Flux-corrected Transport Research Articles

Flux-corrected transport for full-waveform inversion

Flux-corrected transport for full-waveform inversion

Numerical solution of Euler equations employing enthalpy-based equation of state for simulating shock wave propagation in porous materials

Shock wave propagation through porous materials is investigated using numerical simulations of one dimensional Euler equations of inviscid fluid flow, with the basic aim to demonstrate the application of the enthalpy-based equation of state. A one dimensional hydrodynamics solution method employing the flux-corrected transport algorithm, which does not make use of artificial viscosity, is used for all simulations. Benchmark results on the Sedov-von Neumann-Taylor blast wave problem are given thereby showing the capabilities of the algorithm for capturing strong shock profiles. Next, numerical results for plate impact problems are obtained for porous Cu, Al, Fe and W, and compared with experimental data available in the literature. It is shown that shock waves attenuate very fast even in materials of small size and low porosity. Excellent agreement is also found with experimental data on pressure versus particle velocity in the four target materials for different porosities. Thus the enthalpy-based approach, using a few experimental data of solids at normal conditions, is shown to be capable of quantitative prediction of shock wave parameters of porous materials. An accurate but simple approach for in-line calculation of the enthalpy parameters is discussed in the appendix together with numerical results.

The impact of the order of numerical schemes on slug flows modeling

This paper aims to explore the impact of the order of numerical schemes on the simulation of two-phase slug flow with a two-fluid model initiation. The governing equations of the two-fluid model have been solved by a class of Riemann solver. The numerical schemes applied in this paper involve first-order (Lax-Friedrichs and Rusanov), second-order (Ritchmyer), and high-order (flux-corrected transport or FCT and total variance diminishing or TVD). The results suggest that the TVD and FCT are able to predict the slug initiation with high accuracy compared with experimental results. Lax-Friedrichs and Rusanov are both first-order schemes and have second-order truncation error. This second-order truncation error caused numerical diffusion in the solution field and could not predict the slug initiation with high accuracy in contrast to TVD and FCT schemes. Ritchmyer is a second-order scheme and has third-order truncation error. This third-order truncation error caused dispersive results in the solution field and was not a proper scheme.

Enhancement of valley polarization in graphene with an irradiating charged particle

We use a two-dimensional two-component nonlinear hydrodynamic method to study valley-dependent plasmons in bounded strained graphene in the presence of an irradiating proton, realizing electrostatic control of the valley-dependent plasmons in actual space. We use flux-corrected transport to solve the nonlinear hydrodynamic equations numerically and self-consistently. Our results answer the important question of whether full valley polarization can be obtained in a specific valley at the boundary or inside the graphene sheet in the presence of the injected proton. The electrons experience collective excitations due to the proton interaction and the strain-induced pseudomagnetic field. The electron density fluctuation can be much larger than the equilibrium electron density, leading to a strong nonlinear effect and thus full valley polarization. This demonstrates the nature of the nonlinear response of electrons in graphene to strong interactions, a response that originates from the strong nonperturbative interaction between the irradiating proton and the electrons. Thus, our method opens up the possibility of investigating the nonlinear behavior of valley-dependent plasmons in strong modulations. The effects of the proton on the valley polarization are examined. There is K-polarization inside the surface behind the proton, whereas there is K′-polarization at the edge which decays away from the edge, thereby switching the valley polarization. This work establishes a link between actual-space valley-dependent plasmons in graphene and the irradiating proton and provides an alternative way to realize full valley polarization with tunable polarity. Compared to the case with no proton, the valley polarization is enhanced considerably in the presence of a proton.

Multiple harmonics control of edge pseudomagnetoplasmons in strained graphene

Valley-resolved edge plasmons are relevant to nano-optics at subwavelength scales. However, less attention has been paid to their tunable properties in time domain. In this work we investigate edge pseudomagnetoplasmons in a strained graphene modulated by multiple harmonics with frequency in the THz regime. The edge plasmon is described by a set of nonlinear hydrodynamic equations, which are self-consistently solved by the flux-corrected transport method. Without the applied voltage, there exist two unidirectional-propagating edge-plasmon modes with weak valley polarization P. It is demonstrated that by varying the amplitude of multiple harmonics one can alter both the amplitude and the polarity of the valley polarization in the edge plasmon. One can achieve a full valley polarization P=1 at the instant of half cycle of the multiple harmonics and P=-1 at the instant of one cycle. The edge-plasmon density and the transverse velocity vanish for the frozen valley.

