Discretization Of Source Terms Research Articles

Implementation of a high-order spatial discretization into a finite volume solver: Applications to turbomachinery test cases using an eddy-viscosity turbulence closure

In this study, the implementation of a high-order spatial discretization method into a Finite Volume solver is presented. Specific emphasis is put on the analysis of the performance over selected turbomachinery test cases. High-order numerical discretization is achieved by adopting the cell-centered Least-Square reconstruction, which is implemented in the in-house solver HybFlow. The validation of the adopted methodology is performed by assessing the solution of a turbulent flat plate with zero pressure gradient, using a eddy-viscosity transitional model. The test case also evidences the effect of the discretization of gradient-based source terms when a high-order reconstruction methodology is used. In the second part of the paper, the solver is used for the solution of relevant two-dimensional turbomachinery test cases, assessing the impact of 2nd and 3rd order reconstruction on the prediction of the aerodynamics and the heat transfer for respectively a low-pressure blade and a high-pressure turbine vane. It is shown how a high-order reconstruction allows for obtaining a better prediction of turbomachinery aerodynamics, with lower number of elements. The benefits over heat transfer predictions in high Reynolds number conditions are instead limited to the reduction of heat transfer coefficient spikes in under-resolved regions of the blade. Eventually, the methodology is validated for a three-dimensional low-pressure turbine cascade with realistic boundary layer inflow conditions.

GQL-based bound-preserving and locally divergence-free central discontinuous Galerkin schemes for relativistic magnetohydrodynamics

This paper develops novel and robust central discontinuous Galerkin (CDG) schemes of arbitrarily high-order accuracy for special relativistic magnetohydrodynamics (RMHD) with a general equation of state (EOS). These schemes are provably bound-preserving (BP), meaning they consistently preserve the upper bound for subluminal fluid velocity and the positivity of density and pressure, while also (locally) maintaining the divergence-free (DF) constraint for the magnetic field. For 1D RMHD, the standard CDG method is exactly DF, and its BP property is proven under a condition achievable by the BP limiter. For 2D RMHD, we design provably BP and locally DF CDG schemes based on the suitable discretization of a modified RMHD system, which is the relativistic analogue of Godunov's symmetrizable form of the non-relativistic MHD system (Godunov, 1972 [19]). A key novelty in our schemes is the meticulous discretization of additional source terms in the modified RMHD equations, so as to precisely counteract the influence of divergence errors on the BP property across overlapping meshes. Notably, we provide rigorous proofs of the BP property for our CDG schemes and first establish the theoretical connection between BP and discrete DF properties on overlapping meshes for RMHD. Owing to the absence of explicit expressions for primitive variables in terms of conserved variables, the constraints of physical bounds are strongly nonlinear, making the BP proofs highly nontrivial. We overcome these challenges through technical estimates within the geometric quasilinearization (GQL) framework (Wu and Shu, 2023 [49]), which equivalently converts the nonlinear constraints into linear ones. Furthermore, we introduce a new 2D cell average decomposition on overlapping meshes, which relaxes the theoretical BP CFL constraint and reduces the number of internal nodes, thereby enhancing the efficiency of the 2D BP CDG method. Finally, we implement the proposed CDG schemes for extensive RMHD problems with various EOSs, demonstrating their robustness and effectiveness in challenging scenarios like ultra-relativistic blasts and jets in strongly magnetized environments.

A fully well-balanced hydrodynamic reconstruction

Abstract The present work focuses on the numerical approximation of the weak solutions of the shallow water model over a non-flat topography. In particular, we pay close attention to steady solutions with nonzero velocity. The goal of this work is to derive a scheme that exactly preserves these stationary solutions, as well as the commonly preserved lake at rest steady solution. These moving steady states are solution to a nonlinear equation. We emphasize that the method proposed here never requires solving this nonlinear equation; instead, a suitable linearization is derived. To address this issue, we propose an extension of the well-known hydrostatic reconstruction. By appropriately defining the reconstructed states at the interfaces, any numerical flux function, combined with a relevant source term discretization, produces a well-balanced scheme that preserves both moving and non-moving steady solutions. This eliminates the need to construct specific numerical fluxes. Additionally, we prove that the resulting scheme is consistent with the homogeneous system on flat topographies, and that it reduces to the hydrostatic reconstruction when the velocity vanishes. To increase the accuracy of the simulations, we propose a well-balanced high-order procedure, which still does not require solving any nonlinear equation. Several numerical experiments demonstrate the effectiveness of the numerical scheme.

