A new discretization method and its application to solve incompressible Navier–Stokes equations

 A new numerical method is presented in this paper. This method directly solves partial differential equations in the Cartesian coordinate system. It can be easily applied to solve irregular domain problems without introducing the coordinate transformation technique. The concept of the present method is different from the conventional discretization methods. Unlike the conventional numerical methods where the discrete form of the differential equation only involves mesh points inside the solution domain, the new discretization method reduces the differential equation into a discrete form which may involve some points outside the solution domain. The functional values at these points are computed by the approximate form of the solution along a vertical or horizontal line. This process is called extrapolation. The form of the solution along a line can be approximated by Lagrange interpolated polynomial using all the points on the line or by low order polynomial using 3 local points. In this paper, the proposed new discretization method is first validated by its application to solve sample linear and nonlinear differential equations. It is demonstrated that the present method can easily treat different solution domains without any additional programming work. Then the method is applied to simulate incompressible flows in a smooth expansion channel by solving Navier–Stokes equations. The numerical results obtained by the new discretization method agree very well with available data in the literature. All the numerical examples showed that the present method is very efficient, which is suitable for solving irregular domain problems. Received 19 July 2000     

A high-order harmonic polynomial method for solving the Laplace equation with complex boundaries and its application to free-surface flows. Part I: Two-dimensional cases

A high-order harmonic polynomial method (HPM) is developed for solving the Laplace equation with complex boundaries. The “irregular cell” is proposed for the accurate discretization of the Laplace equation, where it is difficult to construct a high-quality stencil. An advanced discretization scheme is also developed for the accurate evaluation of the normal derivative of potential functions on complex boundaries. Thanks to the irregular cell and the discretization scheme for the normal derivative of the potential functions, the present method can avoid the drawback of distorted stencils, that is, the possible numerical inaccuracy/instability. Furthermore, it can involve stationary or moving bodies on the Cartesian grid in an accurate and simple way. With the proper free-surface tracking methods, the HPM has been successfully applied to the accurate and stable modeling of highly nonlinear free-surface potential flows with and without moving bodies, that is, sloshing, water entry, and plunging breaker.     

Boundary integral equation supported differential quadrature method to solve problems over general irregular geometries

Based on the interpolation technique with the aid of boundary integral equations, a new differential quadrature method has been developed (boundary integral equation supported differential quadrature method, BIE-DQM) to solve boundary value problems over generally irregular geometries. The quadrature rule of the BIE-DQM is that the first and the second derivatives of a function with respect to independent variables are approximated by a weighted linear combination of the function values at all discrete nodal points and the corresponding normal derivatives at all boundary points. Several numerical examples are considered to verify the feasibility and effectiveness of the proposed algorithm.     

A priori tests of large eddy simulation of the compressible plane mixing layer

Three important aspects for the assessment of the possibilities of Large Eddy Simulation (LES) of compressible flow are investigated. In particular the magnitude of all subgrid-terms, the role of the discretization errors and the correlation of the turbulent stress tensor with several subgrid-models are studied. The basis of the investigation is a Direct Numerical Simulation (DNS) of the two- and three-dimensional compressible mixing layer, using a finite volume method on a sufficiently fine grid. With respect to the first aspect, the exact filtered Navier-Stokes equations are derived and all terms are classified according to their order of magnitude. It is found that the pressure dilatation subgrid-term in the filtered energy equation, which is usually neglected in the modelling-practice, is as large as e.g. the pressure velocity subgrid-term, which in general is modelled. The second aspect yields the result that second- and fourth-order accurate spatial discretization methods give rise to discretization errors which are larger than the corresponding subgrid-terms, if the ratio between the filter width and the grid-spacing is close to one. Even if an exact representation for the subgrid-scale contributions is assumed, LES performed on a (considerably) coarser grid than required for a DNS, is accurate only if this ratio is sufficiently larger than one. Finally the well-known turbulent stress tensor is investigated in more detail. A priori tests of subgrid-models for this tensor yield poor correlations for Smagorinsky's model, which is purely dissipative, while the non-eddy viscosity models considered here correlate considerably better.     

A Numerical Comparison of Finite Difference and Finite Element Methods for a Stochastic Differential Equation with Polynomial Chaos

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.     