A time‐space FCT‐FE formulation for 1D time dependent advection‐diffusion equation

AbstractWe present a time‐space flux‐corrected transport (FCT) finite element formulation for solving the linear time‐dependent advection dominated advection‐diffusion equation. Solving advection dominated transport equations with conventional finite element (FE) methods suffers from drawbacks of excessive numerical dispersion which results in non‐physical, non‐monotonic solutions. The FCT algorithm is an effective method which suppresses the non‐monotonic behavior of the solution by applying a limited anti‐diffusion operator to a first order scheme. Applying the FCT algorithm to time‐space FE formulation, such as the time‐discontinuous Galerkin (TDG) method, benefits from the advantages of both the TDG scheme and the FCT algorithm. In another word, the time‐space FCT‐FE formulation achieves arbitrary odd order accuracy in time at the discontinuous time nodes. Large time steps can be applied and the scheme ensures monotonic solution when linear interpolation is used for spatial discretization.

Numerical simulation of non-isothermal two-phase flow in pipelines using a two-fluid model

The analysis of two-phase flows through numerical simulations is a very useful tool to design and operate long pipelines. It is also important for monitoring the flow during adverse situations, such as the formation of hydrates and wax deposition that may occur due to temperature variations in the fluids along the pipe. The present work aims to validate a model along with a numerical technique for analyzing non-isothermal stratified-pattern two-phase flows that occur in typical gas-gathering offshore pipelines. The mathematical model is one-dimensional and based on a two-fluid model that comprises two mass and two momentum conservation equations, one for each phase, along with one energy conservation equation for the mixture. The system of equations is discretized in a conservative form within a finite-difference framework and is solved by the flux-corrected transport (FCT) numerical method. Because an ill-posed mathematical problem is obtained if the system of equations is not hyperbolic, a hyperbolicity analysis is performed for all cases simulated in this work, which also indicates how to impose the boundary conditions adequately and avoids the production of non-physical output. The results obtained for the distribution of pressure, fluid velocities, holdup and temperature along the pipeline demonstrated that even relatively small temperature differences between the fluids and the external environment at the pipe inlet cause significant thermal effects on the entire flow. The simulations also predicted results in good agreement with those of a commercial software used for comparison, for steady-state regime, and significantly different results from those produced by an isothermal-flow model, which emphasizes the importance of considering the heat transfer in the problem. In summary, the results showed that the mathematical model and the numerical method form together a reliable tool of second-order accuracy in space that perform accurate numerical simulations of two-phase flows in long pipelines that are subject to non-isothermal conditions.

The barely implicit correction algorithm for low-Mach-Number flows

A new Barely Implicit Correction (BIC) algorithm is presented for the simulation of low-Mach-number flows. This new algorithm is based on the original, introduced by G. Patnaik et al. [G. Patnaik, R. H. Guirguis, J. P. Boris and E. S. Oran, A barely implicit correction for flux-corrected transport. In: Journal of Computational Physics 71.1 (1987), pp. 120], which was a solution procedure including an explicit predictor step to solve the convective portion of the Navier–Stokes equations and an implicit corrector step to remove the acoustic limit on the integration time-step. The explicit predictor uses a high-order monotone algorithm while the implicit corrector solves an elliptic equation for a pressure correction to equilibrate acoustic waves. In this paper, we develop and extend BIC for multidimensional viscous flows. We introduce a new filter to further stabilize the algorithm and clarify the solution procedure for the inclusion of the viscous fluxes. The new algorithm is examined in three test problems with successively increased difficulty. First, a two-dimensional lid-driven cavity flow is simulated to demonstrate the ability of BIC on solving steady-state swirling flows. Using time steps at least 100 times larger than the explicit limit, good agreements are obtained for solutions when compared with an incompressible calculation by a prior work. A two-dimensional (2D) doubly periodic shear layer flow is simulated to examine the algorithm on solving a transient flow with strong vorticity gradients. Finally, vortex breakdown in three-dimensional (3D) swirling flows are used to further test the stability and performance of the new BIC algorithm. Comparisons of explicit and implicit BIC calculations of both the 2D doubly periodic shear layer and 3D vortex breakdown are presented side by side. They demonstrate that the new BIC algorithm is able to predict accurate and robust solutions using time steps varying from near the explicit stability limit to tens and hundreds of times larger. Excellent agreement is also obtained when compared with results from other algorithms. We discuss our observations of these computations and features which were found to be critical for robustly simulating low-speed, highly dynamic flows.