High-order compact gas-kinetic scheme for two-layer shallow water equations on unstructured mesh

For the two-layer shallow water equations, a high-order compact gas-kinetic scheme (GKS) on triangular mesh is proposed. The two-layer shallow water equations have complex source terms in comparison with the single layer equations. The main focus of this study is to construct a time-accurate evolution solution at a cell interface and to design a well-balanced scheme. The evolution model at a cell interface provides not only the numerical fluxes, but also the flow variables. In the development of the well-balanced scheme, the time-dependent flow variables at the cell interfaces surrounding the cell can be utilized to update the cell-averaged gradients. These time-evolving gradients are subsequently applied in the discretization of source terms inside each control volume. Based on the cell-averaged flow variables and their gradients, high-order initial data reconstruction can be achieved with compact stencils. The compact high-order GKS has advantages to simulate the flow evolution in complex domain covered by unstructured mesh. Many test cases are used to validate the accuracy and robustness of the scheme for the two-layer shallow water equations.

Revisiting the second-order convergence of the lattice Boltzmann method with reaction-type source terms

This study analyses an approach to consistently recover the second-order convergence of the lattice Boltzmann method (LBM), which is frequently degraded by an improper discretisation of the required source terms. The current work focuses on advection-diffusion models, in which the source terms are dependent on the intensity of transported fields. Such terms can be observed in reaction-type equations used in heat and mass transfer problems or multiphase flows. The investigated scheme is applicable to a wide range of formulations within the LBM framework. All considered source terms are interpreted as contributions to the zeroth-moment of the distribution function. These account for sources in a scalar field, such as density, concentration, temperature or a phase field. Further application of this work can be found in the method of manufactured solutions or in the immersed boundary method. This paper is dedicated to three aspects regarding proper inclusion of the source term in LBM schemes. Firstly, it identifies the differences observed between the ways in which source terms are included in the LBM schemes present in the literature. The algebraic manipulations are explicitly presented in this paper to clarify the observed differences, and to identify their origin. Secondly, it analyses in full detail, the implicit relation between the value of the transported macroscopic field, and the sum of the LBM densities. This relation is valid for any source term discretization scheme. It is a crucial ingredient for preserving the second-order convergence in the case of complex source terms. Moreover, three equivalent forms of the second-order accurate collision operator are presented. Finally, closed form solutions of this implicit relation are shown for a variety of common models, including general linear and second order terms; population growth models, such as the Logistic or Gompertz model and the Allen-Cahn equation. The second-order convergence of the proposed LBM schemes is verified on both linear and non-linear source terms. The pitfalls of the commonly used acoustic and diffusive scalings are identified and discussed. Furthermore, for a simplified case, the competing errors are shown visually with isolines of error in the space of spatial and temporal resolutions.

A unified consistent source term computational algorithm for the [formula omitted]-based compressible multi-fluid flow model

