of novel methods to handle complex 3D mixed-element meshes and finally the application of the method to the transport of a scalar level set.

Parallel tempering simulation is widely used in enhanced sampling of systems with complex energy surfaces. We hereby introduce generalized canonical ensemble (GCE) instead of the usual canonical ensemble into the parallel tempering to further improve abilities of the simulation technique. GCE utilizes an adapted weight function to obtain a unimodal energy distribution even in phase-coexisting region and then the parallel tempering on GCE yields the steady swap acceptance rates (SARs) instead of the fluctuated SARs in that on canonical ensemble. With the steady SARs, we can facilitate assign the parameters of the parallel tempering simulation to more efficiently reach equilibrium among different phases. We illustrate the parallel tempering simulation on GCE in the phase-coexisting region of 2-dimensional Potts model, a benchmark system for new simulation method developing. The result indicates that the new parallel tempering method is more efficient to estimate statistical quantities (i.e., to sample the conformational space) than the normal parallel tempering, specially in phase-coexisting regions of larger systems.

Explicit time stepping schemes for the immersed boundary method require very small time steps in order to maintain stability. Solving the equations that arise from an implicit discretization is difficult. Recently, several different approaches have been proposed, but a complete understanding of this problem is still emerging. A multigrid method is developed and explored for solving the equations in an implicit-time discretization of a model of the immersed boundary equations. The model problem consists of a scalar Poisson equation with conformation-dependent singular forces on an immersed boundary. This model does not include the inertial terms or the incompressibility constraint. The method is more efficient than an explicit method, but the efficiency gain is limited. The multigrid method alone may not be an effective solver, but when used as a preconditioner for Krylov methods, the speed-up over the explicit-time method is substantial. For example, depending on the constitutive law for the boundary force, with a time step 100 times larger than the explicit method, the implicit method is about 15-100 times more efficient than the explicit method. A very attractive feature of this method is that the efficiency of the multigrid preconditioned Krylov solver is shown to be independent of the number of immersed boundary points.

The most severe limitation of the standard Lattice Boltzmann Method is the use of uniform Cartesian grids especially when there is a need for high resolutions near the body or the walls. Among the recent advances in lattice Boltzmann research to handle complex geometries, a particularly remarkable option is represented by changing the solution procedure from the original “stream and collide” to a finite volume technique. However, most of the presented schemes have stability problems. This paper presents a stable and accurate finite-volume lattice Boltzmann formulation based on a cell-centred scheme. To enhance stability, upwind second order pressure biasing factors are used as flux correctors on a D2Q9 lattice. The resulting model has been tested against a uniform flow past a cylinder and typical free shear flow problems at low and moderate Reynolds numbers: boundary layer, mixing layer and plane jet flows. The numerical results show a very good accuracy and agreement with the exact solution of the Navier-Stokes equation and previous numerical results and/or experimental data. Results in self-similar coordinates are also investigated and show that the time-averaged statistics for velocity and vorticity express self-similarity at low Reynolds numbers. Furthermore, the scheme is applied to simulate the flow around circular cylinder and the Reynolds number range is chosen in such a way that the flow is time dependent. The agreement of the numerical results with previous results is satisfactory.

The computation of compressible flows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time and excessively large numerical viscosity, thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation. In this paper, we develop an all-speed asymptotic preserving (AP) numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers. Our idea is to split the system into two parts: one involves a slow, nonlinear and conservative hyperbolic system adequate for the use of modern shock capturing methods and the other a linear hyperbolic system which contains the stiff acoustic dynamics, to be solved implicitly. This implicit part is reformulated into a standard pressure Poisson projection system and thus possesses sufficient structure for efficient fast Fourier transform solution techniques. In the zero Mach number limit, the scheme automatically becomes a projection method-like incompressible solver. We present numerical results in one and two dimensions in both compressible and incompressible regimes.

In this paper, we review the recent development of phase-field models and their numerical methods for multi-component fluid flows with interfacial phenomena. The models consist of a Navier-Stokes system coupled with a multi-component Cahn-Hilliard system through a phase-field dependent surface tension force, variable density and viscosity, and the advection term. The classical infinitely thin boundary of separation between two immiscible fluids is replaced by a transition region of a small but finite width, across which the composition of the mixture changes continuously. A constant level set of the phase-field is used to capture the interface between two immiscible fluids. Phase-field methods are capable of computing topological changes such as splitting and merging, and thus have been applied successfully to multi-component fluid flows involving large interface deformations. Practical applications are provided to illustrate the usefulness of using a phase-field method. Computational results of various experiments show the accuracy and effectiveness of phase-field models.