Comparison of Numerical Methods for Two-Fluid Model for Gas–Liquid Transient Flow Regime and Its Application in Slug Modeling Initiation

The formation of slug in the horizontal pipes due to the hydrodynamic instabilities has always been of great interest to many researchers. In this research, the effect of various numerical methods on the simulation of slug flow initiation using pressure-free two-fluid model has been investigated. The two-fluid model has been solved by conservative shock-capturing method. When the slug is formed, a strong discontinuity will be developed in the flow stream. Therefore, a numerical method should have the capability to predict this discontinuity with high accuracy and should not have oscillation near the discontinuity. There are three different models to simulate two-phase flow systems: homogeneous equilibrium model, drift-flux model, and two-fluid model. This research used two-fluid model for predicting the slug flow initiation through a pipe. Four different numerical two-fluid methods, namely Lax-Friedrichs, Rusanov, Richtmyer and flux-corrected transport (FCT) have been used in this research. Results show that FCT is the most accurate method for the prediction of the slug flow initiation among other methods where Rusanov and Lax-Friedrichs numerical methods were in the next steps, respectively. Due to the oscillatory nature near the discontinuity caused by formation of slug regime, the Richtmyer numerical method is not an appropriate method for modeling slug flow regime. Results also show that as the numerical diffusion of these methods reduces in the flow field, the slug flow initiation will be predicted with higher accuracy.

One cause of pulse-like anomalies observed at Guza before the Wenchuan earthquake

Many precursor-like anomaly observations prior to the Wenchuan earthquake were reported and analyzed, especially the abnormal strain pulses observed at the Guza station, but there are few discussions of the causes. Stick–slip motion is the basis for description of a great variety of phenomena characterized by the presence of sliding friction. In this article, perturbed Sine–Gordon (SG) equation is established from Bykov’s unsteady-state slip model. Stable solitary solutions of displacement and strain dimension are obtained and nonlinear pulse propagation is simulated using finite-difference modeling, while numerical stability is obtained by the flux-corrected transport method. Considering the solution of SG equation as initial source, a comparison between the modeling results and actual data at the Guza station gives one possible interpretation for this anomaly. During the seismogenic process of the Wenchuan earthquake, faults may likely occur as stick–slip tectonic movements which might be described by SG equation and would generate solitary wave signal. This kind of pulse experiences a forward tilting distortion due to nonlinear effect of the Earth and is received by the borehole strainmeter. Two kinds of nonlinear effects could lead to these special pulses. One is the nonlinear effects in the wave propagation process, and the other is effects of friction and stress in the process of unsteady-state slip. Because of the convergence effect of pulse, the wave would be collapsed at a certain time due to an excessive increase in the tilt angle. Hence, this kind of pulse cannot propagate for a long distance.

Numerical Simulation of Thermoacoustic Wave Induced by Thermal Effects by Using Discontinuous Galerkin Method

This paper describes a numerical model based on discontinuous Galerkin method for thermoacoustic investigation. Numerical investigation was conducted to study the behaviour of thermoacoustic wave propagations induced by thermal effects in 2-dimensional enclosure. The compressible Navier-Stokes equations are used as the governing equations. The spatial domain was discretized by using unstructured discontinuous Galerkin method and the explicit fourth-order Runge-Kutta was used to integrate the temporal domain. The accuracy of the numerical results was assessed against other numerical methods and linear solutions. The comparisons with linear solutions show excellent agreement and comparisons with flux corrected transport (FCT) method show fair agreement.

Flux-corrected transport techniques applied to the radiation transport equation discretized with continuous finite elements

The Flux-Corrected Transport (FCT) algorithm is applied to the unsteady and steady-state particle transport equation. The proposed FCT method employs the following: (1) a low-order, positivity-preserving scheme, based on the application of M-matrix properties, (2) a high-order scheme based on the entropy viscosity method introduced by Guermond [1], and (3) local, discrete solution bounds derived from the integral transport equation. The resulting scheme is second-order accurate in space, enforces an entropy inequality, mitigates the formation of spurious oscillations, and guarantees the absence of negativities. Space discretization is achieved using continuous finite elements. Time discretizations for unsteady problems include theta schemes such as explicit and implicit Euler, and strong-stability preserving Runge–Kutta (SSPRK) methods. The developed FCT scheme is shown to be robust with explicit time discretizations but may require damping in the nonlinear iterations for steady-state and implicit time discretizations.

