Flux-corrected Transport Algorithm Research Articles

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.

Synchronized flux limiting for gas dynamics variables

This work addresses the design of failsafe flux limiters for systems of conserved quantities and derived variables in numerical schemes for the equations of gas dynamics. Building on Zalesak's multidimensional flux-corrected transport (FCT) algorithm, we construct a new positivity-preserving limiter for the density, total energy, and pressure. The bounds for the underlying inequality constraints are designed to enforce local maximum principles in regions of strong density variations and become less restrictive in smooth regions. The proposed approach leads to closed-form expressions for the synchronized correction factors without the need to solve inequality-constrained optimization problems. A numerical study is performed for the compressible Euler equations discretized using a finite element based FCT scheme.

Finite-Element Sea Ice Model (FESIM), version 2

Abstract. The Finite-Element Sea Ice Model (FESIM), used as a component of the Finite-Element Sea ice Ocean Model, is presented. Version 2 includes the elastic-viscous-plastic (EVP) and viscous-plastic (VP) solvers and employs a flux corrected transport algorithm to advect the ice and snow mean thicknesses and concentration. The EVP part also includes a modified approach proposed recently by Bouillon et al. (2013), which is characterized by an improved stability compared to the standard EVP approach. The model is formulated on unstructured triangular meshes. It assumes a collocated placement of ice velocities, mean thicknesses and concentration at mesh vertices, and relies on piecewise-linear (P1) continuous elements. Simple tests for the modified EVP and VP solvers are presented to show that they may produce very close results provided the number of iterations is sufficiently high.

Numerical Simulation of Rod-Plate Gap Streamer Discharge in SF6/N2Gas Mixtures Based on ETG-FCT Method

The Euler-Taylor-Galerkin flux-corrected transport (ETG-FCT) algorithm for the numerical solution of particle transport equations is described, based on the method developed by Lohner to solve conservation equations in fluid mechanics, and its application is extended to gas discharge problems. To improve the efficiency of computing and reduce numerical error, the nonuniform triangular mesh method is introduced, and the continuity equation is solved by ETG-FCT. The new contributions in this paper include the development of the ETG scheme and its application to rod-plate gap streamer discharge in 50~50% SF6/N2gas mixtures problems. Results are obtained: the spatial distributions of electron densities, positive ion densities, negative ion densities, photoelectron densities, and the electric field, respectively. The velocities of streamer propagations and the radius of streamer obtained from the proposed model are in good agreement with experimental and simulation results in literature. The results also prove that the ETG-FCT method is valid.

펄스파워를 이용한 실린더형 전극간 금속 플라즈마 생성현상의 전산유동해석

This computational study features the transient compressible and inviscid flow analysis on a metallic plasma discharge from the opposing composite electrodes which is subjected to pulsed electric power. The computations have been performed using the flux corrected transport algorithm on the axisymmetric two-dimensional domain of electrode gap and outer space along with the calculation of plasma compositions and thermophysical properties such as plasma electrical conductivity. The mass ablation from aluminum electrode surfaces are modeled with radiative flux from plasma column experiencing intense Joule heating. The computational results shows the highly ionized and highly under-expanded supersonic plasma discharge with strong shock structure of Mach disk and blast wave propagation, which is very similar to muzzle blast or axial plasma jet flows. Also, the geometrical effects of composite electrodes are investigated to compare the amount of mass ablation and penetration depth of plasma discharge.

Toward a reduction of mesh imprinting

SUMMARYThis paper is the initial investigation into a new Lagrangian cell‐centered hydrodynamic scheme that is motivated by the desire for an algorithm that resists mesh imprinting and has reduced complexity. Key attributes of the new approach include multidimensional construction, the use of flux‐corrected transport (FCT) to achieve second order accuracy, automatic determination of the mesh motion through vertex fluxes, and vorticity control. Toward this end, vorticity preserving Lax–Wendroff (VPLW) type schemes for the two‐dimensional acoustic equations were analyzed and then implemented in a FCT algorithm. Here, mesh imprinting takes the form of anisotropic dispersion relationships. If the stencil for the LW methods is limited to nine points, four free parameters exist. Two parameters were fixed by insisting that no spurious vorticity be created. Dispersion analysis was used to understand how the remaining two parameters could be chosen to increase isotropy. This led to new VPLW schemes that suffer less mesh imprinting than the rotated Richtmyer method. A multidimensional, vorticity preserving FCT implementation was then sought using the most promising VPLW scheme to address the problem of spurious extrema. A well‐behaved first order scheme and a new flux limiter were devised in the process. The flux limiter is unique in that it acts on temporal changes and does not place a priori bounds on the solution. Numerical results have demonstrated that the vorticity preserving FCT scheme has comparable performance to an unsplit MUSCL‐H algorithm at high Courant numbers but with reduced mesh imprinting and superior symmetry preservation. Copyright © 2014 John Wiley & Sons, Ltd.

