Performance and Comparison of Cell-Centered and Node-Centered Unstructured Finite Volume Discretizations for Shallow Water Free Surface Flows

Finite volume (FV) methods for solving the two-dimensional (2D) nonlinear shallow water equations (NSWE) with source terms on unstructured, mostly triangular, meshes are known for some time now. There are mainly two basic formulations of the FV method: node-centered (NCFV) and cell-centered (CCFV). In the NCFV formulation the finite volumes, used to satisfy the integral form of the equations, are elements of the mesh dual to the computational mesh, while for the CCFV approach the finite volumes are the mesh elements themselves. For both formulations, details are given of the development and application of a second-order well-balanced Godunov-type scheme, developed for the simulation of unsteady 2D flows over arbitrary topography with wetting and drying. The popular approximate Riemann solver of Roe is utilized to compute the numerical fluxes, while second-order spatial accuracy is achieved with a MUSCL-type reconstruction technique. The Green-Gauss (G-G) formulation for gradient computations is implemented for both formulations, in order to maintain a common framework. Two different stencils for the G-G gradient computations in the CCFV formulation are implemented and tested. An edge-based limiting procedure is applied for the control of the total variation of the reconstructed field. This limiting procedure is proved to be effective for the NCFV scheme but inadequate for the CCFV approach. As such, a simple but very effective modification to the reconstruction procedure is introduced that takes into account geometrical characteristics of the computational mesh. In addition, consistent well-balanced second-order discretizations for the topography source term treatment and the wet/dry front treatment are presented for both FV formulations, ensuring absolute mass conservation, along with a stable friction term treatment.     

一类Lagrange坐标系下的ENO有限体积格式

本文首先从积分形式的二维Lagrange流体力学方程组出发,使用ENO高阶插值多项式,推广了四边形结构网格下的一阶有限体积格式,构造得到了一类结构网格下的高精度有限体积格式.该格式针对单介质问题具有良好的计算效果,同时在处理多介质问题时,不会产生物质界面附近强烈的震荡.结合有效的守恒重映方法,用ALE方法进行数值模拟,得到了预期的效果.     

Application of Second Order TVD and WENO Schemes in Internal Aerodynamics

We deal with the comparison of several finite volume TVD schemes and finite difference ENO schemes and we describe a second order finite volume WENO scheme which was developed for the case of general unstructured meshes. The proposed second order WENO reconstruction is much simpler than the original ENO scheme introduced in [Harten and Chakravarthy 1991]. Moreover, the proposed WENO method is very easily extendible for unstructured meshes in 3D. All above mentioned schemes are applied for the solution of 2D and 3D transonic flows in the turbines and channels and the numerical solution is compared to experimental results or to the results obtained by other authors.     

不可压缩流两重稳定有限体积算法应用研究

通过将局部高斯积分稳定化方法和两重网格算法思想紧密结合,提出了粘性不可压缩流体的两重稳定有限体积算法。将该算法的三种迭代格式进行了效率的分析比较。理论分析和数值实验发现：当粗、细网格尺度比例选择适当时,两重算法与传统算法具有相同精度解的同时,效率大大提高;对不同格式的两重有限体积算法进行比较分析发现：Simple格式计算效率最高,Picard格式次之,Newton格式较低。     

Analysis of three stabilized finite volume iterative methods for the steady Navier–Stokes equations

In this work, three stabilized finite volume iterative schemes for the stationary Navier–Stokes equations are considered. Under the finite volume discretization at each iterative step, the iterative scheme I consists in solving the steady Stokes problem, iterative scheme II consists in solving the stationary linearized Navier–Stokes equations and iterative scheme III consists in solving the steady Oseen equations, respectively. We discuss the stabilities and convergence of three iterative methods. The iterative schemes I and II are stable and convergent under some strong uniqueness conditions, while iterative scheme III is unconditionally stable and convergent under the uniqueness condition. Finally, some numerical results are presented to verify the performance of these iterative schemes.     

