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We consider flux-corrected finite element discretizations of 3D convection-dominated transport problems and assess the computational efficiency of algorithms based on such approximations. The methods under investigation include flux-corrected transport schemes and monolithic limiters. We discretize in space using a continuous Galerkin method and ℙ1 or ℚ1 finite elements. Time integration is performed using the Crank–Nicolson method or an explicit strong stability preserving Runge–Kutta method. Nonlinear systems are solved using a fixed-point iteration method, which requires solution of large linear systems at each iteration or time step. The great variety of options in the choice of discretization methods and solver components calls for a dedicated comparative study of existing approaches. To perform such a study, we define new 3D test problems for time dependent and stationary convection–diffusion–reaction equations. The results of our numerical experiments illustrate how the limiting technique, time discretization and solver impact on the overall performance. [ABSTRACT FROM AUTHOR]

We provide a framework for high-order discretizations of nonlinear scalar convection-diffusion equations that satisfy a discrete maximum principle. The resulting schemes can have arbitrarily high order accuracy in time and space, and can be stable and maximum-principle-preserving (MPP) with no step size restriction. The schemes are based on a two-tiered limiting strategy, starting with a high-order limiter-based method that may have small oscillations or maximum-principle violations, followed by an additional limiting step that removes these violations while preserving high order accuracy. The desirable properties of the resulting schemes are demonstrated through several numerical examples. [ABSTRACT FROM AUTHOR]

We introduce a general framework for enforcing local or global maximum principles in high-order space-time discretizations of a scalar hyperbolic conservation law. We begin with sufficient conditions for a space discretization to be bound preserving (BP) and satisfy a semi-discrete maximum principle. Next, we propose a global monolithic convex (GMC) flux limiter which has the structure of a flux-corrected transport (FCT) algorithm but is applicable to spatial semi-discretizations and ensures the BP property of the fully discrete scheme for strong stability preserving (SSP) Runge–Kutta time discretizations. To circumvent the order barrier for SSP time integrators, we constrain the intermediate stages and/or the final stage of a general high-order RK method using GMC-type limiters. In this work, our theoretical and numerical studies are restricted to explicit schemes which are provably BP for sufficiently small time steps. The new GMC limiting framework offers the possibility of relaxing the bounds of inequality constraints to achieve higher accuracy at the cost of more stringent time step restrictions. The ability of the presented limiters to recognize undershoots/overshoots, as well as smooth solutions, is verified numerically for three representative RK methods combined with weighted essentially nonoscillatory (WENO) finite volume space discretizations of linear and nonlinear test problems in 1D. In this context, we enforce global bounds and prove preservation of accuracy for the linear advection equation. [ABSTRACT FROM AUTHOR]

Abstract The refilling of the plasmasphere following geomagnetic storms remains one of the longstanding and interesting problems in ionosphere–magnetosphere coupling research. The objective of this paper is the formulation and development of a one-dimensional (1D) refilling model using the flux-corrected transport method, a numerical method that is well-suited to handling problems with shocks and discontinuities. In this paper, the developed methodology has been validated against exact, analytical benchmarks, and good agreement has been obtained between these analytical benchmarks and numerical results. The objective of this research is the development of a three-dimensional (3D) multi-ion model for ionosphere–magnetosphere coupling problems in open and closed line geometries.

