A 15-point high-order compact scheme with multigrid computation for solving 3D convection diffusion equations

We propose a method with sixth-order accuracy to solve the three-dimensional (3D) convection diffusion equation. We first use a 15-point fourth-order compact discretization scheme to obtain fourth-order solutions on both fine and coarse grids using the multigrid method. Then an iterative mesh refinement technique combined with Richardson extrapolation is used to approximate the sixth-order accurate solution on the fine grid. Numerical results are presented for a variety of test cases to demonstrate the efficiency and accuracy of the proposed method, compared with the standard fourth-order compact scheme.     

Efficient exponential compact higher order difference scheme for convection dominated problems

Present work is the development of a finite difference scheme based on Richardson extrapolation technique. It gives an exponential compact higher order scheme (ECHOS) for two-dimensional linear convection-diffusion equations (CDE). It uses a compact nine point stencil, over which the governing equations are discretized for both fine and coarse grids. The resulting algebraic systems are solved using a line iterative approach with alternate direction implicit (ADI) procedure. Combining the solutions over fine and coarse grids, initially a sixth order solution over coarse grid points is obtained. The resultant solution is then extended to finer grid by interpolation derived from the difference operator. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be monotone. The higher order accuracy and better rate of convergence of the developed algorithm have been demonstrated by solving numerous model problems.     

Structured mesh generation by kriging with local refinement with a new elliptic scheme

Accurate results in finite element analysis are strongly related to mesh quality. In this paper, an automatic quadrilateral mesh generation methodology using kriging interpolation is described, a quality mesh study is conducted, and the development of a new local refinement scheme, called the elliptic scheme, is presented. The new elliptic refinement scheme is evaluated using four standard structural cases, and it is shown that it compares very well with octree-based refinement schemes and other local refinement methods.     

A residual-based compact scheme of optimal order for hyperbolic problems

A new fourth-order dissipative scheme on a compact 3 × 3 stencil is presented for solving 2D hyperbolic problems. It belongs to the family of previously developed residual-based compact schemes and can be considered as optimal since it offers the maximum achievable order of accuracy on the 3 × 3-point stencil. The computation of 2D scalar problems demonstrates the excellent accuracy and efficiency properties offered by this new RBC scheme with respect to existing second- and third-order versions.     

Automatic mesh refinement and data structure for multigrid finite elements techniques

A general algorithm for locally refining any conforming triangulation to generate a new conforming one is presented. The proposed algorithm ensures that all angles in subsequent refined triangulations are greater than, or equal to, half the smallest angle in the original triangulation, the shape regularity of all triangles is maintained and the transition between small and large triangles is smooth. The generated triangulations are nested, so it is possible to implement the approach with adaptive and/or multigrid techniques. A complete algorithm for solving two-dimensional elliptic boundary value problems adaptively by multigrid is presented. The development and implementation of the main parts of this algorithm; automatic mesh generator, a posteriori error estimator, refinement strategy and the multigrid solver are presented in some detail. An appropriate data structure is developed to meet the excess data required for the generation process also to keep track of different grid levels. By the aid of this data structure, it becomes easy to design simple algorithms to store only the non-zero elements of stiffness matrices for different grids and to design a very simple multigrid transfer operator. Numerical examples are presented to show the generated grid sequence for two different boundary value problems.     

Some observations on multigrid convergence for convection–diffusion equations

This paper is concerned with the convergence behaviour of multigrid methods for two- dimensional discrete convection-diffusion equations. In Elman and Ramage (BIT 46:283–299, 2006), we showed that for constant coefficient problems with grid-aligned flow and semiperiodic boundary conditions, the two-grid iteration matrix can be reduced via a set of orthogonal transformations to a matrix containing individual 4 × 4 blocks, enabling a trivial computation of the norm of the iteration matrix. Here we use a similar Fourier analysis technique to investigate the individual contributions from the smoothing and approximation property matrices which form the basis of many standard multigrid analyses. As well as the theoretical results in the semiperiodic case, we present numerical results for a corresponding Dirichlet problem and examine the correlation between the two cases.     