Numerical Convergence of a Parameterisation Method for the Solution of a Highly Anisotropic Two-Dimensional Elliptic Problem

Highly anisotropic two-dimensional elliptic problems lead to severe numerical difficulties. In this paper, starting from a simple finite volume scheme, we present a parameterisation method that allows us to obtain the solution even if the anisotropy ratio is very large. We derive a formal asymptotic limit of the two-dimensional anisotropic problem, in the case where the anisotropy ratio goes to infinity. This formal limit is used as a reference, and we show that the parameterisation method gives similar results, whereas the finite volume scheme fails to give an accurate solution. Numerical results are given, which indicate important parameters to be considered in order to obtain a good precision     

A time semi-exponentially fitted scheme for chemotaxis-growth models

In this work, we develop a new linearized implicit finite volume method for chemotaxis-growth models. First, we derive the scheme for a simplified chemotaxis model arising in embryology. The model consists of two coupled nonlinear PDEs: parabolic convection-diffusion equation with a logistic source term for the cell-density, and an elliptic reaction-diffusion equation for the chemical signal. The numerical approximation makes use of a standard finite volume scheme in space with a special treatment for the convection-diffusion fluxes which are approximated by the classical Il’in fluxes. For the time discretization, we introduce our linearized semi-exponentially fitted scheme. The paper gives a comparison between the proposed scheme and different versions of linearized backward Euler schemes. The existence and uniqueness of a numerical solution to the scheme and its convergence to a weak solution of the studied system are proved. In the last section, we present some numerical tests to show the performance of our method. Our numerical approach is then applied to a chemotaxis-growth model describing bacterial pattern formation.     

A Family of Fourth-Order and Sixth-Order Compact Difference Schemes for the Three-Dimensional Poisson Equation

In this paper a family of fourth-order and sixth-order compact difference schemes for the three dimensional (3D) linear Poisson equation are derived in detail. By using finite volume (FV) method for derivation, the highest-order compact schemes based on two different types of dual partitions are obtained. Moreover, a new fourth-order compact scheme is gained and numerical experiments show the new scheme is much better than other known fourth-order schemes. The outline for the nonlinear problems are also given. Numerical experiments are conducted to verify the feasibility of this new method and the high accuracy of these fourth-order and sixth-order compact difference scheme.     

Towards a more consistent discretization scheme for adaptive implicit RHD computations

We present a new implicit numerical discretization for the equations of radiation hydrodynamics (RHD) which is based on a more geometrical representation of a finite volume scheme suitable for spherical systems. In particular, the motion of the grid points is directly included by appropriate volume changes. Several examples illustrate the accuracy gained by this improved difference scheme.     

A finite difference scheme on nonuniform grids

In the present work, we introduce a finite difference scheme on an nonuniform grid. The truncation errors introduced by the use of this difference scheme is presented. It is shown that the numerical solution in the physical domain on nonuniform grids has some advantages. Finally, we solve some boundary value problems using the introduced scheme and compare the obtained results with that obtained on an uniform grid.     

Curvilinear grids for WENO methods in astrophysical simulations

We investigate the applicability of curvilinear grids in the context of astrophysical simulations and WENO schemes. With the non-smooth mapping functions from Calhoun et al. (2008), we can tackle many astrophysical problems which were out of scope with the standard grids in numerical astrophysics. We describe the difficulties occurring when implementing curvilinear coordinates into our WENO code, and how we overcome them. We illustrate the theoretical results with numerical data. The WENO finite difference scheme works only for high Mach number flows and smooth mapping functions, whereas the finite volume scheme gives accurate results even for low Mach number flows and on non-smooth grids.     

反应流模拟的有限体积法的比较

针对自催化反应流模型的计算,推导了基于有限体积方法的统一通量格式以及十种常用格式的具体形式,并通过数值实验比较了其数值特性。结果表明：无论是一阶精度的迎风格式和Lax-Friedrichs格式,二阶精度的二阶向前差分、Lax-Wendroff、Beam-Warming和Fromm格式还是三阶精度的QUICK格式都会引起较严重的数值耗散和数值震荡,严重降低了数值精度,而带有通量限制器的MTVDLF格式可以消除数值耗散和数值震荡,并且带有Superbee限制器的MTVDLF最适合模拟自催化反应流问题。     

Physical-bound-preserving finite volume methods for the Nagumo equation on distorted meshes

In this paper, we present a boundedness preserving finite volume scheme for the Nagumo equation. In this method, we use the implicit Euler method for the time discretization, and construct a maximum-principle-preserving discrete normal flux for the diffusion term. For the nonlinear reaction term, we design a type of Picard iteration to ensure that at each iterative step it keeps physical boundedness. Moreover we prove that the numerical solution of the resulting scheme can preserve the bound of the solution for the Nagumo equation on distorted meshes. Some numerical results are presented to verify the theoretical analysis.     