Lagrangian methods are widely used in many fields for multi-material compressible flow simulations such as in astrophysics and inertial confinement fusion (ICF), due to their distinguished advantage in capturing material interfaces automatically. In some of these applications, multiple internal energy equations such as those for electron, ion and radiation are involved. In the past decades, several staggered-grid based Lagrangian schemes have been developed which are designed to solve the internal energy equation directly. These schemes can be easily extended to solve problems with multiple internal energy equations. However such schemes are typically not conservative for the total energy. Recently, significant progress has been made in developing cell-centered Lagrangian schemes which have several good properties such as conservation for all the conserved variables and easiness for remapping. However, these schemes are commonly designed to solve the Euler equations in the form of the total energy, therefore they cannot be directly applied to the solution of either the single internal energy equation or the multiple internal energy equations without significant modifications. Such modifications, if not designed carefully, may lead to the loss of some of the nice properties of the original schemes such as conservation of the total energy. In this paper, we establish an equivalency relationship between the cell-centered discretizations of the Euler equations in the forms of the total energy and of the internal energy. By a carefully designed modification in the implementation, the cell-centered Lagrangian scheme can be used to solve the compressible fluid flow with one or multiple internal energy equations and meanwhile it does not lose its total energy conservation property. An advantage of this approach is that it can be easily applied to many existing large application codes which are based on the framework of solving multiple internal energy equations. Several two dimensional numerical examples for both Euler equations and three-temperature hydrodynamic equations in cylindrical coordinates are presented to demonstrate the performance of the scheme in terms of symmetry preserving, accuracy and non-oscillatory performance.

We couple different flow models, i.e. a finite element solver for the Navier-Stokes equations and a Lattice Boltzmann automaton, using the framework Peano as a common base. The new coupling strategy between the meso- and macroscopic solver is presented and validated in a 2D channel flow scenario. The results are in good agreement with theory and results obtained in similar works by Latt et al. In addition, the test scenarios show an improved stability of the coupled method compared to pure Lattice Boltzmann simulations.

With discretized particle velocity space, a multi-scale unified gas-kinetic scheme for entire Knudsen number flows has been constructed based on the kinetic model in one-dimensional case [J. Comput. Phys., vol. 229 (2010), pp. 7747-7764]. For the kinetic equation, to extend a one-dimensional scheme to multidimensional flow is not so straightforward. The major factor is that addition of one dimension in physical space causes the distribution function to become two-dimensional, rather than axially symmetric, in velocity space. In this paper, a unified gas-kinetic scheme based on the Shakhov model in two-dimensional space will be presented. Instead of particle-based modeling for the rarefied flow, such as the direct simulation Monte Carlo (DSMC) method, the philosophical principal underlying the current study is a partial-differential-equation (PDE)-based modeling. Since the valid scale of the kinetic equation and the scale of mesh size and time step may be significantly different, the gas evolution in a discretized space is modeled with the help of kinetic equation, instead of directly solving the partial differential equation. Due to the use of both hydrodynamic and kinetic scales flow physics in a gas evolution model at the cell interface, the unified scheme can basically present accurate solution in all flow regimes from the free molecule to the Navier-Stokes solutions. In comparison with the DSMC and Navier-Stokes flow solvers, the current method is much more efficient than DSMC in low speed transition and continuum flow regimes, and it has better capability than NS solver in capturing of non-equilibrium flow physics in the transition and rarefied flow regimes. As a result, the current method can be useful in the flow simulation where both continuum and rarefied flow physics needs to be resolved in a single computation. This paper will extensively evaluate the performance of the unified scheme from free molecule to continuum NS solutions, and from low speed micro-flow to high speed non-equilibrium aerodynamics. The test cases clearly demonstrate that the unified scheme is a reliable method for the rarefied flow computations, and the scheme provides an important tool in the study of non-equilibrium flow.

A new algorithm, called boosted hybrid method, is proposed for the simulation of chemical reaction systems with scale-separation in time and disparity in species population. For such stiff systems, the algorithm can automatically identify scale-separation in time and slow down the fast reactions while maintaining a good approximation to the original effective dynamics. This technique is called boosting. As disparity in species population may still exist in the boosted system, we propose a hybrid strategy based on coarse-graining methods, such as the tau-leaping method, to accelerate the reactions among large population species. The combination of the boosting strategy and the hybrid method allow for an efficient and adaptive simulation of complex chemical reactions. The new method does not need a priori knowledge of the system and can also be used for systems with hierarchical multiple time scales. Numerical experiments illustrate the versatility and efficiency of the method.

