Flux-corrected Transport Algorithm Research Articles

Numerical investigation of nonlinear effects in a standing wave thermoacoustic engine using the discontinuous Galerkin method

Understanding the nonlinear effects in thermoacoustic heat engines operating under high-pressure amplitude conditions plays a crucial role in improving their operational performance and efficiency. The discontinuous Galerkin (DG) method has several advantages, including high accuracy, adaptability, efficiency, and the ability to handle the coupling of multiple physical fields, making it suitable for accurately simulating the flow and heat transfer characteristics in thermoacoustic engines. The simulation code was developed based on MFEM, an open-source C++ library for modular finite element methods. The fully compressible Navier-Stokes equations and Fourier heat conduction equation were solved to investigate the nonlinear effects, which cannot be captured by the linear theory. To verify the accuracy of the simulation code, the numerical investigation of the thermoacoustic wave propagations induced by thermal effects in a square enclosure was first conducted. The simulation results presented the details of the generation and propagation of thermoacoustic waves, which were largely consistent with the results obtained using the flux-corrected transport (FCT) algorithm. Second, a quarter-wavelength standing wave thermoacoustic engine was simulated. The simulation results captured the process of pressure wave generation in the thermoacoustic engine, which was consistent with the pressure wave generation process in the enclosure. The results revealed the flow and heat transfer characteristics and nonlinear effects in the thermoacoustic engine under high-pressure amplitudes, which were consistent with the results obtained using finite volume method. The simulation results showed that the DG method is an effective numerical method for simulating thermoacoustic engines.

Bound-preserving Flux Limiting for High-Order Explicit Runge–Kutta Time Discretizations of Hyperbolic Conservation Laws

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.

Entropy Stable Flux Correction for Scalar Hyperbolic Conservation Laws

It is known that Flux Corrected Transport algorithms can produce entropy-violating solutions of hyperbolic conservation laws. Our purpose is to design flux correction with maximal antidiffusive fluxes to obtain entropy solutions of scalar hyperbolic conservation laws. To do this we consider a hybrid difference scheme that is a linear combination of a monotone scheme and a scheme of high-order accuracy. The flux limiters for the hybrid scheme are calculated from a corresponding optimization problem. Constraints for the optimization problem consist of inequalities that are valid for the monotone scheme and applied to the hybrid scheme. We apply the discrete cell entropy inequality with the proper numerical entropy flux to single out a physically relevant solution of scalar hyperbolic conservation laws. A nontrivial approximate solution of the optimization problem yields expressions to compute the required flux limiters. We present examples that show that not all numerical entropy fluxes guarantee to single out a physically correct solution of scalar hyperbolic conservation laws.

Fully decoupled, linear and positivity-preserving scheme for the chemotaxis–Stokes equations

In this article, we develop a fully decoupled, linear and positivity-preserving finite element scheme for solving the chemotaxis–Stokes equations, which describe the biological chemotaxis phenomenon in the fluid environment. To deal with the strong coupling of the problem, we first consider a fully decoupled and linear semi-discrete scheme, in which we only need to solve several linearized sub-problems at each time step for the velocity, pressure, oxygen concentration and cell density, respectively. Then, based on the linear finite element method for the spatial discretization, the flux-corrected transport algorithm is extended to the oxygen concentration and cell density sub-problems to preserve their positivity. Finally, we show a series of numerical examples to verify its efficiency. We also find that better simulation can be obtained by regularizing the cut-off function and setting a random initial cell density in the equations.

