Nearly isotropic propagation of spatially discretized waves

The effect of spatial discretization on the isotropy of propagating waves is investigated. A general criterion is given for minimizing the numerical anisotropy and dispersion caused by spatial discretization, and specific discretizations in two and three space dimensions are derived which give, in a well-defined sense, optimally isotropic propagation. We establish the group-theoretic connection between the properties of the spatial discretization and the symmetries of the underlying computational grid. The discretization technique, described here in the context of the scalar wave equation, may also be applied to other partial differential equations containing the Laplacian or gradient operators.     

Travelling Waves for Complete Discretizations of Reaction Diffusion Systems

In this paper we consider the impact that full spatial–temporal discretizations of reaction–diffusion systems have on the existence and uniqueness of travelling waves. In particular, we consider a standard second-difference spatial discretization of the Laplacian together with the six numerically stable backward differentiation formula methods for the temporal discretization. For small temporal time-steps and a fixed spatial grid-size, we establish some useful Fredholm properties for the operator that arises after linearizing the system around a travelling wave. In particular, we perform a singular perturbation argument to lift these properties from the natural limiting operator. This limiting operator is associated to a lattice differential equation, where space has been discretized but time remains continuous. For the backward-Euler temporal discretization, we also obtain travelling waves for arbitrary time-steps. In addition, we show that in the anti-continuum limit, in which the temporal time-step and the spatial grid-size are both very large, wave speeds are no longer unique. This is in contrast to the situation for the original continuous system and its spatial semi-discretization. This non-uniqueness is also explored numerically and discussed extensively away from the anti-continuum limit.     

High-order Lagrangian cell-centered conservative scheme on unstructured meshes

A high-order Lagrangian cell-centered conservative gas dynamics scheme is presented on unstructured meshes. A high-order piecewise pressure of the cell is intro- duced. With the high-order piecewise pressure of the cell, the high-order spatial discretiza- tion fluxes are constructed. The time discretization of the spatial fluxes is performed by means of the Taylor expansions of the spatial discretization fluxes. The vertex velocities are evaluated in a consistent manner due to an original solver located at the nodes by means of momentum conservation. Many numerical tests are presented to demonstrate the robustness and the accuracy of the scheme.     

A stabilization for three‐dimensional discontinuous Galerkin discretizations applied to nonhydrostatic atmospheric simulations

A discontinuous Galerkin nonhydrostatic atmospheric model is used for two‐dimensional and three‐dimensional simulations. There is a wide range of timescales to be dealt with. To do so, two different implicit/explicit time discretizations are implemented. A stabilization, based upon a reduced‐order discretization of the gravity term, is introduced to ensure the balance between pressure and gravity effects. While not affecting significantly the convergence properties of the scheme, this approach allows the simulation of anisotropic flows without generating spurious oscillations, as it happens for a classical discontinuous Galerkin discretization. This approach is shown to be less diffusive than usual spatial filters. A stability analysis demonstrates that the use of this modified scheme discards the instability associated with the usual discretization. Validation against analytical solutions is performed, confirming the good convergence and stability properties of the scheme. Numerical results demonstrate the attractivity of the discontinuous Galerkin method with implicit/explicit time integration for large‐scale atmospheric flows. Copyright © 2015 John Wiley & Sons, Ltd.     

A multigrid pseudo-spectral method for incompressible Navier–Stokes flows

A pseudo-spectral solver with multigrid acceleration for the numerical prediction of incompressible non-isothermal flows is presented. The spatial discretization is based on a Chebyshev collocation method on Gauss–Lobatto points and for the discretization in time the second-order backward differencing scheme (BDF2) is employed. The multigrid method is invoked at the level of algebraic system solving within a pressure-correction method. The approach combines the high accuracy of spectral methods with efficient solver properties of multigrid methods. The capabilities of the proposed scheme are illustrated by a buoyancy driven cavity flow as a standard benchmark case. To cite this article: K. Krastev, M. Schäfer, C. R. Mecanique 333 (2005).     

拉格朗日高阶中心型守恒气体动力学格式

提出了拉格朗日高阶中心型守恒气体动力学格式。用产生于当前时刻子网格密度和当前时刻网格声速的子网格压力构造了子网格力,用加权本质无震荡方法构造的高阶子网格力构造了高阶空间通量,借助时间中点通量的泰勒展开完成了高阶时间通量离散,利用动量守恒条件使得格点速度以与网格面的数值通量相容的方式计算。编制了拉格朗日高阶中心型守恒气体动力学格式,对Saltzman活塞问题进行了数值模拟,数值结果表明,拉格朗日高阶中心型守恒气体动力学格式的有效性和精确性.     