非等距网格上一维对流扩散方程的保正格式

对流扩散方程广泛存在于很多领域,为适应一些实际问题模型的求解,对离散格式,不仅要求满足一些基本性质,如稳定性和解的存在唯一性等,还要求离散格式的保正性．采用有限体积格式求解对流扩散方程的工作较少,但在保正性方面所做的工作不多．本文构造了任意非等距网格上一维对流扩散方程的非线性保正有限体积格式．其中,扩散通量的离散,在等距网格上,当扩散系数为标量时可退化为标准的二阶中心差分格式．而对流通量的离散,为避免数值振荡而使其保持迎风特性,提出一种新的方法使格式精度提高到二阶．该方法在上游单元中心处作泰勒级数展开,通过相关辅助未知量来完成梯度的重构,并对出负情形作正性校正,使得格式满足保正性要求．新格式只含有区间单元中心未知量,并满足区间端点处通量的局部守恒性．数值结果表明,本文所提格式是有效的,对于处理扩散占优、对流占优问题,扩散系数连续和间断情形均具有良好的适应性,并且保持二阶精度．另外,新格式适用于扩散系数间断问题的求解．     

A finite volume nodal point scheme for solving two dimensional Navier-Stokes equations

Summary A finite volume nodal point spatial discretization scheme for the computation of viscous fluxes in two dimensional Navier-Stokes equations has been presented here. The present scheme gives second order accurate first derivatives and at least first order accurate second derivatives even for stretched and skewed grid. It takes almost the same numerical efforts to solve full Navier-Stokes equations as that for using thin layer approximation. The scheme has been implemented to solve laminar viscous flows past NACA0012 aerofoil. To advance the solution in time a five stage Runge-Kutta scheme has been used. To accelerate the rate of convergence to steady state, local time stepping, residual averaging and enthalpy damping have been employed. Only a fourth order artificial dissipation has been used here for global stability of the solution. A comparative study of the results obtained by the present scheme for full Navier-Stokes equations and for thin layer approximation have been made with other numerical methods developed earlier.     

A comparative study of the behaviour of a direct solver and a preconditioned iterative solver for the equations arising from the discretization by Chebyshev collocation of a second-order partial differential equation on a square

The behaviour is compared of two solvers for the discrete equations arising from the discretization using Chebyshev collocation of a second-order linear partial differential equation on a square. The alternative solvers considered are a direct solver and an iterative solver based on preconditioning with the matrix arising from finite-difference discretization of the governing equation. The total error of the collocation derivatives and the separate contributions from round-off and discretization error are examined. The efficiency of the two solvers is compared. The iterative solver is more efficient than the direct solver on fine grids for equations similar to the Poisson equation, provided that there are Dirichlet boundary conditions on at least three of the sides of the square.     

An adaptive multiresolution method for parabolic PDEs with time‐step control

We present an efficient adaptive numerical scheme for parabolic partial differential equations based on a finite volume (FV) discretization with explicit time discretization using embedded Runge–Kutta (RK) schemes. A multiresolution strategy allows local grid refinement while controlling the approximation error in space. The costly fluxes are evaluated on the adaptive grid only. Compact RK methods of second and third order are then used to choose automatically the new time step while controlling the approximation error in time. Non‐admissible choices of the time step are avoided by limiting its variation. The implementation of the multiresolution representation uses a dynamic tree data structure, which allows memory compression and CPU time reduction. This new numerical scheme is validated using different classical test problems in one, two and three space dimensions. The gain in memory and CPU time with respect to the FV scheme on a regular grid is reported, which demonstrates the efficiency of the new method. Copyright © 2008 John Wiley & Sons, Ltd.     

A three‐dimensional hybrid meshfree‐Cartesian scheme for fluid–body interaction

A numerical method based on a hybrid meshfree‐Cartesian grid is developed for solving three‐dimensional fluid–solid interaction (FSI) problems involving solid bodies undergoing large motion. The body is discretized and enveloped by a cloud of meshfree nodes. The motion of the body is tracked by convecting the meshfree nodes against a background of Cartesian grid points. Spatial discretization of second‐order accuracy is accomplished by the combination of a generalized finite difference (GFD) method and conventional finite difference (FD) method, which are applied to the meshfree and Cartesian nodes, respectively. Error minimization in GFD is carried out by singular value decomposition. The discretized equations are integrated in time via a second‐order fractional step projection method. A time‐implicit iterative procedure is employed to compute the new/evolving position of the immersed bodies together with the dynamically coupled solution of the flow field. The present method is applied on problems of free falling spheres and tori in quiescent flow and freely rotating spheres in simple shear flow. Good agreement with published results shows the ability of the present hybrid meshfree‐Cartesian grid scheme to achieve good accuracy. An application of the method to the self‐induced propulsion of a deforming fish‐like swimmer further demonstrates the capability and potential of the present approach for solving complex FSI problems in 3D. Copyright © 2011 John Wiley & Sons, Ltd.     