A new scheme for mesh generation and mesh refinement using graph theory

There are an extensive number of algorithms available from graph theory, some of which, for problems with geometric content, make graphs an attractive framework in which to model an object from its geometry to its discretization into a finite element mesh. This paper presents a new scheme for finite element mesh generation and mesh refinement using concepts from graph theory. This new technique, which is suitable for an interactive graphical environment, can also be used efficiently for fully automatic remeshing in association with self-adaptive schemes. Problems of mesh refinement around holes and local mesh refinement are treated. The suitability of the algorithms presented in this paper is demonstrated by some examples.     

A finite volume discontinuous Galerkin scheme¶for nonlinear convection–diffusion problems

A popular method for the discretization of conservation laws is the finite volume (FV) method, used extensively in CFD, based on piecewise constant approximation of the solution sought. However, the FV method has problems with the approximation of diffusion terms. Therefore, in several works [17–19, 1, 12, 16, 2], a combination of the FV and FE methods is used. To this end, it is necessary to construct various combinations of simplicial FE meshes with suitable associated FV grids. This is rather complicated from the point of view of the mesh refinement, particularly in 3D problems [20, 21]. It is desirable to use only one mesh. The combination of FV and FE discretizations on the same triangular grid is proposed in [39]. Another possibility is to use the DG method (see [7] or [9] (and the references there) for a general survey). Here we shall use a compromise between the DG FE method and the FV method using piecewise linear discontinuous finite elements over the grid ? _h and piecewise constant approximation of convective terms on the same grid. Dedicated to Professor Ivo Babuška on the occasion of his 75th birthday Received: May 2001 / Accepted: September 2001     

Towards algebraic multigrid for elliptic problems of second order

An algebraic multigrid method is developed which can be used as a preconditioner for the solution of linear systems of equations with postitive definite matrices. The method is directed to equations which arise from the discretization of elliptic equations of second order, but only the matrix is the source for the information used by the algorithm. One has only to know whether the matrix stems from a 2-dimensional or 3-dimensional problem and whether the elliptic equations are scalar equations or belong to a system.     

Finite volume methods with local refinement for convection-diffusion problems

Computing - Based on approximation of the balance relation for convection-diffusion problems, finite difference schemes on rectangular locally refined grids are derived and studied. A priori...     

Development of adaptive mesh refinement scheme for engine spray simulations

Spray modeling is a critical component to engine combustion and emissions simulations. Accurate spray modeling often requires a fine computational mesh for better numerical resolutions. However, computations with a fine mesh will require extensive computer time. This study developed a methodology that uses a locally refined mesh in the spray region. The fine mesh virtually moves with the liquid spray. Such adaptive mesh refinement can enable greater resolution of the liquid-gas interaction while incurring only a small increase in the total number of computational cells. The present study uses an h-refinement adaptive method. A face-based approach is used for the inter-level boundary condition. The prolongation and restriction procedure preserves conservation of properties in performing grid refinement/coarsening. The refinement criterion is based on the total mass of liquid drops and fuel vapor in each cell. The efficiency and accuracy of the present adaptive mesh refinement scheme is described in the paper. Results show that the present scheme can achieve the same level of accuracy in modeling sprays with significantly lower computational cost as compared to a uniformly fine mesh.     

A methodology for adaptive mesh refinement in optimum shape design problems

This work presents a methodology based on the use of adaptive mesh refinement (AMR) techniques in the context of shape optimization problems analyzed by the Finite Element Method (FEM). A suitable and very general technique for the parametrization of the optimization problem using B-splines to define the boundary is first presented. Then, mesh generation using the advancing front method, the error estimation and the mesh refinement criteria are dealt with in the context of a shape optimization problems. In particular, the sensitivities of the different ingredients ruling the problem (B-splines, finite element mesh, design behaviour, and error estimator) are studied in detail. The sensitivities of the finite element mesh coordinates and the error estimator allow their projection from one design to the next, giving an “a priori knowledge” of the error distribution on the new design. This allows to build up a finite element mesh for the new design with a specified and controlled level of error. The robustness and reliability of the proposed methodology is checked out with some 2D examples.     

Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems

An algebraic multigrid algorithm for symmetric, positive definite linear systems is developed based on the concept of prolongation by smoothed aggregation. Coarse levels are generated automatically. We present a set of requirements motivated heuristically by a convergence theory. The algorithm then attempts to satisfy the requirements. Input to the method are the coefficient matrix and zero energy modes, which are determined from nodal coordinates and knowledge of the differential equation. Efficiency of the resulting algorithm is demonstrated by computational results on real world problems from solid elasticity, plate bending, and shells.     

High-order residual-based compact schemes for advection–diffusion problems

A residual-based compact scheme, previously developed to compute viscous compressible flows with 2nd or 3rd-order accuracy [Lerat A, Corre C. A residual-based compact scheme for the compressible Navier–Stokes equations. J Comput Phys 2001; 170(2): 642–75], is generalized to very high-orders of accuracy. Compactness is retained since for instance a 5th-order accurate dissipative approximation of a d-dimensional advection–diffusion problem can be achieved on a 5^d stencil, without requiring the linear system solutions associated with usual compact schemes. Applications to 1D and 2D model problems are presented and demonstrate that the theoretical orders of accuracy can be achieved in practice.     

High order well-balanced scheme for river flow modeling

We propose a new numerical scheme based on the finite volumes to simulate the river flow in the presence of a variable bottom surface. Our approach is based on the Riemann solver designed for the augmented quasilinear homogeneous formulation. The scheme has general semidiscrete wave-propagation form and can be extended to an arbitrary high order accuracy. The main goal is to construct the scheme, which is well-balanced, i.e. maintains not only some special steady states, but all steady states which can occur.     

Adaptive mesh refinement for shells with modified Ahmad elements

This paper presents a simple scheme for the generation of a quadrilateral element mesh for shells with arbitrary three-dimensional geometry. The present mesh generation scheme incorporates a normal mesh generator for generating a mesh in the two-dimensional plane and a specific mapping technique which maps the two-dimensional mesh onto the three-dimensional curved surface. As the mapping is a one-to-one mapping between the mesh in the plane and that on the curved surface, the resulting surface discretization is compatible with the local mesh parameters in two dimensions. This scheme is further combined, both with a sophisticated error estimate determined by using the best guess values of bending moments and membrane and transverse shear forces obtained from a previous solution, and an effective mesh refinement strategy established at an element level in order to complete an adaptive analysis for shell structures. Numerical examples are shown to illustrate the principles and procedure of the present adaptive analysis.     

Finite elements on locally uniform meshes for convection–diffusion problems with boundary layers

The layer-adapted meshes used to achieve robust convergence results for problems with layers are not locally uniform. We discuss concepts of almost robust convergence and some realizations of locally-uniform meshes.     

求解双曲守恒律及其对流扩散方程的中心迎风方法

提出了一种新的求解双曲守恒律方程(组)的四阶半离散中心迎风差分方法．空间导数项的离散采用四阶CWENO(central weighted essentially non—oscillatory)的构造方法,使所得到的新方法在提高精度的同时,具有更高的分辨率．使用该方法产生的数值粘性要比交错的中心格式小,而且由于数值粘性与时间步长无关,从而时间步长可根据稳定性需要尽可能的小．     

Discontinuous Galerkin time stepping with local projection stabilization for transient convection–diffusion-reaction problems

A time-dependent convection–diffusion-reaction problem is discretized in space by a continuous finite element method with local projection stabilization and in time by a discontinuous Galerkin method. We present error estimates for the semidiscrete problem after discretizing in space only and for the fully discrete problem. Numerical tests confirm the theoretical results.