An improved global-direction stencil based on the face-area-weighted centroid for the gradient reconstruction of unstructured finite volume methods

The accuracy of unstructured finite volume methods is greatly influenced by the gradient reconstruction, for which the stencil selection plays a critical role. Compared with the commonly used face-neighbor and vertex-neighbor stencils, the global-direction stencil is independent of the mesh topology, and characteristics of the flow field can be well reflected by this novel stencil. However, for a high-aspect-ratio triangular grid, the grid skewness is evident, which is one of the most important grid-quality measures known to affect the accuracy and stability of finite volume solvers. On this basis and inspired by an approach of using face-area-weighted centroid to reduce the grid skewness, we explore a method by combining the global-direction stencil and face-area-weighted centroid on high-aspect-ratio triangular grids, so as to improve the computational accuracy. Four representative numerical cases are simulated on high-aspect-ratio triangular grids to examine the validity of the improved global-direction stencil. Results illustrate that errors of this improved methods are the lowest among all methods we tested, and in high-mach-number flow, with the increase of cell aspect ratio, the improved global-direction stencil always has a better stability than commonly used face-neighbor and vertex-neighbor stencils. Therefore, the computational accuracy as well as stability is greatly improved, and superiorities of this novel method are verified.

Finite Volume Methods for Solving Electromagnetic Problems in Nonhomogeneous Media

This article shall review some of the recent advances on finite volume methods for solving electromagnetic problems in nonhomogeneous media. The stability, convergence and applications of the finite volume methods will be discussed.

High order finite volume methods for singular perturbation problems

In this paper we establish a high order finite volume method for the fourth order singular perturbation problems.In conjunction with the optimal meshes,the numerical solutions resulting from the method have optimal convergence order.Numerical experiments are presented to verify our theoretical estimates.

Solving Two-Mode Shallow Water Equations Using Finite Volume Methods

In this paper,we develop and study numerical methods for the two-mode shallow water equations recently proposed in[S.STECHMANN,A.MAJDA,and B.KHOUIDER,Theor.Comput.Fluid Dynamics,22(2008),pp.407–432].Designing a reliable numerical method for this system is a challenging task due to its conditional hyperbolicity and the presence of nonconservative terms.We present several numerical approaches—two operator splitting methods(based on either Roe-type upwind or central-upwind scheme),a central-upwind scheme and a path-conservative centralupwind scheme—and test their performance in a number of numerical experiments.The obtained results demonstrate that a careful numerical treatment of nonconservative terms is crucial for designing a robust and highly accurate numerical method。

Finite Volume Methods for Wave Propagation in Stratified Magneto-Atmospheres

We present a model for simulating wave propagation in stratified magnetoatmospheres. The model is based on equations of ideal MHD together with gravitational source terms. In addition, we present suitable boundary data and steady statesto model wave propagation. A finite volume framework is developed to simulate thewaves. The framework is based on HLL and Roe type approximate Riemann solversfor numerical fluxes, a positivity preserving fractional steps method for discretizingthe source and modified characteristic and Neumann type numerical boundary conditions. Second-order spatial and temporal accuracy is obtained by using an ENO piecewise linear reconstruction and a stability preserving Runge-Kutta method respectively.The boundary closures are suitably modified to ensure mass balance. The numericalframework is tested on a variety of test problems both for hydrodynamic as well asmagnetohydrodynamic configurations. It is observed that only suitable choices of HLLsolvers for the numerical fluxes and balanced Neumann type boundary closures yieldstable results for numerical wave propagation in the presence of complex magneticfields.

MixedMultiscale Finite Volume Methods for Elliptic Problems in Two-Phase Flow Simulations

We develop a framework for constructing mixed multiscale finite volume methods for elliptic equations with multiple scales arising from flows in porous media.Some of the methods developed using the framework are already known[20];others are new.New insight is gained for the known methods and extra flexibility is provided by the new methods.We give as an example a mixed MsFV on uniform mesh in 2-D.This method uses novel multiscale velocity basis functions that are suited for using global information,which is often needed to improve the accuracy of the multiscale simulations in the case of continuum scales with strong non-local features.The method efficiently captures the small effects on a coarse grid.We analyze the new mixed MsFV and apply it to solve two-phase flow equations in heterogeneous porous media.Numerical examples demonstrate the accuracy and efficiency of the proposed method for modeling the flows in porous media with non-separable and separable scales.

Cell Conservative Flux Recovery and A Posteriori Error Estimate of Vertex-Centered Finite Volume Methods

A cell conservative flux recovery technique is developed here for vertexcentered finite volume methods of second order elliptic equations.It is based on solving a local Neumann problem on each control volume using mixed finite element methods.The recovered flux is used to construct a constant free a posteriori error estimator which is proven to be reliable and efficient.Some numerical tests are presented to confirm the theoretical results.Our method works for general order finite volume methods and the recovery-based and residual-based a posteriori error estimators is the first result on a posteriori error estimators for high order finite volume methods.