Numerical modeling of the waves evolution generated by the depressurization of the vessels containing a supercritical parameters coolant

The development of power plants focuses on increasing the parameters of water coolants up to a supercritical level. Depressurization of the unit circuits with such a coolant leads to emergency situations. Their scenarios can change significantly with the variation of initial pressure and temperature before the start of depressurization. When the pressure drops from the supercritical single-phase region of the initial thermodynamic parameters of the coolant, either the liquid boils up, or the vapor is condensed. Because of the rapid pressure decrease, the phase transition can be non-equilibrium that must be taken into account in the simulation. In the present study, an axisymmetric problem of the outflow of a water coolant from the pipe butt-end is considered. The equations of continuity, momentum and energy for a two-phase homogeneous mixture are solved numerically. The vapor and liquid properties are calculated using the TTSE software package (The Tabular Taylor Series Expansion Method). On the basis of the computer complex LCPFCT (The Flux-Corrected Transport Algorithm) the program code was developed for solving numerous problems on the depressurization of vessels or pipelines, containing superheated water or gas under high pressure. Different variants of outflow in the external model atmosphere and generation of waves are analyzed. The calculated data on the interaction of pressure waves with a barrier are calculated. To describe phase transitions, an asymptotic relaxation model of nonequilibrium evaporation and condensation has been created and tested.

On the effect of different flux limiters on the performance of an engine gas exchange gas-dynamic model

A suitable tool for the design of intake and exhaust systems of internal combustion engines is provided by time-domain non-linear finite volume models. These models, however, are affected by overshoots at discontinuities and numerical dispersion unless some flux limiter is used. In this paper, the effect of the most relevant of such flux limiters on a non-linear staggered-mesh finite-volume model is evaluated. Flux-Corrected-Transport (FCT) and Total Variation Diminishing (TVD) schemes, together with a Momentum Diffusion Term (MDT) are presented for such a model, and the performance of the resulting methods is checked in different problems representative of the influence of engine gas exchange flows on engine performance and intake and exhaust noise. First, two one-dimensional cases are considered: the shock-tube problem, and the propagation of a finite amplitude pressure pulse. Secondly, a simple but representative three-dimensional geometry is studied. From the results obtained, it can be concluded that, even if none of the methods is able to handle properly the three problems considered, the FCT method provides the best overall performance.

Flux-corrected transport algorithms preserving the eigenvalue range of symmetric tensor quantities

This paper presents a new approach to constraining the eigenvalue range of symmetric tensors in numerical advection schemes based on the flux-corrected transport (FCT) algorithm and a continuous finite element discretization. In the context of element-based FEM-FCT schemes for scalar conservation laws, the numerical solution is evolved using local extremum diminishing (LED) antidiffusive corrections of a low order approximation which is assumed to satisfy the relevant inequality constraints. The application of a limiter to antidiffusive element contributions guarantees that the corrected solution remains bounded by the local maxima and minima of the low order predictor.The FCT algorithm to be presented in this paper guarantees the LED property for the maximal and minimal eigenvalues of the transported tensor at the low order evolution step. At the antidiffusive correction step, this property is preserved by limiting the antidiffusive element contributions to all components of the tensor in a synchronized manner. The definition of the element-based correction factors for FCT is based on perturbation bounds for auxiliary tensors which are constrained to be positive semidefinite to enforce the generalized LED condition. The derivation of sharp bounds involves calculating the roots of polynomials of degree up to 3. As inexpensive and numerically stable alternatives, limiting techniques based on appropriate estimates are considered. The ability of the new limiters to enforce local bounds for the eigenvalue range is confirmed by numerical results for 2D advection problems.

A single-stage flux-corrected transport algorithm for high-order finite-volume methods

We present a new limiter method for solving the advection equation using a high-order, finite-volume discretization. The limiter is based on the flux-corrected transport algorithm. We modify the classical algorithm by introducing a new computation for solution bounds at smooth extrema, as well as improving the pre-constraint on the high-order fluxes. We compute the high-order fluxes via a method of lines approach with fourth order Runge-Kutta as the time integrator. For computing low-order fluxes, we select the corner transport upwind method due to its improved stability over donor-cell upwind. Several spatial differencing schemes are investigated for the high-order flux computation, including centered difference and upwind schemes. We show that the upwind schemes perform well on account of the dissipation of high wavenumber components. The new limiter method retains high-order accuracy for smooth solutions and accurately captures fronts in discontinuous solutions. Further, we need only apply the limiter once per complete time step.