A process of the removal of dissolved elements in the ocean by adsorption onto settling particulate matters (scavenging) is studied analytically and using the Lagrangian and Eulerian numerical methods. The generalized model of scavenging in a multicomponent reactive medium with first-order kinetics consisting of water and multifraction suspended particulate matter was developed. Two novel numerical schemes were used to solve the transport–diffusion–reaction equations for transport dominated flows. The particle tracking algorithm based on the method of moments was developed. The modified flux-corrected transport method for the Eulerian transport–diffusion–reaction equations is a flux-limiter method based on a convex combination of low-order and high-order schemes. The flux limiters in the developed approach are obtained as an approximate solution of a corresponding optimization problem with a linear objective function. This approach allows the construction of the flux limiters with desired properties. The similarity solutions of the model equations for an idealized case of instantaneous release of reactive radionuclide on the ocean surface were obtained. It was found analytically that the dispersion of reactive contamination caused by reversible phase transition with increase of settling velocity, concentration of particulate matter and distribution coefficient can be much greater than that caused by diffusion, whereas an increase in the desorption rate results in a decrease of the dispersion caused by the phase transfer. The solutions using both numerical schemes are consistent with the analytical similarity solution even at zero diffusivity. The scavenging of the 239 , 240 Pu that was introduced to the ocean surface due to the fallout from past nuclear weapon testing was simulated. The simulation results were in agreement with the observations in the northern Pacific. It was shown that even if the concentration of the 239 , 240 Pu on the particulate matter does not exceed 2% of the total concentration, settling of particulate matter plays a crucial role in the vertical transport and dispersion of the reactive radionuclide. The importance of the scavenging by both the large fast-settling particles and small particles slowly settling and dissolving with depth due to the biochemical processes was demonstrated. For large particles, the "pseudodiffusivity" caused by phase transfer was 60 times greater than the diffusivity. [ABSTRACT FROM AUTHOR]

Summary: We present a new event‐driven approach that combines a shallow water flow model with a practical sedimentation technique to simulate the formation of turbidite depositional systems at a stratigraphic scale. The equations that govern turbidity currents dynamics are solved using a new finite element flux‐corrected transport scheme. In this sense, the low‐order formulation is built by adding a novel Rusanov‐like scalar dissipation scaled by a shock‐capturing operator to standard Galerkin equations. From it, the high‐order system is obtained by including antidiffusive fluxes linearized around the low‐order solution and limited with the Zalesak's algorithm, following a minmod prelimiter. Implicit time integration with adaptive time steps is performed with an iterative nonlinear scheme that linearizes source terms. Sedimentation is implemented by carrying five granulometric fractions (clay, silt, and fine, medium, and coarse sands) along evolved streaklines and radially scattering sediments that deposit filling the available depositional space and compacting the underneath sediment layers. The flow is computed while an event discharge into an area of interest is active, or the inflow current has not reached an equilibrium state. Afterward, the event deposition step is executed. Numerical results of our flow solver presented a good agreement with available exact and literature solutions, and the simulated sediment deposits suggest that our approach is well suited for stratigraphic scale simulations. [ABSTRACT FROM AUTHOR]

The refilling of the plasmasphere following geomagnetic storms remains one of the longstanding and interesting problems in ionosphere–magnetosphere coupling research. The objective of this paper is the formulation and development of a one-dimensional (1D) refilling model using the flux-corrected transport method, a numerical method that is well-suited to handling problems with shocks and discontinuities. In this paper, the developed methodology has been validated against exact, analytical benchmarks, and good agreement has been obtained between these analytical benchmarks and numerical results. The objective of this research is the development of a three-dimensional (3D) multi-ion model for ionosphere–magnetosphere coupling problems in open and closed line geometries. [ABSTRACT FROM AUTHOR]

Conventional full-waveform inversion (FWI) often fails to retrieve the unknown model parameters from noisy seismic data. A successful FWI implementation usually requires to follow a multistage recovery approach, starting from the retrieval of the lower model wavenumbers (tomography) to those with the higher resolutions (migration). Here, we propose a new method based on the flux-corrected transport (FCT) technique often used in computational fluid dynamics for the removal of instabilities in a shock profile. FCT involves three finite-difference steps: a transport, a diffusion and an antidiffusion process. This third step involves non-linear operators such as maximum and minimum, which are non-differentiable in a classic sense. However, since the seismic source wavelet and the corresponding wavefield are relatively smooth and continuous in nature without any strong ripples like shock waves, we exclude the non-linear step from FCT, which allows us to evaluate the novel FWI gradient efficiently. As a result, we achieve a converging FWI model by gradually reducing the diffusive flux-correction. We demonstrate the versatility of FCT-based FWI on a noisy synthetic data set from the Marmousi II model and a marine field data set from offshore Australia. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