A simple algorithm to improve the performance of the WENO scheme on non-uniform grids

This paper presents a simple approach for improving the performance of the weighted essentially nonoscillatory(WENO) finite volume scheme on non-uniform grids. This technique relies on the reformulation of the fifthorder WENO-JS(WENO scheme presented by Jiang and Shu in J. Comput. Phys. 126:202–228, 1995) scheme designed on uniform grids in terms of one cell-averaged value and its left and/or right interfacial values of the dependent variable.The effect of grid non-uniformity is taken into consideration by a proper interpolation of the interfacial values. On nonuniform grids, the proposed scheme is much more accurate than the original WENO-JS scheme, which was designed for uniform grids. When the grid is uniform, the resulting scheme reduces to the original WENO-JS scheme. In the meantime,the proposed scheme is computationally much more efficient than the fifth-order WENO scheme designed specifically for the non-uniform grids. A number of numerical test cases are simulated to verify the performance of the present scheme.

Extending the global-direction stencil with“face-area-weighted centroid”to unstructured finite volume discretization from integral formfinitevolumediscretization from integral form

Accuracy of unstructured finite volume discretization is greatly influenced by the gradient reconstruction.For the commonly used k-exact reconstruction method,the cell centroid is always chosen as the reference point to formulate the reconstructed function.But in some practical problems,such as the boundary layer,cells in this area are always set with high aspect ratio to improve the local field resolution,and if geometric centroid is still utilized for the spatial discretization,the severe grid skewness cannot be avoided,which is adverse to the numerical performance of unstructured finite volume solver.In previous work[Kong,et al.Chin Phys B 29(10):100203,2020],we explored a novel global-direction stencil and combined it with the face-area-weighted centroid on unstructured finite volume methods from differential form to realize the skewness reduction and a better reflection of flow anisotropy.Greatly inspired by the differential form,in this research,we demonstrate that it is also feasible to extend this novel method to the unstructured finite volume discretization from integral form on both second and third-order finite volume solver.Numerical examples governed by linear convective,Euler and Laplacian equations are utilized to examine the correctness as well as effectiveness of this extension.Compared with traditional vertex-neighbor and face-neighbor stencils based on the geometric centroid,the grid skewness is almost eliminated and computational accuracy as well as convergence rate is greatly improved by the global-direction stencil with face-area-weighted centroid.As a result,on unstructured finite volume discretization from integral form,the method also has superiorities on both computational accuracy and convergence rate.

The Roper resonance in a finite volume

We calculate the energy levels corresponding to the Roper resonance based on a two-flavor chiral effective Lagrangian for pions,nucleons,deltas,and the Roper resonance at the leading one-loop order.We show that the Roper mass can be extracted from these levels for lattice volumes of moderate size.

A Discontinuous Galerkin Extension of the Vertex-Centered Edge-Based Finite Volume Method

The finite volume(FV)method is the dominating discretization technique for computational fluid dynamics(CFD),particularly in the case of compressible fluids.The discontinuous Galerkin(DG)method has emerged as a promising highaccuracy alternative.The standard DG method reduces to a cell-centered FV method at lowest order.However,many of today’s CFD codes use a vertex-centered FV method in which the data structures are edge based.We develop a new DG method that reduces to the vertex-centered FV method at lowest order,and examine here the new scheme for scalar hyperbolic problems.Numerically,the method shows optimal-order accuracy for a smooth linear problem.By applying a basic hp-adaption strategy,the method successfully handles shocks.We also discuss how to extend the FV edge-based data structure to support the new scheme.In this way,it will in principle be possible to extend an existing code employing the vertex-centered and edge-based FV discretization to encompass higher accuracy through the new DG method.

A Nominally Second-Order Cell-Centered Finite Volume Scheme for Simulating Three-Dimensional Anisotropic Diffusion Equations on Unstructured Grids

We present a finite volume based cell-centered method for solving diffusion equations on three-dimensional unstructured grids with general tensor conduction.Our main motivation concerns the numerical simulation of the coupling between fluid flows and heat transfers.The corresponding numerical scheme is characterized by cell-centered unknowns and a local stencil.Namely,the scheme results in a global sparse diffusion matrix,which couples only the cell-centered unknowns.The space discretization relies on the partition of polyhedral cells into sub-cells and on the partition of cell faces into sub-faces.It is characterized by the introduction of sub-face normal fluxes and sub-face temperatures,which are auxiliary unknowns.A sub-cellbased variational formulation of the constitutive Fourier law allows to construct an explicit approximation of the sub-face normal heat fluxes in terms of the cell-centered temperature and the adjacent sub-face temperatures.The elimination of the sub-face temperatures with respect to the cell-centered temperatures is achieved locally at each node by solving a small and sparse linear system.This system is obtained by enforcing the continuity condition of the normal heat flux across each sub-cell interface impinging at the node under consideration.The parallel implementation of the numerical algorithm and its efficiency are described and analyzed.The accuracy and the robustness of the proposed finite volume method are assessed by means of various numerical test cases.