Reverse time migration from irregular surface by flattening surface topography

Abstract Prestack reverse time migration (RTM) is a powerful tool to provide high resolution structure images of crust and upper mantle in wide-angle seismic experiments. As a standard migration algorithm assumes that data are recorded on a flat surface, RTM meets severe challenge on the issue of complicated topography where many seismic surveys are carried out. Up till now, there are two popular techniques for handling elevation changes, i.e., elevation-static correction (or time shift) and wave equation datuming. Numerous studies demonstrate that the elevation-static correction is inaccurate for non-vertically traveling energy and subsequently may migrate reflectors to incorrect positions. Wave equation datuming is more accurate but more sophisticated and too computationally demanding for routine use. Here, we present an alternative scheme to deal with topography in prestack RTM that can migrate seismic data acquired in areas with irregular topography to the reflector positions directly without datuming or elevation static corrections. In this scheme, surface topography is flattened by a transformation of coordinates from Cartesian to curvilinear. Such a transformation is equivalent to map a rectangular grid onto a curved one, and is implemented by the introduction of boundary conforming grid. Then we use an explicit finite-difference scheme with second-order accuracy to discretize the elastic wave equations and perform the RTM in the curvilinear coordinate system. Moreover, a flux-corrected transport (FCT) algorithm, adapted from hydrodynamics, is incorporated in the RTM implementation, which can not only overcome problems of numerical dispersion that arise in finite difference algorithms, but also offer the opportunity to use a coarse grid to obtain the same accuracy as conventional finite difference methods on a dense grid. The FCTRTM method presented here has no limitations on surface irregularity, complexity of the subsurface structure, or velocity function. Extensive numerical examples demonstrate the feasibility and effectiveness of our method.

Reprint of: A conservative multi-tracer transport scheme for spectral-element spherical grids

Atmospheric models used for practical climate simulation must be capable handling the transport of hundreds of tracers. For computational efficiency conservative multi-tracer semi-Lagrangian type transport schemes are appropriate. Global models based on high-order Galerkin approach employ highly non-uniform spectral-element grids, and semi-Lagrangian transport is a challenge on those grids. A conservative semi-Lagrangian scheme (SPELT — SPectral-Element Lagrangian Transport) employing a multi-moment compact reconstruction procedure is developed for non-uniform quadrilateral grids. The scheme is based on a characteristic semi-Lagrangian method that avoids complex and expensive upstream area computations. The SPELT scheme has been implemented in the High-Order Method Modeling Environment (HOMME), which is based on a cubed-sphere grid with spectral-element spatial discretization. Additionally, we show the (strong) scalability and multi-tracer efficiency using several benchmark tests. The SPELT solution can be made monotonic (positivity preserving) by combining the flux-corrected transport algorithm, which is demonstrated on a uniform resolution grid. In particular, SPELT can be efficiently used for non-uniform grids and provides accurate and stable results for high-resolution meshes.

A large-scale wave-current coupled module with wave diffraction effect on unstructured meshes

Based on the extended mild-slope equation, a large-scale wave module is developed. By combining the eikonal equation and the modified wave action equation, the wave model can account for diffraction in most situations such as in the lee of islands and breakwaters, and using unstructured meshes provides great flexibility for modelling the wave in the complex geomorphology of barriers and islands, also allowing for refinement of the grid resolution within computationally important domains. The numerical implementation of the module is based on the explicit second-order upwind finite-volume schemes in geographic space, the Flux-Corrected Transport (FCT) algorithm in frequency space and the implicit Crank-Nicolson method in directional space. The three-dimensional hydrodynamic module is then modified to couple with the wave model, where the wave readily provides the depth-dependent radiation stress and the wave-induced turbulence coefficient for the current fields, and the wave propagation takes into account the current-induced advection, refraction and diffraction of wave energy and the effect of water level. The applicability of the proposed model to calculate Snell’s Law, wave transformation over the breakwaters and the elliptic shoal, wave propagation over the rip current field and the undertow on a sloping beach is evaluated. Numerical results show that the present model makes better predictions of the near-shore wave propagation and complex three-dimensional (3D) near-shore circulation driven by the waves, considering analytical solutions and experimental values.