Using the criterion of one-fluid preservation, we developed a consistent algorithm for the source term to solve the γ-based compressible multi-fluid flow model with three approximate Riemann solvers, namely the Lax–Friedrichs (LxF), Kurganov, and Harten–Lax–van Leer contact (HLLC) solvers. The consistent algorithm comprises a standard Godunov solver with a high-order reconstruction and a consistent source term integration part. We prove that the present algorithm is consistent with Abgrall's criterion of the moving-material-interface property in the finite volume method framework. The cell boundary velocity for the source term discretization is found to be the same as HLLC solvers from five-equation model, but for the first time for LxF and Kurganov Riemann solvers. We develop 12 compressible multi-fluid solvers by combining four reconstruction schemes and three approximate Riemann solvers together with their consistent-source-term integration algorithms. All 12 solvers can maintain the one-fluid preservation and moving-material-interface properties numerically in several one- and two-dimensional example cases. The simulation of an underwater explosion demonstrates that a boundary-variation-diminishing reconstruction predicts an interface with width controlled within approximately three cells which is independent of the Riemann solver. The simulation of an explosion near a free surface demonstrates that the proposed model can simulate a severe compressible multi-fluid flow involving an interface across which there are large differences in density, pressure and parameters in the equation of state. In conclusion, the proposed consistent algorithm provides a unified framework for one kind of non-conservative hyperbolic system into a conservative hyperbolic system and a source term with velocity divergence, where the former can be computed by classical Godunov-type algorithms and the latter can be solved by the proposed consistent algorithm.

Discrete effects on the source term for the lattice Boltzmann modelling of one-dimensional reaction–diffusion equations

This work presents a detailed numerical analysis of one-dimensional, time-dependent (linear) reaction–diffusion type equations modelled with the lattice Boltzmann method (LBM), using the two-relaxation-time (TRT) scheme, for the D1Q3 lattice. The interest behind this study is twofold. First, because it applies to the description of many engineering problems, such as the mass transport in membranes, the heat conduction in fins, or the population growth in biological systems. Second, because this study also permits understanding the general effect of solution-dependent sources in LBM, where this problem offers a simple, yet non-trivial, canonical groundwork. Without recurring to perturbative techniques, such as the Chapman-Enskog expansion, we exactly derive the macroscopic numerical scheme that is solved by the LBM-TRT model with a solution-dependent source and show that it obeys a four-level explicit finite difference structure. In the steady-state limit, this scheme reduces to a second-order finite difference approximation of the stationary reaction–diffusion equation that, due to artefacts from the source term discretization, may operate with an effective diffusion coefficient of negative value, although still remaining stable. Such a surprising result is demonstrated through an exact stability analysis that proves the unconditional stability of the LBM-TRT model with a solution-dependent source, in line with the already proven source-less pure diffusion case (Lin et al., 2021). This proof enlarges the confidence over the LBM-TRT model robustness also for the (linear) reaction–diffusion problem class. Finally, a truncation error analysis is performed to disclose the structure of the leading order errors. From this knowledge, two strategies are proposed to improve the scheme accuracy from second- to fourth-order. One exclusively based on the tuning of the LBM-TRT scheme free-parameters, namely the two relaxation rates and the lattice weight coefficient, and the other based on the redefinition of the structure of the relaxation rates, where the leading order truncation error is absorbed into one of the relaxation rates, liberating the other to improve additional features of the scheme. Numerical tests presented in the last part of the work support the ensemble of theoretical findings.

Extending EB3 scheme for the differential conservation law from node-centered to cell-centered control volumes I: Basic formula on regular cells

The third-order edge-based (EB3) scheme for the differential conservation law is widely used on node-centered control volumes. It is much efficient and easy to implement, since the second derivatives for both fluxes and solutions are not required to compute and store, and the single quadrature point at per edge is enough to preserve the third-order solution accuracy on triangular grids. In this paper, the scheme is extended to the regular cell-centered control volume in two dimensions, and by analogy, it is named FB3, where “F” denotes the face of cell-centered elements. Compatible source term discretization is carefully derived, and based on the “collapsing-cell” idea, accuracy preserving boundary scheme is developed. More importantly, within the scope of FB3 scheme, the regularity of triangular grids is specifically discussed from the analytical and numerical perspectives. Numerical experiments based on the linear convective and Euler equations are implemented, and results indicate that the third-order solution accuracy is achieved at both internal and boundary fields, as long as the control volumes consist of regular triangles.