Matrix-free subcell residual distribution for Bernstein finite elements: Monolithic limiting

This paper is focused on the aspects of limiting in residual distribution (RD) schemes for high-order finite element approximations to advection problems. Both continuous and discontinuous Galerkin methods are considered in this work. Discrete maximum principles are enforced using algebraic manipulations of element contributions to the global nonlinear system. The required modifications can be carried out without calculating the element matrices and assembling their global counterparts. The components of element vectors associated with the standard Galerkin discretization are manipulated directly using localized subcell weights to achieve optimal accuracy. Low-order nonlinear RD schemes of this kind were originally developed to calculate local extremum diminishing predictors for flux-corrected transport (FCT) algorithms. In the present paper, we incorporate limiters directly into the residual distribution procedure, which makes it applicable to stationary problems and leads to well-posed nonlinear discrete problems. To circumvent the second-order accuracy barrier, the correction factors of monolithic limiting approaches and FCT schemes are adjusted using smoothness sensors based on second derivatives. The convergence behavior of presented methods is illustrated by numerical studies for two-dimensional test problems.

Limiting and divergence cleaning for continuous finite element discretizations of the MHD equations

This work introduces a new type of constrained algebraic stabilization for continuous piecewise-linear finite element approximations to the equations of ideal magnetohydrodynamics (MHD). At the first step of the proposed flux-corrected transport (FCT) algorithm, the Galerkin element matrices are modified by adding graph viscosity proportional to the fastest characteristic wave speed. At the second step, limited antidiffusive corrections are applied and divergence cleaning is performed for the magnetic field. The limiting procedure developed for this stage is designed to enforce local maximum principles, as well as positivity preservation for the density and thermodynamic pressure. Additionally, it adjusts the magnetic field in a way which penalizes divergence errors without violating conservation laws or positivity constraints. Numerical studies for 2D test problems are performed to demonstrate the ability of the proposed algorithms to accomplish this task in applications to ideal MHD benchmarks.

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

Using the theoretical framework of algebraic flux correction and invariant domain preserving schemes, we introduce a monolithic approach to convex limiting in continuous finite element schemes for linear advection equations, nonlinear scalar conservation laws, and hyperbolic systems. In contrast to flux-corrected transport (FCT) algorithms that apply limited antidiffusive corrections to bound-preserving low-order solutions, our new limiting strategy exploits the fact that these solutions can be expressed as convex combinations of bar states belonging to a convex invariant set of physically admissible solutions. Each antidiffusive flux is limited in a way which guarantees that the associated bar state remains in the invariant set and preserves appropriate local bounds. There is no free parameter and no need for limit fluxes associated with the consistent mass matrix of time derivative term separately. Moreover, the steady-state limit of the nonlinear discrete problem is well defined and independent of the pseudo-time step. In the case study for the Euler equations, the components of the bar states are constrained sequentially to satisfy local maximum principles for the density, velocity, and specific total energy in addition to positivity preservation for the density and pressure. The results of numerical experiments for standard test problems illustrate the ability of built-in convex limiters to resolve steep fronts in a sharp and nonoscillatory manner.

Bound-preserving flux limiting schemes for DG discretizations of conservation laws with applications to the Cahn–Hilliard equation

Many mathematical models of computational fluid dynamics involve transport of conserved quantities, which must lie in a certain range to be physically meaningful. The analytical or numerical solution u of a scalar conservation law is said to be bound-preserving if global bounds u∗ and u∗ exist such that u∗≤u≤u∗ holds in the domain of definition. These bounds must be known a priori. To enforce such inequality constraints at least for element averages in the context of discontinuous Galerkin (DG) methods, the numerical fluxes must be defined and constrained in an appropriate manner. In this work, we introduce a general framework for calculating fluxes that produce non-oscillatory DG approximations and preserve relevant global bounds for element averages even if the analytical solution of the PDE violates them due to modeling errors. The proposed methodology is based on a combination of flux and slope limiting. The (optional) slope limiter adjusts the gradients to impose local bounds on pointwise values of the high-order DG solution, which is used to calculate the fluxes. The flux limiter constrains changes of element averages so as to prevent violations of global bounds. Since manipulations of the target flux may introduce a consistency error, it is essential to guarantee that physically admissible fluxes remain unchanged. We propose two kinds of flux limiters, which meet this requirement. The first one is of monolithic type and its time-implicit version requires the iterative solution of a nonlinear problem. Only a fully converged solution is provably bound-preserving. The time-explicit version of this limiter is subject to a time step restriction, which we derive in this article. The second limiter is an iterative version of the multidimensional flux-corrected transport (FCT) algorithm and works as postprocessed correction scheme. This fractional step limiting approach guarantees that each iterate is bound-preserving but avoidable consistency errors may occur if the iterative process is terminated too early. Each iterate depends only on local information of the previous iterate. This concept of limiting the numerical fluxes is also applicable to finite volume methods. Practical applicability of the proposed flux limiters as well as the benefits of using an optional slope limiter are demonstrated by numerical studies for the advection equation (hyperbolic, linear) and the Cahn–Hilliard equation (parabolic, nonlinear) for first-order polynomials. While both flux limiters work for arbitrary order polynomials, we discuss the construction of bound-preserving slope limiters, and show numerical studies only for first-order polynomials.