A variational formulation for exterior problems in linear hydrodynamics

This paper describes a method of coupling between finite elements and integral representation, where the numerical scheme is obtained by means of a finite element discretization of a continuous variational problem. A numerical study of the accuracy of this method precedes its application to two classical naval hydrodynamics problems, and we show that the results are very accurate even with a small number of elements.     

Augmented Riemann solver for urethra flow modeling

In this paper we propose a new conservative numerical scheme for the male urethra and ureter fluid flow simulations. We use finite volume method based on technique of augmented system. Main goal is to construct such conservative scheme which maintains not only some special steady states but all possible ones. Furthermore this scheme should preserve non-negativity of essentially nonnegative quantities from their physical fundamental (here it is the cross-section of the urethra and ureter). Our scheme can be also modified to the high order scheme. At the end we present some numerical experiments.     

求解带有间断系数泊松方程的修正有限体积

利用修正的有限体积方法求解带有间断系数的泊松方程,改进是对基于笛卡尔坐标系下的调和平均系数进行的。数值实验表明新格式二阶逐点收敛并且在界面处具有二阶精度,新方法较已有的求解不连续扩散系数的算术平均法和调和平均法,特别是在系数跳跃较大的情况下更具优势。     

Local Discontinuous Galerkin Finite Element Method and Error Estimates for One Class of Sobolev Equation

In this paper we present a numerical scheme based on the local discontinuous Galerkin (LDG) finite element method for one class of Sobolev equations, for example, generalized equal width Burgers equation. The proposed scheme will be proved to have good numerical stability and high order accuracy for arbitrary nonlinear convection flux, when time variable is continuous. Also an optimal error estimate is obtained for the fully discrete scheme, when time is discreted by the second order explicit total variation diminishing (TVD) Runge-Kutta time-marching. Finally some numerical results are given to verify our analysis for the scheme.     

Equivolumetric partition of solid spheres with applications to orientation workspace analysis of robot manipulators

Orientation workspace analysis is a critical issue in the design of robot manipulators, especially the spherical manipulators. However, there is a lack of effective methods for such analysis, because the orientation workspace of a robot manipulator is normally a subset of SO(3) (the special orthogonal group) with a complex boundary. Numerical approaches appear more practical in actual implementations. For numerical analysis, a finite partition of the orientation workspace in its parametric domain is necessary. It has been realized that the exponential coordinates parameterization is more appropriate for finite partition. With such a parameterization, the rigid body rotation group, i.e., SO(3), can be mapped to a solid sphere D/sup 3/ of radius /spl pi/ with antipodal points identified. A novel partition scheme is proposed to geometrically divide the parametric domain, i.e., the solid sphere D/sup 3/ of radius /spl pi/, into finite elements with equal volume. Subsequently, the volume of SO(3) can be numerically computed as a weighted volume sum of the equivolumetric elements, in which the weightages are the element-associated integration measures. In this way, we can simplify the partition scheme and also reduce the computation efforts, as the elements in the same partition layer (along the radial direction) have the same integration measure. The effectiveness of the partition scheme is demonstrated through analysis of the orientation workspace of a three-degree-of-freedom spherical parallel manipulator. Numerical convergence on various orientation workspace measures, such as the workspace volume and the global condition index, are obtained based on this partition scheme.     

On the simulation of wave propagation with a higher-order finite volume scheme based on Reproducing Kernel Methods

In this work we study the dispersion and dissipation characteristics of a higher-order finite volume method based on Moving Least Squares approximations (FV-MLS), and we analyze the influence of the kernel parameters on the properties of the scheme. Several numerical examples are included. The results clearly show a significant improvement of dispersion and dissipation properties of the numerical method if the third-order FV-MLS scheme is used compared with the second-order one. Moreover, with the explicit fourth-order Runge–Kutta scheme the dispersion error is lower than with the third-order Runge–Kutta scheme, whereas the dissipation error is similar for both time-integration schemes. It is also shown than a CFL number lower than 0.8 is required to avoid an unacceptable dispersion error.     

Finite element implementation of nearly-incompressible rheological models based on multiplicative decompositions

The present work is concerned with the numerical integration of finite viscoelastic or viscoplastic models. A numerical integration scheme based on the definition of a flow direction and a flow amplitude as in elastoplasticity is proposed. The most original feature of this approach resides in a local correction of the direction and amplitude with a sub-stepping strategy. Comparisons with the results obtained using a classical tensorial integrator based on a Runge-Kutta–Fehlberg scheme are provided. The reliability of the present numerical scheme is investigated with three rheological models two are viscoelastic (Zener and Poynting–Thomson) and one is viscoplastic.