Theory and applications in stability of free‐surface time‐domain boundary element models

A method is presented for examining the stability of a free‐surface time‐domain boundary element model based on B‐splines. Effects of a non‐uniform discretization occurring in practical applications are included. It is demonstrated that instabilities may occur, even in situations where earlier stability analyses predicted the scheme to be stable. These instabilities are due to non‐uniformities in the spatial discretization, which have until now not been included in the stability analyses. Copyright © 2001 John Wiley & Sons, Ltd.     

Nonlinear Multigrid Methods for Numerical Solution of the Variably Saturated Flow Equation in Two Space Dimensions

The need of accurate and efficient numerical schemes to solve Richards’ equation is well recognized. This study is carried out to examine the numerical performances of the nonlinear multigrid method for numerical solving of the two-dimensional Richards’ equation modeling water flow in variably saturated porous media. The numerical approach is based on an implicit, second-order accurate time discretization combined with a vertex centered finite volume method for spatial discretization. The test problems simulate infiltration of water in 2D saturated–unsaturated soils with hydraulic properties described by van Genuchten–Mualem models. The numerical results obtained are compared with those provided by the modified Picard–preconditioned conjugated gradient (Krylov subspace) approach.     

非定常不可压N-S方程的最小二乘算子分裂有限元法数值求解

采用最小二乘算子分裂有限元法求解非定常不可压N-S(Navier-Stokes)方程,即在每个时间层上采用算子分裂法将N-S方程分裂成扩散项和对流项,这样既能考虑对流占优特点又能顾及方程的扩散性质。扩散项是一个抛物型方程,时间离散采用向后差分格式,空间离散采用标准Galerkin有限元法。对流项的时间项采用后向差分格式,非线性部分用牛顿法进行线性化处理,再用最小二乘有限元法进行空间离散,得到对称正定的代数方程组系数矩阵。采用Re=1000的方腔流对该算法的有效性进行检验,表明其具有较高的精度,能够很好地捕捉流场中的涡结构。同时,对圆柱层流绕流进行了数值研究,通过流线图、压力场、阻力系数、升力系数及斯特劳哈数等结果的分析与对比,表明本文算法对于模拟圆柱层流绕流是准确和可靠的。     

The solution of non-linear hyperbolic equation systems by the finite element method

The difficulties experienced in the treatment of hyperbolic systems of equations by the finite element method (or other) spatial discretization procedures are well known. In this paper a temporal discretization precedes the spatial one which in principle is considered along the characteristics to achieve a self adjoint form. By a suitable expansion, the original co-ordinates are preserved and combined with the use of a standard Galerkin process to achieve an accurate discretization. It is shown that the process is equivalent to the Taylor-Galerkin methods of Donea.¹⁷ Several examples illustrate the accuracy and efficiency attainable in such problems as transport, shallow water equations, transonic flow etc.     

Nonlinear Multigrid Methods for Numerical Solution of the Unsaturated Flow Equation in Two Space Dimensions

Picard and Newton iterations are widely used to solve numerically the nonlinear Richards’ equation (RE) governing water flow in unsaturated porous media. When solving RE in two space dimensions, direct methods applied to the linearized problem in the Newton/Picard iterations are inefficient. The numerical solving of RE in 2D with a nonlinear multigrid (MG) method that avoids Picard/Newton iterations is the focus of this work. The numerical approach is based on an implicit, second-order accurate time discretization combined with a second-order accurate finite difference spatial discretization. The test problems simulate infiltration of water in 2D unsaturated soils with hydraulic properties described by Broadbridge–White and van Genuchten–Mualem models. The numerical results show that nonlinear MG deserves to be taken into consideration for numerical solving of RE.     