SUMMARY This 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. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we discuss the numerical treatment of three-dimensional mixture models for (semi-)dilute and concentrated suspensions of particles in incompressible fluids. The generalized Navier–Stokes system and the continuity equation for the volume fraction of the disperse phase are discretized using an implicit high-resolution finite element scheme, and maximum principles are enforced using algebraic flux correction. To prevent the volume fractions from exceeding the maximum packing limit, a conservative overshoot limiter is applied to the converged convective fluxes at the end of each time step. A numerical study of the proposed approach is performed for 3D particulate flows over a backward-facing step and in a lid-driven cavity. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

In this work we construct a finite element method for shallow water flows with focus on problems with evolutionary coastlines. The method does not utilize ad hoc techniques to track the water-terrain interfaces and fulfills global conservation and positivity. The basis of the procedure is a generalized flux correction technique, and its application can be extended to diverse finite element solutions of the shallow water equations. Experiments are mainly concentrated on frictional flows with fronts propagating over dry regions. [ABSTRACT FROM PUBLISHER]

Abstract: 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. [Copyright &y& Elsevier]

A monotonic transport algorithm for a high-order time integration scheme is described in this paper. The algorithm is a modified version of an existing high-order time integration scheme, and is tested using a simple one-dimensional pulse and two-dimensional deformational flows. It is found that the new formulation can remove the error, caused by new maxima/minima and excessive smoothing, which occurs in scalar transport using the original extrapolation scheme. The results show that there may be potential for the high-order time integration scheme to be applied in numerical weather prediction models. [ABSTRACT FROM AUTHOR]

This 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper is concerned with the extension of the algebraic flux-correction (AFC) approach (Kuzmin in Computational fluid and solid mechanics, Elsevier, Amsterdam, pp 887-888, ; J Comput Phys 219:513-531, ; Comput Appl Math 218:79-87, ; J Comput Phys 228:2517-2534, ; Flux-corrected transport: principles, algorithms, and applications, 2nd edn. Springer, Berlin, pp 145-192, ; J Comput Appl Math 236:2317-2337, ; Kuzmin et al. in Comput Methods Appl Mech Eng 193:4915-4946, ; Int J Numer Methods Fluids 42:265-295, ; Kuzmin and Möller in Flux-corrected transport: principles, algorithms, and applications. Springer, Berlin, ; Kuzmin and Turek in J Comput Phys 175:525-558, ; J Comput Phys 198:131-158, ) to nonconforming finite element methods for the linear transport equation. Accurate nonoscillatory approximations to convection-dominated flows are obtained by stabilizing the continuous Galerkin method by solution-dependent artificial diffusion. Its magnitude is controlled by a flux limiter. This concept dates back to flux-corrected transport schemes. The unique feature of AFC is that all information is extracted from the system matrices which are manipulated to satisfy certain mathematical constraints. AFC schemes have been devised with conforming $$P_1$$ and $$Q_1$$ finite elements in mind but this is not a prerequisite. Here, we consider their extension to the nonconforming Crouzeix-Raviart element (Crouzeix and Raviart in RAIRO R3 7:33-76, ) on triangular meshes and its quadrilateral counterpart, the class of rotated bilinear Rannacher-Turek elements (Rannacher and Turek in Numer Methods PDEs 8:97-111, ). The underlying design principles of AFC schemes are shown to hold for (some variant of) both elements. However, numerical tests for a purely convective flow and a convection-diffusion problem demonstrate that flux-corrected solutions are overdiffusive for the Crouzeix-Raviart element. Good resolution of smooth and discontinuous profiles is attested to $$Q_1^\mathrm{nc}$$ approximations on quadrilateral meshes. A synthetic benchmark is used to quantify the artificial diffusion present in conforming and nonconforming high-resolution schemes of AFC-type. Finally, the implementation of efficient sparse matrix-vector multiplications is addressed. [ABSTRACT FROM AUTHOR]