Extension of a Roe-type Riemann solver scheme to model non-hydrostatic pressure shallow flows

The aim of this work is, first of all, to extend a finite volume numerical scheme, previously designed for hydrostatic Shallow Water (SWE) formulation, to Non Hydrostatic Pressure (NHP) depth averaged model. The second objective is focused on exploring two available options in the context of previous work in this field: Hyperbolic-Elliptic (HE-NHP) formulations solved with a Pressure-Correction technique (PCM) and Hyperbolic Relaxation formulations (HR-NHP). Thus, besides providing an extension of a robust and well-proved Roe-type scheme developed for hydrostatic SWE to solve NHP systems, the work assesses the use of first order numerical schemes in the kind of phenomena typically solved with higher order methods. In particular, the relative performance and differences of both NHP numerical models are explored and analysed in detail. The performance of the models is compared using a steady flow test case with quasi-analytical solution and another unsteady case with experimental data, in which frequencies are analysed in experimental and computational results. The results highlight the need to understand the behaviour of a parameter-dependent model when using it as a prediction tool, and the importance of a proper discretization of non-hydrostatic source terms to ensure stability. On the other hand, it is proved that the incorporation of a non-hydrostatic model to a shallow water Roe solver provides good results.

A robust hybrid unstaggered central and Godunov-type scheme for Saint-Venant–Exner equations with wet/dry fronts

We aim to propose a robust hybrid unstaggered central and Godunov-type scheme based on hydrostatic reconstruction (HR) for Saint-Venant–Exner equations with wetting and drying transitions. The discretization of the bed slope source term is based on the HR method to preserve the still water steady-state solution. The hydrodynamic model described by the Saint-Venant system is numerically solved using the well-balanced unstaggered central scheme proposed in Dong and Li (2020). The morphodynamic model described by the Exner equation is numerically solved by using a Lax–Friedrichs method. In solving the morphodynamic model, we proposed a novel “cut-off” function to guarantee the nonlinear stability and preserve the stationary solution. The present scheme can exactly obtain the stationary solution and is capable of guaranteeing the positivity of the water depth. The key contributions of this work are the well-balanced property at the wet-dry fronts and the stability of the current scheme in solving two physical models that have relatively either strong or weak interactions. Finally, we use several classical problems of the system to demonstrate these properties.

On the Verification of the Pedestrian Evacuation Model

In this article we deal with numerical solution of macroscopic models of pedestrian movement. From a macroscopic point of view, pedestrian movement can be described by a system of first order hyperbolic equations similar to 2D compressible inviscid flow. For the Pedestrian Flow Equations (PFEs) the density ρ and the velocity v are considered as the unknown variables. In PFEs, the social force is also taken into account, which replaces the outer volume force term used in the fluid flow formulation, e.g., the pedestrian movement is influenced by the proximity of other pedestrians. To be concrete, the desired direction μ of the pedestrian movement is density dependent and is incorporated in the source term. The system of fluid dynamics equations is thus coupled with the equation for μ. The main message of this paper is the verification of this model. Firstly, we propose two approaches for the source term discretization. Secondly, we propose two splitting schemes for the numerical solution of the coupled system. This leads us to four different numerical methods for the PFEs. The novelty of this work is the comparative study of the numerical solutions, which shows, that all proposed methods are in the good agreement.

A well-balanced ADER discontinuous Galerkin method based on differential transformation procedure for shallow water equations

This article develops a new discontinuous Galerkin (DG) method on structured meshes for solving shallow water equations. The method here applies the one-stage ADER (Arbitrary DERivatives in time and space) approach for the temporal discretization and employs the differential transformation procedure to express spatiotemporal expansion coefficients of the solution through low order spatial expansion coefficients recursively. Numerical fluxes using the hydrostatic reconstruction together with a simple source term discretization results in a well-balanced method accordingly. Compared with the Runge-Kutta DG (RKDG) methods, the proposed method needs less computer memory storage due to no intermediate stages. In comparison with traditional ADER schemes, this method avoids solving the generalized Riemann problems at inter-cells. The differential transformation procedure used here is cheaper than the Cauchy-Kowalewski procedure in traditional ADER schemes. In summary, this method is one-step, one-stage, fully-discrete, and easily achieves arbitrary high order accuracy in time and space. Theoretically as well as numerically, the proposed method is verified to be well-balanced. Numerical results illustrate the high order accuracy as well as good resolutions for discontinuous solutions. Moreover, this method is more efficient than the Runge-Kutta DG (RKDG) method.