DSMC Simulation of Low Knudsen Micro/Nanoflows Using Small Number of Particles per Cells

Direct simulation Monte Carlo (DSMC) method in low Knudsen rarefied flows at micro/nanoscales remains a big challenge for researchers due to large computational requirements. In this article, the application of the simplified Bernoulli-trials (SBT)/dual grid collision scheme is extended for solving low Knudsen/low speed and low Knudsen/high gradient rarefied micro/nanoflows. The main advantage of the SBT algorithm is to provide accurate calculations using much smaller number of particles per cell, i.e., 〈N〉 ≈ 2, which is quite beneficial for near continuum DSMC simulations where the requirement of fine meshes faces the simulation with serious memory restrictions. Comparing the results of the SBT/dual grid scheme with the no time counter (NTC) scheme and majorant frequency scheme (MFS), it is shown that the SBT/dual grid scheme could successfully predict the thermal pattern and hydrodynamics field as well as surface parameters such as velocity slip, temperature jump and wall heat fluxes. Therefore, we present SBT/dual grid algorithm as a suitable alternative of the standard collision schemes in the DSMC method for typical micro/nanoflows solution. Nonlinear flux-corrected transport (FCT) algorithm is also employed as a filter to extract the smooth solution from the noisy DSMC calculation for low speed/low Knudsen number DSMC calculations.

A conservative multi-tracer transport scheme for spectral-element spherical grids

Atmospheric models used for practical climate simulation must be capable handling the transport of hundreds of tracers. For computational efficiency conservative multi-tracer semi-Lagrangian type transport schemes are appropriate. Global models based on high-order Galerkin approach employ highly non-uniform spectral-element grids, and semi-Lagrangian transport is a challenge on those grids. A conservative semi-Lagrangian scheme (SPELT — SPectral-Element Lagrangian Transport) employing a multi-moment compact reconstruction procedure is developed for non-uniform quadrilateral grids. The scheme is based on a characteristic semi-Lagrangian method that avoids complex and expensive upstream area computations. The SPELT scheme has been implemented in the High-Order Method Modeling Environment (HOMME), which is based on a cubed-sphere grid with spectral-element spatial discretization. Additionally, we show the (strong) scalability and multi-tracer efficiency using several benchmark tests. The SPELT solution can be made monotonic (positivity preserving) by combining the flux-corrected transport algorithm, which is demonstrated on a uniform resolution grid. In particular, SPELT can be efficiently used for non-uniform grids and provides accurate and stable results for high-resolution meshes.

Implicit large-eddy simulation of passive scalar mixing in statistically stationary isotropic turbulence

Turbulent mixing of a passive scalar by forced isotropic turbulence with a prescribed mean scalar gradient is studied in the context of implicit large-eddy simulation. The simulation strategy uses a multi-dimensional compressible flux-corrected transport algorithm, with low wavenumber momentum forcing imposed separately for the solenoidal and dilatational velocity components. Effects of grid resolution on the flow and scalar mixing are investigated at turbulent Mach numbers 0.13 and 0.27. Turbulence metrics are used to show that an implicit large-eddy simulation can accurately capture the mixing transition and asymptotic self-similar behaviors predicted by previous theoretical, laboratory, and direct numerical simulation studies, including asymptotically constant scalar variance and increasing velocity-to-scalar Taylor micro-scales ratio as function of effective Reynolds number determined by grid resolution. The results demonstrate the feasibility of predictive under-resolved simulations of high Reynolds number turbulent scalar mixing using implicit large-eddy simulation.

Numerical simulation of chemotaxis models on stationary surfaces

In this paper we present an implicit finite element method for a class of chemotaxis models, where a new linearized flux-corrected transport (FCT) algorithm is modified in such a way as to keep the density of on-surface living cells nonnegative. Level set techniques are adopted for an implicit description of the surface and for the numerical treatment of the corresponding system of partial differential equations. The presented scheme is able to deliver a robust and accurate solution for a large class of chemotaxis-driven models. The numerical behavior of the proposed scheme is tested on the blow-up model on a sphere and an ellipsoid and on the pattern-forming dynamics model of Escherichia coli on a sphere.