In this work the Laguerre basis for the biharmonic equation introduced by Jie Shen is employed in the spectral solution of self-similar problems of the boundary layer theory. An original Petrov-Galerkin formulation of the Falkner-Skan equation is presented which is based on a judiciously chosen special basis function to capture the asymptotic behaviour of the unknown. A spectral method of remarkable simplicity is obtained for computing Falkner-Skan-Cooke boundary layer flows. The accuracy and efficiency of the Laguerre spectral approximation is illustrated by determining the linear stability of nonseparated and separated flows according to the Orr-Sommerfeld equation. The pentadiagonal matrices representing the derivative operators are explicitly provided in an Appendix to aid an immediate implementation of the spectral solution algorithms.

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 volume-conservation 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.

In this paper, we propose a strong stability-preserving predictor-corrector (SSPC) method based on an implicit Runge-Kutta method to solve the acoustic- and elastic-wave equations. We first transform the wave equations into a system of ordinary differential equations (ODEs) and apply the local extrapolation method to discretize the spatial high-order derivatives, resulting in a system of semi-discrete ODEs. Then we use the SSPC method based on an implicit Runge-Kutta method to solve the semi-discrete ODEs and introduce a weighting parameter into the SSPC method. On top of such a structure, we develop a robust numerical algorithm to effectively suppress the numerical dispersion, which is usually caused by the discretization of wave equations when coarse grids are used or geological models have large velocity contrasts between adjacent layers. Meanwhile, we investigate the performance of the SSPC method including numerical errors and convergence rate, numerical dispersion, and stability criteria with different choices of the weighting parameter to solve 1-D and 2-D acoustic- and elastic-wave equations. When the SSPC is applied to seismic simulations, the computational efficiency is also investigated by comparing the SSPC, the fourth-order Lax-Wendroff correction (LWC) method, and the staggered-grid (SG) finite difference method. Comparisons of synthetic waveforms computed by the SSPC and analytic solutions for acoustic and elastic models are given to illustrate the accuracy and the validity of the SSPC method. Furthermore, several numerical experiments are conducted for the geological models including a 2-D homogeneous transversely isotropic (TI) medium, a two-layer elastic model, and the 2-D SEG/EAGE salt model. The results show that the SSPC can be used as a practical tool for large-scale seismic simulation because of its effectiveness in suppressing numerical dispersion even in the situations such as coarse grids, strong interfaces, or high frequencies.

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 treat more 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 freedom of the immersed structure move 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.

The present paper provides a numerical investigation of the decoherence effect induced on a quantum heavy particle by the scattering with a light one. The time dependent two-particle Schrödinger equation is solved by means of a time-splitting method. The damping undergone by the non-diagonal terms of the heavy particle density matrix is estimated numerically as well as the error in the Joos-Zeh approximation formula.

We introduce a lattice-free hard sphere exclusion stochastic process. The resulting stochastic rates are distance based instead of cell based. The corresponding Markov chain build for this many particle system is updated using an adaptation of the kinetic Monte Carlo method. It becomes quickly apparent that due to the lattice-free environment, and because of that alone, the dynamics behave differently than those in the lattice-based environment. This difference becomes increasingly larger with respect to particle densities/temperatures. The well-known packing problem and its solution (Palasti conjecture) seem to validate the resulting lattice-free dynamics.

An algorithm for computing wavefronts, based on the high frequency approximation to the wave equation, is presented. This technique applies the level set method to underwater acoustic wavefront propagation in the time domain. The level set method allows for computation of the acoustic phase function using established numerical techniques to solve a first order transport equation to a desired order of accuracy. Traditional methods for solving the eikonal equation directly on a fixed grid limit one to only the first arrivals, so these approaches are not useful when multi-path propagation is present. Applying the level set model to the problem allows for the time domain computation of the phase function on a fixed grid, without having to restrict to first arrival times. The implementation presented has no restrictions on range dependence or direction of travel, and offers improved efficiency over solving the full wave equation which under the high frequency assumption requires a large number of grid points to resolve the highly oscillatory solutions. Boundary conditions are discussed, and an approach is suggested for producing good results in the presence of boundary reflections. An efficient method to compute the amplitude from the level set method solutions is also presented. Comparisons to analytical solutions are presented where available, and numerical results are validated by comparing results with exact solutions where available, a full wave equation solver, and with wavefronts extracted from ray tracing software.