Matrix-free subcell residual distribution for Bernstein finite element discretizations of linear advection equations

In this work, we introduce a new residual distribution (RD) framework for the design of bound-preserving high-resolution finite element schemes. The continuous and discontinuous Galerkin discretizations of the linear advection equation are modified to construct local extremum diminishing (LED) approximations. To that end, we perform mass lumping and redistribute the element residuals in a manner which guarantees the LED property. The hierarchical correction procedure for high-order Bernstein finite element discretizations involves localization to subcells and definition of bound-preserving weights for subcell contributions. Using strong stability preserving (SSP) Runge–Kutta methods for time integration, we prove the validity of discrete maximum principles under CFL-like time step restrictions. The low-order version of our method has roughly the same accuracy as the one derived from a piecewise (multi)-linear approximation on a submesh with the same nodal points. In high-order extensions, we use an element-based flux-corrected transport (FCT) algorithm which can be interpreted as a nonlinear RD scheme. The proposed LED corrections are tailor-made for matrix-free implementations which avoid the rapidly growing cost of matrix assembly for high-order Bernstein elements. The results for 1D, 2D, and 3D test problems compare favorably to those obtained with the best matrix-based approaches.

A flux-corrected RBF-FD method for convection dominated problems in domains and on manifolds

Abstract In this work, we present a Flux-Corrected Transport (FCT) algorithm for enforcing discrete maximum principles in Radial Basis Function (RBF) generalized Finite Difference (FD) methods for convection-dominated problems. The algorithm is constructed to guarantee mass conservation and to preserve positivity of the solution for irregular data nodes. The method can be applied both for problems defined in a domain or if equipped with level set techniques, on a stationary manifold. We demonstrate the numerical behavior of the method by performing numerical tests for the solid-body rotation benchmark in a unit square and for a transport problem along a curve implicitly prescribed by a level set function. Extension of the proposed method to higher dimensions is straightforward and easily realizable.

An efficient operator-splitting FEM-FCT algorithm for 3D chemotaxis models

An efficient operator-splitting finite element method (FEM) combined with the flux-corrected transport (FCT) algorithm is presented to reduce the computation and storage of solving three-dimensional (3D) chemotaxis models. Firstly, the 3D coupled and positivity-preserving problem is split into a series of 1D subproblems in three spatial directions. Then each 1D subproblem is solved by the FEM-FCT algorithm which guarantees the positivity of numerical solutions. As the 1D subproblems in one direction are spatially independent, they can be calculated in parallel. Additionally, the accuracy and efficiency of the proposed method are investigated by numerical tests. Furthermore, we employ the proposed method to simulate 3D chemotaxis phenomena, including the typical blow-up effect, the more complex pattern formation and aggregating behavior of cell distribution.