Well‐balanced nonstaggered central schemes based on hydrostatic reconstruction for the shallow water equations with Coriolis forces and topography

The objective of this paper is to present a nonstaggered central scheme based on hydrostatic reconstruction (HR) for the shallow water equations with Coriolis forces and nonflat bottom topography. The discretization of the bed slope source term is based on the HR method to ensure the well‐balanced property for the stationary solution. We use the cell average of the discharge on the staggered cell to discretize the source term due to the effect of Coriolis forces. The positivity‐preserving property of the water depth is obtained by providing an appropriate CFL condition with the help of a “draining” time‐step technique. Finally, several two‐dimensional numerical results of classical problems of the system are presented to test these properties of the current scheme.

A steady-state-preserving scheme for shallow water flows in channels

A steady-state-preserving numerical model is developed for the shallow water equations in channels in the current study. The proposed numerical model is based on the finite volume method and capable to preserve both the moving-water and still-water steady states. In order to preserve the general steady states, the source terms in the momentum equation are incorporated into a global flux term; and the new momentum equation with global flux term is then relaxed in order to avoid the non-linearity in the computation. One can thus develop an upwind/modified-HLL hybrid Riemann solver to estimate the fluxes in different flow regimes. The developed numerical model can preserve the general steady states, and one avoids: (1) non-trivial root-finding for point values of wetted cross-sectional area A at cell interfaces; (2) complex discretization of geometric source terms incorporating artificial conservation corrections. The developed numerical model has second order accuracy in both space and time. Numerical solutions are presented to demonstrate that the proposed numerical model is capable to exactly preserve both moving-water and still-water steady states.

A novel well-balanced scheme for spatial and temporal bed evolution in rapidly varying flow

This paper presents a novel method of preserving well-balanced conditions for dam-break flows on irregular beds. The research on preservation of well-balanced conditions commonly deals only with spatial variation of beds. It is challenging to design a well-balanced scheme that can model both spatial and temporal variation of beds particularly on domains discretized with unstructured meshes. This challenge arises due to the non-trivial nature of the governing equations modeling hydro-morpho dynamics. This study proposes a new and simple well-balanced method applicable for both spatial and temporal bed variations without requiring additional treatment and complicated discretization of slope source terms. The work relies on a cell-centered Godunov-type finite volume method for solving coupled Saint Venant and sediment continuity equations. The two most important issues, i.e., the ‘lake at rest condition’ and ‘water depth positivity’ of shallow-water flows over irregular beds, are tested and verified in the proposed numerical scheme. The authors employed Riemann solvers to compute fluxes through the control volume surfaces. The proposed scheme is evaluated for various benchmark cases of dam-break flows on rigid and erodible beds. Comparisons between the results of the proposed numerical scheme, analytical solution, experimental data, and previous numerical observations show significant improvement for some cases and a good agreement in the rest.

A depth-averaged non-cohesive sediment transport model with improved discretization of flux and source terms

This paper presents novel flux and source term treatments within a Godunov-type finite volume framework for predicting the depth-averaged shallow water flow and sediment transport with enhanced the accuracy and stability. The suspended load ratio is introduced to differentiate between the advection of the suspended load and the advection of water. A modified Harten, Lax and van Leer Riemann solver with the contact wave restored (HLLC) is derived for the flux calculation based on the new wave pattern involving the suspended load ratio. The source term calculation is enhanced by means of a novel splitting-point implicit discretization. The slope effect is introduced by modifying the critical shear stress, with two treatments being discussed. The numerical scheme is tested in five examples that comprise both fixed and movable beds. The model predictions show good agreement with measurement, except for cases where local three-dimensional effects dominate.