Stability properties of the Discontinuous Galerkin Material Point Method for hyperbolic problems in one and two space dimensions

In this paper, stability conditions are derived for the Discontinuous Galerkin Material Point Method (DGMPM) on the scalar linear advection equation for the sake of simplicity and without loss of generality for linear problems. The discrete systems resulting from the application of the DGMPM discretization in one and two space dimensions are first written. For these problems, a second-order Runge-Kutta and the forward Euler time discretizations are respectively considered. Moreover, the numerical fluxes are computed at cell faces by means of either the Donor-Cell Upwind or the Corner Transport Upwind methods for multidimensional problems. Second, the discrete scheme equations are derived assuming that all cells of a background grid contain at least one particle. Although a Cartesian grid is considered in two space dimensions, the results can be extended to regular grids. The von Neumann linear stability analysis then allows the computation of the critical Courant number for a given space discretization. Although the DGMPM is equivalent to the first-order finite volume method if one particle lies in each element, so that the Courant number can be set to unity, other distributions of particles may restrict the stability region of the scheme. The study of several configurations is then proposed.     

Comparing RBF‐FD approximations based on stabilized Gaussians and on polyharmonic splines with polynomials

In this work, we are concerned with radial basis function–generated finite difference (RBF‐FD) approximations. Numerical error estimates are presented for stabilized flat Gaussians (RBF(SGA)‐FD) and polyharmonic splines with supplementary polynomials (RBF(PHS)‐FD) using some analytical solutions of the Poisson equation in a square domain. Both structured and unstructured point clouds are employed for evaluating the influence of cloud refinement, size of local supports, and maximal permissible degree of the polynomials in RBF(PHS)‐FD. High order of accuracy was attained with both RBF(SGA)‐FD and RBF(PHS)‐FD especially for unstructured clouds. Absolute errors in the first and second derivatives were also estimated at all points of the domain using one of the analytical solutions. For RBF(SGA)‐FD, this test showed the occurrence of improprieties of some decentered supports localized on boundary neighborhoods. This phenomenon was not observed with RBF(PHS)‐FD.     

A locally analytic technique applied to grid generation by elliptic equations

One technique for obtaining grids for irregular geometries is to solve sets of elliptic partial differential equations. The solution of the partial differential equations yields a grid which discretizes the physical solution domain and also a transformation for the irregular physical domain to a regular computational domain. Expressing the governing equation of interest in the computational domain requires the derivatives of the physical to computational domain transformation, i.e., the metrics. These metrics are typically determined by numerical differentiation, which is a potential source of error. The locally analytic method uses the analytic solution of the locally linearized equation to develop numerical stencils. Thus, the locally analytic method allows numerical differentiation with no loss of accuracy. In this paper, the locally analytic method is applied to the solution of the Poisson and Brackbill–Saltzman equations. Comparison with an exact solution shows the locally analytic method to be more accurate than the finite difference method, both in solving the partial differential equation and evaluating the metrics. However, it is more computationally expensive.     

Determination of second-order derivatives of a skew ray with respect to the variables of its source ray in optical prism systems

The second-order derivative of a scalar function with respect to a variable vector is known as the Hessian matrix. We present a computational scheme based on the principles of differential geometry for determining the Hessian matrix of a skew ray as it travels through a prism system. A comparison of the proposed method and the conventional finite difference (FD) method is made at last. It is shown that the proposed method has a greater inherent accuracy than FD methods based on ray-tracing data. The proposed method not only provides a convenient means of investigating the wavefront shape within complex prism systems, but it also provides a potential basis for determining the higher order derivatives of a ray by further taking higher order differentiations.     

Velocity potential computation by finite differences for compressible flow in ducts

Preliminary studies of computation with velocity potential are made with a view to the analysis of complex three-dimensional flows. The methods used are applicable more generally to quasilinear elliptic problems with derivative boundary conditions on irregular domains. Second order finite difference approximations are constructed in simple form for plane ducts of general shape by using an irregular net. Derivative boundary conditions are handled quite easily. An iterative method is described which corresponds to freezing the coefficients in the quasilinear differential equation for velocity potential. The discretization is such that this is a ‘generalized Newton’ method for the non-linear algebraic equations. Good convergence has been found in practice even when there are small supersonic zones. The discretization accuracy is tested by comparisons with the exact solution for incompressible flow between confocal hyperbolas.     

