Time-splitting Algorithm Research Articles

Simulation and analysis of elastic waves in partially saturated double-porosity media based on finite difference method

Double-porosity poroelastic model takes into account the effect of mesoscopic flow induced by rock heterogeneity on dispersion and attenuation of elastic waves, and has obtained good application results in the quantitative explanation of seismic data in heterogeneous reservoirs. Wavefield simulation based on double-porosity model not only helps visualize the propagation characteristics of the elastic waves but also lays the foundation for seismic imaging. In this work, we perform wavefield simulation and analysis based on the Santos-Rayleigh model which incorporates mesoscopic and global flow in a partially-saturated double-porosity medium. Specifically, the mesoscopic flow mechanism is represented with a Zener viscoelastic model. The comparison shows that the Zener model can accurately capture the propagation characteristics of fast P-wave, but fails to describe the attenuation characteristics of slow P3 wave in the low-frequency band. It implies that Zener viscoelastic model and slow wave modes follow different mechanisms. Then the staggered grid finite-difference method is used to simulate wave propagation in a double-porosity medium, and the stiff problem is solved with a time-splitting algorithm, which can significantly improve computational efficiency. Based on the above methods, the correctness of our algorithm is verified with derived analytical solution for a P-wave source in a uniform partially saturated poroelastic medium. Analytical and numerical solutions are in good agreement and mean error is 0.33%. We provide some examples of wavefield snapshots and seismograms in homogeneous and layered heterogeneous media at seismic and ultrasonic frequencies. The simulation results demonstrate the strong attenuation of fast P-wave and no change of S-wave in the seismic band due to mesoscopic flow mechanism, which is consistent with the theoretical prediction of double-porosity model. Moreover, the energy of fast P-wave is concentrated in solid phase while slow waves are stronger in fluid phase. This work contributes to the understanding of broadband elastic wave propagation in a heterogeneous partially saturated porous medium and can be applied to the reservoir imaging with broadband geophysical data.

Heat exchange enhancement of ferrofluid flow into rectangular channel in the presence of a magnetic field

Heat exchange and Fe3O4/water nanofluid flow behaviors into horizontal channel subjected to the effect of a magnetic field was studied numerically. The basic equations were solved using the finite volume method with the time-splitting algorithm. The results were represented by temperature field, streamlines, velocity field, averaged and normalized Nusselt numbers, for various nanoparticles volume fractions, Reynolds numbers and magnetic numbers. Results show that the isotherms, the streamlines and the heat exchange rate are strongly changed by applying a magnetic field. A recirculation region is created near the magnetic source where the thermal boundary layer is removed enhancing so the local heat exchange. The overall heat exchange is enhanced by suspending nanoparticles and/or by increasing magnetic field strength. In the absence of a magnetic field, suspending nanoparticles can enhance heat exchange rate up to 20%. Under the only action of magnetic field a maximum 60% heat exchange enhancement is obtained. It can reach up to 86% as the combined effects of both nanoparticles and magnetic field are considered.

Simulation of wave propagation in linear thermoelastic media

We have developed a numerical algorithm for simulation of wave propagation in linear thermoelastic media, based on a generalized Fourier law of heat transport in analogy with a Maxwell model of viscoelasticity. The wavefield is computed by using a grid method based on the Fourier differential operator and two time-integration algorithms to cross-check solutions. Because the presence of a slow quasistatic mode (the thermal mode) makes the differential equations stiff and unstable for explicit time-stepping methods, first, a second-order time-splitting algorithm solves the unstable part analytically and a Runge-Kutta method the regular equations. Alternatively, a first-order explicit Crank-Nicolson algorithm yields more stable solutions for low values of the thermal conductivity. These time-stepping methods are second- and first-order accurate, respectively. The Fourier differential provides spectral accuracy in the calculation of the spatial derivatives. The model predicts three propagation modes, namely, a fast compressional or (elastic) P-wave, a slow thermal P diffusion/wave (the T-wave), having similar characteristics to the fast and slow P-waves of poroelasticity, respectively, and an S-wave. The thermal mode is diffusive for low values of the thermal conductivity and wave-like for high values of this property. Three velocities define the wavefront of the fast P-wave, i.e., the isothermal velocity in the uncoupled case, the adiabatic velocity at low frequencies, and a higher velocity at high frequencies.