A parameter-free smoothness indicator for high-resolution finite element schemes

AbstractThis paper presents a postprocessing technique for estimating the local regularity of numerical solutions in high-resolution finite element schemes. A derivative of degree p ≥ 0 is considered to be smooth if a discontinuous linear reconstruction does not create new maxima or minima. The intended use of this criterion is the identification of smooth cells in the context of p-adaptation or selective flux limiting. As a model problem, we consider a 2D convection equation discretized with bilinear finite elements. The discrete maximum principle is enforced using a linearized flux-corrected transport algorithm. The deactivation of the flux limiter in regions of high regularity makes it possible to avoid the peak clipping effect at smooth extrema without generating spurious undershoots or overshoots elsewhere.

A flux-corrected transport algorithm for handling the close-packing limit in dense suspensions

Convection of a scalar quantity by a compressible velocity field may give rise to unbounded solutions or nonphysical overshoots at the continuous and discrete level. In this paper, we are concerned...

An $$hp$$ -adaptive flux-corrected transport algorithm for continuous finite elements

This paper presents an $$hp$$ -adaptive flux-corrected transport algorithm for continuous finite elements. The proposed approach is based on a continuous Galerkin approximation with unconstrained higher-order elements in smooth regions and constrained $$P_1/Q_1$$ elements in the neighborhood of steep fronts. Smooth elements are found using a hierarchical smoothness indicator based on discontinuous higher-order reconstructions. A gradient-based error indicator determines the local mesh size $$h$$ and polynomial degree $$p$$ . The discrete maximum principle for linear/bilinear finite elements is enforced using a linearized flux-corrected transport (FCT) algorithm. The same limiting strategy is employed when it comes to constraining the $$L^2$$ projection of data from one finite-dimensional space into another. The new algorithm is implemented in the open-source software package Hermes. The use of hierarchical data structures that support arbitrary-level hanging nodes makes the extension of FCT to $$hp$$ -FEM relatively straightforward. The accuracy of the proposed method is illustrated by a numerical study for a two-dimensional benchmark problem with a known exact solution.

Low speed/low rarefaction flow simulation in micro/nano cavity using DSMC method with small number of particles per cell

The aim of this study is to extend the validity of the simplified Bernoulli-trials (SBT)/dual grid algorithm, newly proposed by Stefanov [1], as a suitable alternative of the standard collision scheme in the direct simulation Monte Carlo (DSMC) method, for solving low speed/low Knudsen number rarefied micro/nano flows. The main advantage of the SBT algorithm is to provide accurate calculations using much smaller number of particles per cell, i.e., < N > ≈ 1. Compared to the original development of SBT [1], we extend the application of the SBT scheme to the near continuum rarefied flows, i.e., Kn = 0.005, where NTC scheme requires a relatively large sample size. Comparing the results of the SBT/dual grid scheme with NTC, it is shown that the SBT/dual grid scheme could successfully predict the thermal pattern and hydrodynamics field as well as surface parameters such as velocity slip and temperature jump. Nonlinear flux-corrected transport algorithm (FCT) is also employed as a filter to extract the smooth solution from the noisy DSMC calculation for low-speed/low-Knudsen number DSMC calculations. The results indicate that combination of SBT/dual grid and FTC filtering can decrease the total sample size needed to reach smooth solution without losing significant accuracy.

One-dimensional numerical simulation of thermoacoustic engine with flux-corrected transport algorithm

The generation and propagation characteristics of the thermoacoustic wave in thermoacoustic engines have been numerically investigated with two-dimensional implicit methods in most of previous works. While there are natural limitations in time, scale and precision, in the explicit numerical method, the flux corrected transport (FCT) algorithm features easy calculation and high precision when solving large gradient problems. Using the FCT algorithm, we simulated the thermoacoustic generation and propagation in a cavity. The results agreed well with previous works, showing the stability and reliability of the present method in addressing thermoacoustic wave problems. Then, combining the timestep splitting technique, oscillation boundary conditions and simplified one-dimensional thermoacoustic engines model, we used the FCT algorithm to simulate the nonlinear characteristics of thermoacoustic engines. We obtained nonlinear wave profiles caused by large parameters and plate extreme edge effects. The profiles qualitatively agreed with previous results, thus exhibiting the potential of this method in simulating thermoacoustic engines.