Flux-corrected transport algorithms for continuous Galerkin methods based on high order Bernstein finite elements

This work extends the flux-corrected transport (FCT) methodology to arbitrary order continuous finite element discretizations of scalar conservation laws on simplex meshes. Using Bernstein polynomials as local basis functions, we constrain the total variation of the numerical solution by imposing local discrete maximum principles on the Bézier net. The design of accuracy-preserving FCT schemes for high order Bernstein–Bézier finite elements requires the development of new algorithms and/or generalization of limiting techniques tailored for linear and multilinear Lagrange elements. In this paper, we propose (i) a new discrete upwinding strategy leading to local extremum bounded low order approximations with compact stencils, (ii) high order variational stabilization based on the difference between two gradient approximations, and (iii) new localized limiting techniques for antidiffusive element contributions. The optional use of a smoothness indicator, based on a second derivative test, makes it possible to potentially avoid unnecessary limiting at smooth extrema and achieve optimal convergence rates for problems with smooth solutions. The accuracy of the proposed schemes is assessed in numerical studies for the linear transport equation in 1D and 2D.

An FCT finite element scheme for ideal MHD equations in 1D and 2D

This paper presents an implicit finite element (FE) scheme for solving the equations of ideal magnetohydrodynamics in 1D and 2D. The continuous Galerkin approximation is constrained using a flux-corrected transport (FCT) algorithm. The underlying low-order scheme is constructed using a Rusanov-type artificial viscosity operator based on scalar dissipation proportional to the fast wave speed. The accuracy of the low-order solution can be improved using a shock detector which makes it possible to prelimit the added viscosity in a monotonicity-preserving iterative manner. At the FCT correction step, the changes of conserved quantities are limited in a way which guarantees positivity preservation for the density and thermal pressure. Divergence-free magnetic fields are extracted using projections of the FCT predictor into staggered finite element spaces forming exact sequences. In the 2D case, the magnetic field is projected into the space of Raviart–Thomas finite elements. Numerical studies for standard test problems are performed to verify the ability of the proposed algorithms to enforce relevant constraints in applications to ideal MHD flows.

Implementation of a conservative two-step shape-preserving advection scheme on a spherical icosahedral hexagonal geodesic grid

An Eulerian flux-form advection scheme, called the Two-step Shape-Preserving Advection Scheme (TSPAS), was generalized and implemented on a spherical icosahedral hexagonal grid (also referred to as a geodesic grid) to solve the transport equation. The C grid discretization was used for the spatial discretization. To implement TSPAS on an unstructured grid, the original finite-difference scheme was further generalized. The two-step integration utilizes a combination of two separate schemes (a low-order monotone scheme and a high-order scheme that typically cannot ensure monotonicity) to calculate the fluxes at the cell walls (one scheme corresponds to one cell wall). The choice between these two schemes for each edge depends on a pre-updated scalar value using slightly increased fluxes. After the determination of an appropriate scheme, the final integration at a target cell is achieved by summing the fluxes that are computed by the different schemes. The conservative and shape-preserving properties of the generalized scheme are demonstrated. Numerical experiments are conducted at several horizontal resolutions. TSPAS is compared with the Flux Corrected Transport (FCT) approach to demonstrate the differences between the two methods, and several transport tests are performed to examine the accuracy, efficiency and robustness of the two schemes.

Preserving Nonnegativity in Discontinuous Galerkin Approximations to Scalar Transport via Truncation and Mass Aware Rescaling (TMAR)

Abstract An accurate nonnegativity preserving limiter is presented for use with discontinuous Galerkin (DG) discretizations of scalar advection equations. The nonnegativity of the tracer field is preserved through the application of a mass conservative limiter that truncates negatives within each element and linearly rescales the resulting DG polynomials to preserve element-mean mass. As a preliminary step, the DG fluxes through each side of the element are limited in a manner similar to flux-corrected transport to ensure that the element-mean mass remains nonnegative during each individual stage of the time integration. In this paper, it is proven that such a truncation and mass aware rescaling (TMAR) does not change the order of accuracy of the underlying unlimited DG approximation. Numerical tests with two-dimensional deforming flows confirm that the method remains accurate and efficient while preserving nonnegativity. In comparison to some popular previous approaches, TMAR limiting is particularly well suited to approximations that use high-degree polynomial expansions (quartics or higher) to capture features that are only moderately well resolved.