A non-hydrostatic non-Boussinesq algorithm for free-surface ocean modelling

A three-mode time-split algorithm is proposed to simulate non-hydrostatic ocean processes in a natural way: fast pressure adjustments via acoustic waves are explicitly represented. The full set of compressible Navier–Stokes equations is integrated for a free-surface ocean with three time-steps respectively associated to the internal, external and compressible (non Boussinesq) modes. The resulting three-mode algorithm can be implemented either in a fully compressible configuration or in a “pseudo-compressible” configuration by artificially lowering acoustic-wave velocities to reduce computational costs. The present description of the three-mode algorithm focuses in the first place on the simulation of non-hydrostatic processes in free-surface ocean models.Several test simulations related to linear and non-linear acoustic, surface and internal gravity waves are studied in details with the proposed three-mode algorithm.

Effects of Lewis numbers and kinetics on spontaneous ignition of hydrogen jets

The transient process following a hydrogen leak into the atmosphere initiates as a contact surface appears separating air heated by the leading shock from hydrogen cooled by expansion. Diffusion of heat and species produces reactive mixture, potentially leading to ignition. Reactions being very temperature-sensitive, their rate peaks close to the hot air-rich side of the interface, where the small fuel concentration depends upon the fuel Lewis number. If the Lewis number is less than unity, diffusion brings in more fuel than temperature-controlled chemistry consumes. If greater, diffusion does not bring in as much fuel as chemistry would burn. Results from the current analysis for multistep kinetics also show that the role of the H Lewis number is crucial. In the short time limit, the evolution of the diffusion layer appears as a perturbation superimposed to the self-similar non-reactive diffusion solution. Initiation is very slow compared with steps consuming one reactant and an intermediate species, while some steps associated with termination are extremely fast. The approximation being made, assuming initiation much smaller than most rates, and termination much faster, is very accurate since the corresponding ratios are of the order of 105. The resulting problem still requires a numerical solution. A time-splitting algorithm combines exact solutions to three subproblems, avoiding issues such as stiffness of the kinetics. The problem is formulated not in physical space but in the similarity variable of the diffusion problem, hence avoiding difficulties associated with the initial singularity when the layer has zero thickness, and the resulting uncertainties. These are distinct advantages over numerical simulations, which however will retain a reasonable accuracy at later times. Results confirm the role of the Lewis number. They also show two distinct regimes: an early one mainly controlled by initiation, and a later one controlled by chain-branching, with a sharp transition.

A time-implicit numerical method and benchmarks for the relativistic Vlasov–Ampere equations

We present a time-implicit numerical method to solve the relativistic Vlasov–Ampere system of equations on a two dimensional phase space grid. The time-splitting algorithm we use allows the generalization of the work presented here to higher dimensions keeping the linear aspect of the resulting discrete set of equations. The implicit method is benchmarked against linear theory results for the relativistic Landau damping for which analytical expressions using the Maxwell-Jüttner distribution function are derived. We note that, independently from the shape of the distribution function, the relativistic treatment features collective behaviours that do not exist in the nonrelativistic case. The numerical study of the relativistic two-stream instability completes the set of benchmarking tests.

Eulerian simulations of collisional effects on electrostatic plasma waves

The problem of collisions in a plasma is a wide subject with a huge historical literature. In fact, the description of realistic plasmas is a tough problem to attack, both from the theoretical and the numerical point of view. In this paper, a Eulerian time-splitting algorithm for the study of the propagation of electrostatic waves in collisional plasmas is presented. Collisions are modeled through one-dimensional operators of the Fokker-Planck type, both in linear and nonlinear forms. The accuracy of the numerical code is discussed by comparing the numerical results to the analytical predictions obtained in some limit cases when trying to evaluate the effects of collisions in the phenomenon of wave plasma echo and collisional dissipation of Bernstein-Greene-Kruskal waves. Particular attention is devoted to the study of the nonlinear Dougherty collisional operator, recently used to describe the collisional dissipation of electron plasma waves in a pure electron plasma column [M. W. Anderson and T. M. O'Neil, Phys. Plasmas 14, 112110 (2007)]. Finally, for the study of collisional plasmas, a recipe to set the simulation parameters in order to prevent the filamentation problem can be provided, by exploiting the property of velocity diffusion operators to smooth out small velocity scales.