Non-linear modal properties of non-shallow cables

A non-linear mechanical model of non-shallow linearly elastic suspended cables is employed to investigate the non-linear modal characteristics of the free planar motions. An asymptotic analysis of the equations of motion is carried out directly on the partial-differential equations overcoming the drawbacks of a discretization process. The direct asymptotic treatment delivers the approximation of the individual non-linear normal modes. General properties about the non-linearity of the in-plane modes of different type—geometric, elasto-static and elasto-dynamic—are unfolded. The spatial corrections to the considered linear mode shape caused by the quadratic geometric forces are investigated for modes belonging to the three mentioned classes. Moreover, the convergence of Galerkin reduced-order models is discussed and the influence of passive modes is highlighted.     

Shape Identification for Fluid-Structure Interaction Problem Using Improved Bubble Element

Numerical solutions for identification of the shape of a circular cylinder are addressed in this paper. The Sakawa-Shindo method is used to minimize the algorithm. A unified computational approach for simulation of flow and shape identification is presented. As a numerical approach for spatial discretization, mixed interpolation by the bubble and linear elements is used for the velocity and pressure fields, respectively.     

Upwind discretization of the steady Navier–Stokes equations

A discretization method is presented for the full, steady, compressible Navier–Stokes equations. The method makes use of quadrilateral finite volumes and consists of an upwind discretization of the convective part and a central discretization of the diffusive part. In the present paper the emphasis lies on the discretization of the convective part. The solution method applied solves the steady equations directly by means of a non-linear relaxation method accelerated by multigrid. The solution method requires the discretization to be continuously differentiable. For two upwind schemes which satisfy this requirement (Osher's and van Leer's scheme), results of a quantitative error analysis are presented. Osher's scheme appears to be increasingly more accurate than van Leer's scheme with increasing Reynolds number. A suitable higher-order accurate discretization of the convection terms is derived. On the basis of this higher-order scheme, to preserve monotonicity, a new limiter is constructed. Numerical results are presented for a subsonic flat plate flow and a supersonic flat plate flow with oblique shock wave–boundary layer interaction. The results obtained agree with the predictions made. Useful properties of the discretization method are that it allows an easy check of false diffusion and that it needs no tuning of parameters.     

Finite‐difference 4th‐order compact scheme for the direct numerical simulation of instabilities of shear layers

An algorithm based on the 4th‐order finite‐difference compact scheme is developed and applied in the direct numerical simulations of instabilities of channel flow. The algorithm is illustrated in the context of stream function formulation that leads to field equation involving 4th‐order spatial derivatives. The finite‐difference discretization in the wall‐normal direction uses five arbitrarily spaced points. The discretization coefficients are determined numerically, providing a large degree of flexibility for grid selection. The Fourier expansions are used in the streamwise direction. A hybrid Runge–Kutta/Crank–Nicholson low‐storage scheme is applied for the time discretization. Accuracy tests demonstrate that the algorithm does deliver the 4th‐order accuracy. The algorithm has been used to simulate the natural instability processes in channel flow as well as processes occurring when the flow is spatially modulated using wall transpiration. Extensions to three‐dimensional situations are suggested. Copyright © 2005 John Wiley & Sons, Ltd.     

A FINITE VOLUME SOLVER FOR THE SHALLOW WATER EQUATIONS IN VARYING BED ENVIRONMENTS

Abstract

A numerical scheme for solving the shallow-water equations is presented. An analogy is made between flows governed by shallow-water equations and the Euler system of equations used in gas dynamics. An emphasis is placed on the difference presented by the bathymetry in hydraulic systems. The discretization of the governing equations is based on Roe's flux difference-splitting solver, initially developed for solving inviscid compressible flows. The spatial discretization is handled within a finite-volume context by using triangles or quadrilaterals as the basic control-volume cells. This approach enables an easy and flexible treatment of general geometries. A development of the boundary conditions tailored for the current scheme is given. Fundamental validation tests are presented.     

Fully dispersive dynamic models for surface water waves above varying bottom, Part 1: Model equations

In this paper we formulate relatively simple models to describe the propagation of coastal waves from deep parts in the ocean to shallow parts near the coast. The models have good dispersive properties that are based on smooth quasi-homogeneous interpolation of the exact dispersion above flat bottom. This dispersive quality is then maintained in the second order nonlinear terms of uni-directional equations as known from the AB-equation. A linear coupling is employed to obtain bi-directional propagation which includes (interactions with) reflected waves.The derivation of the models is consistent with the basic variational formulation of surface waves without rotation. A subsequent spatial discretization that takes this variational structure into account leads to efficient and accurate codes, as will be shown in Part 2.