Study on divergence approximation formula for pressure calculation in particle method

The moving particle semi-implicit method is a meshless particle method for incompressible fluid and has proven useful in a wide variety of engineering applications of free-surface flows. Despite its wide applicability, the moving particle semi-implicit method has the defects of spurious unphysical pressure oscillation. Three various divergence approximation formulas, including basic divergence approximation formula, difference divergence approximation formula, and symmetric divergence approximation formula are proposed in this paper. The proposed three divergence approximation formulas are then applied for discretization of source term in pressure Poisson equation. Two numerical tests, including hydrostatic pressure problem and dam-breaking problem, are carried out to assess the performance of different formulas in enhancing and stabilizing the pressure calculation. The results demonstrate that the pressure calculated by basic divergence approximation formula and difference divergence approximation formula fluctuates severely. However, application of symmetric divergence approximation formula can result in a more accurate and stabilized pressure.

Comment on “An Efficient and Stable Hydrodynamic Model With Novel Source Term Discretization Schemes for Overland Flow and Flood Simulations” by Xilin Xia et al.

AbstractXia et al. (2017) proposed a novel, fully implicit method for the discretization of the bed friction terms for solving the shallow‐water equations. The friction terms contain h−7∕3 (h denotes water depth), which may be extremely large, introducing machine error when h approaches zero. To address this problem, Xia et al. (2017) introduces auxiliary variables (their equations (37) and (38)) so that h−4∕3 rather than h−7∕3 is calculated and solves a transformed equation (their equation (39)). The introduced auxiliary variables require extra storage. We implemented an analysis on the magnitude of the friction terms to find that these terms on the whole do not exceed the machine floating‐point number precision, and thus we proposed a simple‐to‐implement technique by splitting h−7∕3 into different parts of the friction terms to avoid introducing machine error. This technique does not need extra storage or to solve a transformed equation and thus is more efficient for simulations. We also showed that the surface reconstruction method proposed by Xia et al. (2017) may lead to predictions with spurious wiggles because the reconstructed Riemann states may misrepresent the water gravitational effect.

Reply to Comment by Lu et al. on “An Efficient and Stable Hydrodynamic Model With Novel Source Term Discretization Schemes for Overland Flow and Flood Simulations”

AbstractThis document addresses the comments raised by Lu et al. (2017). Lu et al. (2017) proposed an alternative numerical treatment for implementing the fully implicit friction discretization in Xia et al. (2017). The method by Lu et al. (2017) is also effective, but not necessarily easier to implement or more efficient. The numerical wiggles observed by Lu et al. (2017) do not affect the overall solution accuracy of the surface reconstruction method (SRM). SRM introduces an antidiffusion effect, which may also lead to more accurate numerical predictions than hydrostatic reconstruction (HR) but may be the cause of the numerical wiggles. As suggested by Lu et al. (2017), HR may perform equally well if fine enough grids are used, which has been investigated and recognized in the literature. However, the use of refined meshes in simulations will inevitably increase computational cost and the grid sizes as suggested are too small for real‐world applications.

Assessment of USM3D Hierarchical Adaptive Nonlinear Method Preconditioners for Three-Dimensional Cases

Enhancements to the previously reported hierarchical adaptive nonlinear iteration methods implemented in a mixed-element cell-centered framework have been made to improve robustness, efficiency, and accuracy of computational-fluid-dynamic simulations. The key enhancements include a multicolor line-implicit preconditioner, a discretely consistent symmetry boundary condition, and a line-mapping method for the turbulence source term discretization. The Reynolds-averaged Navier–Stokes solutions have been computed using a Spalart–Allmaras turbulence model and families of uniformly refined nested grids on two three-dimensional configurations, namely, a bump in a channel and a hemisphere cylinder. Iterative convergence of two hierarchical adaptive nonlinear iteration method approaches using line- and point-implicit preconditioners has been compared with the iterative convergence of the point-implicit preconditioner-alone method that broadly represents the baseline solver technology. The line-implicit hierarchical adaptive nonlinear iteration method shows superior iterative convergence with progressively increasing benefits on finer grids.