Practical and Efficient Split Decomposition via Graph-Labelled Trees

Split decomposition of graphs was introduced by Cunningham (under the name join decomposition) as a generalization of the modular decomposition. This paper undertakes an investigation into the algorithmic properties of split decomposition. We do so in the context of graph-labelled trees (GLTs), a new combinatorial object designed to simplify its consideration. GLTs are used to derive an incremental characterization of split decomposition, with a simple combinatorial description, and to explore its properties with respect to Lexicographic Breadth-First Search (LBFS). Applying the incremental characterization to an LBFS ordering results in a split decomposition algorithm that runs in time O(n+m)?(n+m), where ? is the inverse Ackermann function, whose value is smaller than 4 for any practical graph. Compared to Dahlhaus' linear time split decomposition algorithm (Dahlhaus in J. Algorithms 36(2):205---240, 2000), which does not rely on an incremental construction, our algorithm is just as fast in all but the asymptotic sense and full implementation details are given in this paper. Also, our algorithm extends to circle graph recognition, whereas no such extension is known for Dahlhaus' algorithm. The companion paper (Gioan et al. in arXiv:1104.3284 , 2011) uses our algorithm to derive the first sub-quadratic circle graph recognition algorithm.

Solving the Vlasov equation for one-dimensional models with long range interactions on a GPU

We present a GPU parallel implementation of the numeric integration of the Vlasov equation in one spatial dimension based on a second order time-split algorithm with a local modified cubic-spline interpolation. We apply our approach to three different systems with long-range interactions: the Hamiltonian mean field, ring and self-gravitating sheet models. Speedups and accuracy for each model and different grid resolutions are presented.

ASAMgpu V1.0 – a moist fully compressible atmospheric model using graphics processing units (GPUs)

Abstract. In this work the three dimensional compressible moist atmospheric model ASAMgpu is presented. The calculations are done using graphics processing units (GPUs). To ensure platform independence OpenGL and GLSL are used, with that the model runs on any hardware supporting fragment shaders. The MPICH2 library enables interprocess communication allowing the usage of more than one GPU through domain decomposition. Time integration is done with an explicit three step Runge-Kutta scheme with a time-splitting algorithm for the acoustic waves. The results for four test cases are shown in this paper. A rising dry heat bubble, a cold bubble induced density flow, a rising moist heat bubble in a saturated environment, and a DYCOMS-II case.

Numerical simulation of trapped dipolar quantum gases: Collapse studies and vortex dynamics

We numerically study the three dimensional Gross-Pitaevskii equationfor dipolar quantum gases using a time-splitting algorithm. We aremainly concerned with numerical investigations of the possibleblow-up of solutions, i.e. collapse of the condensate, and thedynamics of vortices.

Numerical resolution of the Vlasov equation for the Hamiltonian Mean-Field model

We present in this paper detailed numerical Vlasov simulations of the Hamiltonian Mean-Field model. This model is used as a representative of the class of systems under long-range interactions. We check existing results on the stability of the homogeneous situation and analyze numerical properties of the semi-Lagrangian time-split algorithm for solving the Vlasov equation. We also detail limitations due to finite resolution of the method.

Nonlinear dynamics of viscoelastic Taylor–Couette flow: effect of elasticity on pattern selection, molecular conformation and drag