An efficient complex‐frequency shifted‐perfectly matched layer for second‐order acoustic wave equation

In the context of simulations of wave propagations in unbounded domain, absorbing boundary conditions are often used to truncate the simulation domain to a finite space. Perfectly matched layer (PML) has proven to be an excellent absorbing boundary conditions. However, as this technique was primarily designed for the first‐order equation system, it cannot be applied to the second‐order equation system directly. In this paper, based on a complex‐coordinate stretching technique, we developed a novel, efficient auxiliary‐differential equation form of the complex‐frequency shifted‐PML for the second‐order equation system. This facilitates the use of complex‐frequency shifted‐PML in acoustic simulations based upon wave equations of second‐order form. Compared with previous state‐of‐the‐art methods, the proposed one has the advantage of simpler implementation. It is an unsplit‐field scheme that can be extended to higher‐order discretization schemes conveniently. Numerical results from both homogeneous and heterogeneous computational domains are provided to illustrate the validity of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

An adaptive solution of the 3-D Euler equations on an unstructured grid

Summary A new algorithm to generate the unstructured grid on a curved surface is developed. The advancing front method is used to generate the tetrahedral meshes in the space. An adaptive grid technique is used to enhance the calculation efficiency. The AUSM⁺ (Advection Upstream Splitting Method) scheme which was developed on a structured grid has been extended to be used to the spatial discretization of a cell-centered finite volume formulation on the unstructured grid. A second order spatial accuracy is achieved by applying a novel cell reconstruction procedure which can prevent the solution from exhibiting spurious oscillations without adding a limiter. A 3-D Euler solver for an adaptive tetrahedral grid and numerical results for several cases are presented.     

Corrected discrete least-squares meshless method for simulating free surface flows

In this paper, a modified version of discrete least-squares meshless (DLSM) method is used to simulate free surface flows with moving boundaries. DLSM is a newly developed meshless approach in which a least-squares functional of the residuals of the governing differential equations and its boundary conditions at the nodal points is minimized with respect to the unknown nodal parameters. The meshless shape functions are also derived using the Moving Least Squares (MLS) method of function approximation. The method is, therefore, a truly meshless method in which no integration is required in the computations. Since the second order derivative of the MLS shape function are known to contain higher errors compared to the first derivative, a modified version of DLSM method referred to as corrected discrete least-squares meshless (corrected DLSM) is proposed in which the second order derivatives are evaluated more accurately and efficiently by combining the first order derivatives of MLS shape functions with a finite difference approximation of the second derivatives. The governing equations of fluid flow (Navier–Stokes) are solved by the proposed method using a two-step pressure projection method in a Lagrangian form. Three benchmark problems namely; dam break, underwater rigid landslide and Scott Russell wave generator problems are used to test the accuracy of the proposed approach. The results show that proposed corrected DLSM can be employed to simulate complex free surface flows more accurately.     

Numerical solution of the Helmholtz equation with high wavenumbers

This paper investigates the pollution effect, and explores the feasibility of a local spectral method, the discrete singular convolution (DSC) algorithm for solving the Helmholtz equation with high wavenumbers. Fourier analysis is employed to study the dispersive error of the DSC algorithm. Our analysis of dispersive errors indicates that the DSC algorithm yields a dispersion vanishing scheme. The dispersion analysis is further confirmed by the numerical results. For one‐ and higher‐dimensional Helmholtz equations, the DSC algorithm is shown to be an essentially pollution‐free scheme. Furthermore, for large‐scale computation, the grid density of the DSC algorithm can be close to the optimal two grid points per wavelength. The present study reveals that the DSC algorithm is accurate and efficient for solving the Helmholtz equation with high wavenumbers. Copyright © 2003 John Wiley & Sons, Ltd.     

The time-marching method of fundamental solutions for wave equations

In this paper, a meshless numerical algorithm is developed for the solution of multi-dimensional wave equations with complicated domains. The proposed numerical method, which is truly meshless and quadrature-free, is based on the Houbolt finite difference (FD) scheme, the method of the particular solutions (MPS) and the method of fundamental solutions (MFS). The wave equation is transformed into a Poisson-type equation with a time-dependent loading after the time domain is discretized by the Houbolt FD scheme. The Houbolt method is used to avoid the difficult problem of dealing with time evolution and the initial conditions to form the linear algebraic system. The MPS and MFS are then coupled to analyze the governing Poisson equation at each time step. In this paper we consider six numerical examples, namely, the problem of two-dimensional membrane vibrations, the wave propagation in a two-dimensional irregular domain, the wave propagation in an L-shaped geometry and wave vibration problems in the three-dimensional irregular domain, etc. Numerical validations of the robustness and the accuracy of the proposed method have proven that the meshless numerical model is a highly accurate and efficient tool for solving multi-dimensional wave equations with irregular geometries and even with non-smooth boundaries.