Abstract: A new high-resolution scheme is developed for convection–diffusion problems in domains with moving boundaries. A finite element approximation of the governing equation is designed within the framework of a conservative Arbitrary Lagrangian Eulerian (ALE) formulation. An implicit flux-corrected transport (FCT) algorithm is implemented to suppress spurious undershoots and overshoots appearing in convection-dominated problems. A detailed numerical study is performed for P 1 finite element discretizations on fixed and moving meshes. Simulation results for a Taylor dispersion problem (moderate Peclet numbers) and for a convection-dominated problem (large Peclet numbers) are presented to give a flavor of practical applications. [Copyright &y& Elsevier]

Abstract: A new approach to flux limiting for systems of conservation laws is presented. The Galerkin finite element discretization/L 2 projection is equipped with a failsafe mechanism that prevents the birth and growth of spurious local extrema. Within the framework of a synchronized flux-corrected transport (FCT) algorithm, the velocity and pressure fields are constrained using node-by-node transformations from the conservative to the primitive variables. An additional correction step is included to ensure that all the quantities of interest (density, velocity, pressure) are bounded by the physically admissible low-order values. The result is a conservative and bounded scheme with low numerical diffusion. The new failsafe FCT limiter is integrated into a high-resolution finite element scheme for the Euler equations of gas dynamics. Also, bounded L 2 projection operators for conservative interpolation/initialization are designed. The performance of the proposed limiting strategy and the need for a posteriori control of flux-corrected solutions are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: This study considers the development and assessment of a flux-corrected transport (FCT) algorithm for simulating high-speed flows on structured overlapping grids. This class of algorithm shows promise for solving some difficult highly-nonlinear problems where robustness and control of certain features, such as maintaining positive densities, is important. Complex, possibly moving, geometry is treated through the use of structured overlapping grids. Adaptive mesh refinement (AMR) is employed to ensure sharp resolution of discontinuities in an efficient manner. Improvements to the FCT algorithm are proposed for the treatment of strong rarefaction waves as well as rarefaction waves containing a sonic point. Simulation results are obtained for a set of test problems and the convergence characteristics are demonstrated and compared to a high-resolution Godunov method. The problems considered are an isolated shock, an isolated contact, a modified Sod shock tube problem, a two-shock Riemann problem, the Shu–Osher test problem, shock impingement on single cylinder, and irregular Mach reflection of a strong shock striking an inclined plane. [Copyright &y& Elsevier]

Abstract: A new approach to the design of flux-corrected transport (FCT) algorithms for continuous (linear/multilinear) finite element approximations of convection-dominated transport problems is pursued. The algebraic flux correction paradigm is revisited, and a family of nonlinear high-resolution schemes based on Zalesak’s fully multidimensional flux limiter is considered. In order to reduce the cost of flux correction, the raw antidiffusive fluxes are linearized about an auxiliary solution computed by a high- or low-order scheme. By virtue of this linearization, the costly computation of solution-dependent correction factors is to be performed just once per time step, and there is no need for iterative defect correction if the governing equation is linear. A predictor–corrector algorithm is proposed as an alternative to the hybridization of high- and low-order fluxes. Three FEM-FCT schemes based on the Runge–Kutta, Crank–Nicolson, and backward Euler time-stepping are introduced. A detailed comparative study is performed for linear convection–diffusion equations. [Copyright &y& Elsevier]

This paper proposes a new lattice Boltzmann (LB) method for the study of shock wave propagation in elastic solids. The method, which implements a flux-corrected transport (FCT) algorithm, contains three stages: collision, streaming, and correction. In the collision stage, distribution functions are updated. In the streaming stage, the distribution functions are shifted between lattice points. Generally, a partial differential equation (PDE) is solved in the streaming stage, and finite element methods are employed to support the use of unstructured meshes in the LB method. The FCT algorithm is used in the correction stage to revise the distribution functions at lattice points, so fluctuations behind shock wave fronts can be eliminated efficiently. In this method, schemes for shock wave reflection at fixed and free boundaries are developed based on the bounce-back technique. A similar technique is used to treat wave reflection and transmission at material interfaces of composites. Several one-dimensional examples show that this LB-FCT method can provide ideal depictions of shock wave propagation in structures, especially composite structures. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]