Three-dimensional and time-dependent simulations of viscoelastic Taylor–Couette flow of dilute polymer solutions are performed using a fully implicit parallel spectral time-splitting algorithm to discover flow patterns with various spatio-temporal symmetries, namely rotating standing waves (RSWs), disordered oscillations (DOs) and solitary vortex structures referred to as oscillatory strips (OSs) and diwhirls (DWs). A detailed account of the impact of flow transitions on molecular conformation and viscoelastic stress, velocity profiles, hydrodynamic drag force and energy spectra of time-dependent flow states is presented. Overall, predicted pattern selection and flow features compare very favourably with experimental observations. For elasticity number E, that signifies the ratio of elastic to viscous forces, >0.1, and when the shear rate (cylinder rotation speed) is increased above the linear stability threshold, the circular Couette flow (CCF) becomes unstable to RSWs which are characterized by a checkerboard-like pattern in the space–time plot of radial velocity, implying symmetry between inflow/outflow (I/O) regions. As the shear rate is further increased, perturbations that break the I/O symmetry are amplified leading to DOs and/or flame-like patterns with spectral mechanical energy transfer reminiscent of elastically induced low-Reynolds-number turbulence. However, when the shear rate is decreased from those at which such chaotic states are observed, the radially inward acting polymer body force created by flow-induced molecular stretching causes the development of narrow inflow regions surrounded by much broader weak outflow domains. This promotes the formation of solitary vortex structures, which can be stationary and axisymmetric (DWs) or time-dependent (OSs). The dynamics of the formation of these structures by merging and coalescence of vortex pairs and the implication of such events on instantaneous hydrodynamic force are studied. For O(1) values of E, OSs and DWs appear approximately at constant values of the We, defined as the ratio of polymer relaxation time to the inverse shear rate in the gap. As shear rate is decreased further, DWs decay to CCF although at We values less than the linear stability threshold. The flow transitions are hysteretic with respect to We, as evidenced by a plot of drag force versus We.

Modeling Viscoelastic Behavior of Arterial Walls and Their Interaction with Pulsatile Blood Flow

Fluid-structure interaction describing wave propagation in arteries driven by the pulsatile blood flow is a complex problem. Whenever possible, simplified models are called for. One-dimensional models are typically used in arterial sections that can be approximated by the cylindrical geometry allowing axially symmetric flows. Although a good first approximation to the underlying problem, the one-dimensional model suffers from several drawbacks: the model is not closed (an ad hoc velocity profile needs to be prescribed to obtain a closed system) and the model equations are quasi-linear hyperbolic (oversimplifying the viscous fluid dissipation), typically producing shock wave solutions not observed in healthy humans. In this manuscript we derived a simple, closed reduced model that accounts for the viscous fluid dissipation to the leading order. The resulting fluid-structure interaction system is of hyperbolic-parabolic type. Arterial walls were modeled by a novel, linearly viscoelastic cylindrical Koiter shell model and the flow of blood by the incompressible, viscous Navier-Stokes equations. Kelvin-Voigt-type viscoelasticity was used to capture the hysteresis behavior observed in the measurements of the arterial stress-strain response. Using the a priori estimates obtained from an energy inequality, together with the asymptotic analysis and ideas from homogenization theory for porous media flows, we derived an effective model which is an � 2-approximation to the three-dimensional axially symmetric problem, whereis the aspect ratio of the cylindrical arterial section. Our model shows two interesting features of the underlying problem: bending rigidity, often times neglected in the arterial wall models, plays a nonnegligible role in the � 2 -approximation of the original problem, and the viscous fluid dissipation imparts long-term viscoelastic memory effects on the motion of the arterial walls. This does not, to the leading order, influence the hysteresis behavior of arterial walls. The resulting model, although two-dimensional, is in the form that allows the use of one-dimensional finite element method techniques producing fast numerical solutions. We devised a version of the Douglas-Rachford time-splitting algorithm to solve the underlying hyperbolic-parabolic problem. The results of the numerical simulations were compared with the experimental flow measurements performed at the Texas Heart Institute, and with the data corresponding to the hysteresis of the human femoral artery and the canine abdominal aorta. Excellent agreement was observed.

High-order finite volume methods for viscoelastic flow problems

In this paper accurate and stable finite volume schemes for solving viscoelastic flow problems are presented. Two contrasting finite volume schemes are described: a hybrid cell-vertex scheme and a pure cell-centred counterpart. Both schemes employ a time-splitting algorithm to evolve the solution through time towards steady state. In the case of the hybrid scheme, a semi-implicit formulation is employed in the momentum equation, based on the Taylor–Galerkin approach with a pressure-correction step to enforce incompressibility. The basis of the pure finite volume approach is a backward Euler scheme with a semi-Lagrangian step to treat the convection terms in the momentum and constitutive equations. Two distinct finite volume schemes are presented for solving the systems of partial differential equations describing the flow of viscoelastic fluids. The schemes are constructed to be second-order accurate in space. The issue of stability is also addressed with respect to the treatment of convection. Numerical examples are presented illustrating the performance of these schemes on some steady and transient problems that possess analytical solutions.