Quasi-Lagrangian Acceleration of Eulerian Methods

We present a simple and efficient strategy for the acceleration of explicit Eulerian methods for multidimensional hyperbolic systems of conservation laws.The strategy is based on the Galilean invariance of dynamic equations and optimization of the reference frame,in which the equations are numerically solved.The optimal reference frame moves(locally in time)with the average characteristic speed of the system,and,in this sense,the resulting method is quasi-Lagrangian.This leads to the acceleration of the numerical computations thanks to the optimal CFL condition and automatic adjustment of the computational domain to the evolving part of the solution.We show that our quasi-Lagrangian acceleration procedure may also reduce the numerical dissipation of the underlying Eulerian method.This leads to a significantly enhanced resolution,especially in the supersonic case.We demonstrate a great potential of the proposed method on a number of numerical examples.

NumericalMethods for Balance Laws with Space Dependent Flux:Application to Radiotherapy Dose Calculation

The present work is concerned with the derivation of numerical methods to approximate the radiation dose in external beam radiotherapy.To address this issue,we consider a moment approximation of radiative transfer,closed by an entropy minimization principle.The model under consideration is governed by a system of hyperbolic equations in conservation form supplemented by source terms.The main difficulty coming from the numerical approximation of this system is an explicit space dependence in the flux function.Indeed,this dependence will be seen to be stiff and specific numerical strategiesmust be derived in order to obtain the needed accuracy.A first approach is developed considering the 1D case,where a judicious change of variables allows to eliminate the space dependence in the flux function.This is not possible in multi-D.We therefore reinterpret the 1D scheme as a scheme on two meshes,and generalize this to 2D by alternating transformations between separate meshes.We call this procedure projection method.Several numerical experiments,coming from medical physics,illustrate the potential applicability of the developed method.

Interaction between the atmospheric boundary layer and a standalone wind turbine in Gansu—Part II： Numerical analysis

To analyze the interaction between wind turbines and the atmospheric boundary layer, we integrated a large-eddy simulation with an actuator line model and examined the characteristics of wind-turbine loads and wakes with reference to a corresponding experiment in Gansu. In the simulation, we set the wind turbine to have a rotor diameter of 14.8 m and a tower height of 15.4 m in the center of an atmospheric boundary layer with a 10.6° yaw angle. The results reveal an obviously skewed wake structure behind the rotor due to the thrust component normal to the flow direction. The power spectra of the inflow fluctuation velocity exhibit a region of-5/3 slope, which confirms the ability of large-eddy simulations to reproduce the energy cascade from larger to smaller scales. We found there to be more energy in the power spectrum of the axial velocity, which shows that coherent turbulence structures have more energy in the horizontal direction. By the conjoint analysis of atmospheric turbulence and windturbine loads, we found that when the inflow wind direction changes rapidly, the turbulence kinetic energy and coherent turbulence kinetic energy in the atmospheric turbulence increase, which in turn causes fluctuations in the wind turbine load.Furthermore, anisotropic atmospheric turbulence causes an asymmetric load cycle, which imposes a strike by the turbine blade on the shaft, thereby increasing the fatigue load on the shaft. Our main conclusion is that the atmospheric boundary layer has a strong effect on the evolution of the wake and the structural response of the turbine.

Numerical simulations of high enthalpy flows around entry bodies

Ablation flows around entry bodies are at highly nonequilibrium states. This paper pre- sents comprehensive computational fluid dynamics simulations of such hypersonic flow examples. The computational scheme adopted in this study is based on the Navie-Stokes equations, and it is capable of simulating multiple-dimensional, non-equilibrium, chemically reacting gas flows with multiple species. Finite rate chemical reactions, multiple temperature relaxation processes, and ion- izations phenomena with electrons are modeled. Simulation results of several hypersonic gas flows over axisymmetric bodies are presented and compared with results in the literature. It confirms that some past treatment of adopting less species for hypersonic flows is acceptable, and the differences from more species and more chemical reactions are not significant.

On a Shallow Water Model for the Simulation of Turbidity Currents

We present a model for hyperpycnal plumes or turbidity currents that takes into account the interaction between the turbidity current and the bottom,considering deposition and erosion effects as well as solid transport of particles at the bed load due to the current.Water entrainment from the ambient water in which the turbidity current plunges is also considered.Motion of ambient water is neglected and the rigid lid assumption is considered.The model is obtained as a depth-average system of equations under the shallow water hypothesis describing the balance of fluid mass,sediment mass and mean flow.The character of the system is analyzed and numerical simulations are carried out using finite volume schemes and path-conservative Roe schemes.

Numerical Approximation of a Reaction-Diffusion System with Fast Reversible Reaction

The authors consider the finite volume approximation of a reaction-diffusion system with fast reversible reaction.It is deduced from a priori estimates that the approximate solution converges to the weak solution of the reaction-diffusion problem and satisfies estimates which do not depend on the kinetic rate.It follows that the solution converges to the solution of a nonlinear diffusion problem,as the size of the volume elements and the time steps converge to zero while the kinetic rate tends to infinity.