Algebraic flux correction schemes preserving the eigenvalue range of symmetric tensor fields

This work extends the algebraic flux correction (AFC) paradigm to finite element discretizations of conservation laws for symmetric tensor fields. The proposed algorithms are designed to enforce discrete maximum principles and preserve the eigenvalue range of evolving tensors. To that end, a continuous Galerkin approximation is modified by adding a linear artificial diffusion operator and a nonlinear antidiffusive correction. The latter is decomposed into edge-based fluxes and constrained to prevent violations of local bounds for the minimal and maximal eigenvalues. In contrast to the flux-corrected transport (FCT) algorithm developed previously by the author and existing slope limiting techniques for stress tensors, the admissible eigenvalue range is defined implicitly and the limited antidiffusive terms are incorporated into the residual of the nonlinear system. In addition to scalar limiters that use a common correction factor for all components of a tensor-valued antidiffusive flux, tensor limiters are designed using spectral decompositions. The new limiter functions are analyzed using tensorial extensions of the existing AFC theory for scalar convection-diffusion equations. The proposed methodology is backed by rigorous proofs of eigenvalue range preservation and Lipschitz continuity. Convergence of pseudo time-stepping methods to stationary solutions is demonstrated in numerical studies.

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.

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.

The role of low air pressure in the variation of negative corona‐generated space charge in a rod to plane electrode

The corona-generated space charge may vary in different air pressures, and the quantitative law between the corona-generated space charge density and air pressure has never been investigated. This work utilised computational approaches to explore the role of air pressure in the variation of negative corona-generated space charge. A negative fluid model in which the air pressure could be varied was built up to calculate the charged-particle density and electric field under different applied voltages. The continuity equations were solved by flux corrected transport algorithm and the Poisson equation was computed by the successive over relaxation method. The variations of important parameters due to air pressure were analysed. The particles densities of electrons, positive ions and negative ions at standard and low atmosphere pressures were computed, which increased markedly due to the decrease of atmosphere pressure. At fixed applied voltages, the exponential decay law between the maximal values of electron/positive ion densities at the head of the streamer and air pressure could be obtained. The quantitative law between the maximal values of electron/positive ion densities at the head of the streamer and the applied voltage, relative air density was also achieved.

Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems

The objective of this paper is to present a local bounds preserving stabilized finite element scheme for hyperbolic systems on unstructured meshes based on continuous Galerkin (CG) discretization in space. A CG semi-discrete scheme with low order artificial dissipation that satisfies the local extremum diminishing (LED) condition for systems is used to discretize a system of conservation equations in space. The low order artificial diffusion is based on approximate Riemann solvers for hyperbolic conservation laws. In this case we consider both Rusanov and Roe artificial diffusion operators. In the Rusanov case, two designs are considered, a nodal based diffusion operator and a local projection stabilization operator. The result is a discretization that is LED and has first order convergence behavior. To achieve high resolution, limited antidiffusion is added back to the semi-discrete form where the limiter is constructed from a linearity preserving local projection stabilization operator. The procedure follows the algebraic flux correction procedure usually used in flux corrected transport algorithms. To further deal with phase errors (or terracing) common in FCT type methods, high order background dissipation is added to the antidiffusive correction. The resulting stabilized semi-discrete scheme can be discretized in time using a wide variety of time integrators. Numerical examples involving nonlinear scalar Burgers equation, and several shock hydrodynamics simulations for the Euler system are considered to demonstrate the performance of the method. For time discretization, Crank–Nicolson scheme and backward Euler scheme are utilized.

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.

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.

Relativistic Vlasov-Maxwell modelling using finite volumes and adaptive mesh refinement

The dynamics of collisionless plasmas can be modelled by the Vlasov-Maxwell system of equations. An Eulerian approach is needed to accurately describe processes that are governed by high energy tails in the distribution function, but is of limited efficiency for high dimensional problems. The use of an adaptive mesh can reduce the scaling of the computational cost with the dimension of the problem. Here, we present a relativistic Eulerian Vlasov-Maxwell solver with block-structured adaptive mesh refinement in one spatial and one momentum dimension. The discretization of the Vlasov equation is based on a high-order finite volume method. A flux corrected transport algorithm is applied to limit spurious oscillations and ensure the physical character of the distribution function. We demonstrate a speed-up by a factor of 7 × in a typical scenario involving laser pulse interaction with an underdense plasma due to the use of an adaptive mesh.Graphical abstract