A FINITE-VOLUME-BASED TIME-SPLITTING SCHEME FOR COMPUTATION OF ELECTRODEPOSITION

The LIGA process is emerging as a viable contender in the microfabrication of metallic high-aspect-ratio microstructures. The final step in the LIGA process is the electroplating of the metal part. In this step, metallic ions are transported to the cathode by the fluid and are deposited on it through chemical reactions. There are two kinds of chemical reactions: homogeneous dissociation reactions in the electrolyte bath, and finite-rate surface reactions on the electrode. In this article we develop a finite-volume method for computing electrodeposition in the LIGA process. We employ a time-splitting algorithm for coupling the bulk and surface reactions. First there is an instantaneous equilibration step in which the species existing in a cell are equilibrated to yield an intermediate concentration distribution. This serves as the initial condition to the transport step in which slow reactions and convective and diffusive transport are computed. As an alternative, the species concentration field is solved over the whole domain using a direct method. The two methods are compared to establish the validity and efficiency of the time-splitting algorithm.

A Numerical Method for the Accurate Solution of the Fokker–Planck–Landau Equation in the Nonhomogeneous Case

A new approach for the accurate numerical solution of the Fokker–Planck–Landau (FPL) equation in the nonhomogeneous case is presented. The method couples, through a time-splitting algorithm, a finite-volume scheme for the transport with a fast spectral solver for the efficient solution of the collision operator recently introduced. The scheme allows the use of different grids in the velocity space for the transport and the collision phases. The use of a suitable explicit Runge–Kutta solver for the time integration of the collision phase permits avoid once of excessive small time steps induced by the stiffness of the diffusive collision operator. Numerical results for both space homogeneous and space nonhomogeneous situations show the efficiency and the accuracy of the present method.

An improved projection scheme applied to pseudospectral methods for the incompressible Navier-Stokes equations

SUMMARY An improved projection scheme is proposed and applied to pseudospectral collocation-Chebyshev approximation for the incompressible Navier‐Stokes equations. It consists of introducing a correct predictor for the pressure, one which is consistent with a divergence-free velocity field at each time step. The main objective is to allow a time variation of the pressure gradient at boundaries. From different test problems, it is shown that this method, associated with a multistep second-order time scheme, provides a time accuracy of the same order as the temporal scheme used for the pressure, and also improves the prediction of the velocity slip. Moreover, it does not exhibit any numerical boundary layer mentioned as a drawback of fractional steps algorithm, and does not require the use of staggered grids for the velocity and the pressure. Its effectiveness is validated by comparison with a previous time-splitting algorithm proposed by Goda (K. Goda, J. Comput. Phys., 30, 76‐95 (1979)) and implemented by Gresho (P. Gresho, Int. j. numer. methods fluids, 11, 587‐620 (1990)) to finite element approximations. Steady and unsteady solutions for the regularized driven cavity and the rotating cavity submitted to throughflow are also used to assess the efficiency of this algorithm. © 1998 John Wiley & Sons, Ltd.

A Versatile Lattice Simulator for Fluid−Solid Noncatalytic Reactions in Complex Media

A versatile lattice simulator for fluid−solid noncatalytic reactions is developed in detail in order to study linear, nonlinear, and nonisothermal kinetics in complex media, in the presence of multicomponent diffusion and n-solid reactant species. The simulator is based on a time-splitting algorithm for the diffusion and for the reaction steps. The simulator is quantitatively checked in many different cases involving initially nonporous particles constituted by a solid reactant dispersed in an inert matrix. The influence of spatial correlation properties in the case of uniform and nonuniform solid reactant distribution is analyzed in detail.

STEADY VISCOELASTIC FLOW PAST A SPHERE USING SPECTRAL ELEMENTS

The steady flow of a viscoelastic fluid past a sphere in a cylindrical tube is considered. A spectral element method is used to solve the system of coupled non-linear partial differential equations governing the flow. The spectral element method combines the flexibility of the traditional finite element method with the accuracy of spectral methods. A time-splitting algorithm is used to determine the solution to the steady problem. Results are presented for the Oldroyd B model. These show excellent agreement with the literature. The results converge with mesh refinement. A limiting Deborah number of approximately 0⋅6 is found, irrespective of the spatial resolution.