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.     

Variational subgrid scale formulations for the advection–diffusion-reaction equation

The exact variational multiscale (VMS) and the subgrid scale (SGS) methods have been developed for the advection-reaction and the advection–diffusion-reaction equations. From the element Green's function, approximate intrinsic time scale parameters have been derived for these cases and are shown to be similar to other expressions obtained in the literature out of the maximum principle and convergence/error analysis. The methods have been compared with typical stabilized finite element methods. As expected, the VMS is nodally exact for the one-dimensional case.     

Uniformly high-order schemes on arbitrary unstructured meshes for advection–diffusion equations

The paper presents a linear high-order method for advection–diffusion conservation laws on three-dimensional mixed-element unstructured meshes. The key ingredient of the method is a reconstruction procedure in local computational coordinates. Numerical results illustrate the convergence rates for the linear equation and a non-linear hyperbolic system with diffusion terms for various types of meshes.     

Piecewise cubic monotone interpolation with assigned slopes

An algorithm to construct a monotonicity preserving cubicC ¹ interpolant without modification of the assigned slopes is proposed. AnO(h ⁴) convergence result is obtained when exact function and derivative values are available andO(h ^p ) convergence can be obtained withp=min(4,q) forO(h ^q ) accurate function and derivative values. Numerical experiments carried out on data coming from functions with very different behaviours are presented. The results show that the method can interpolate monotone data in a visually pleasing way, even for data which present rapid variations.     

Numerical method using cubic B-spline for the heat and wave equation

In the present paper, the cubic B-splines method is considered for solving one-dimensional heat and wave equations. A typical finite difference approach had been used to discretize the time derivative while the cubic B-spline is applied as an interpolation function in the space dimension. The accuracy of the method for both equations is discussed. The efficiency of the method is illustrated by some test problems. The numerical results are found to be in good agreement with the exact solution.     

Exact solutions of discrete complex cubic Ginzburg–Landau equation via extended tanh-function approach

In this paper, we obtained rich solutions for the discrete complex cubic Ginzburg–Landau equation by means of the extended tanh-function approach. These solutions include chirpless bright soliton, chirpless dark soliton, triangular function solutions and some solutions with alternating phases, and so on. Meanwhile, the range of parameters where some exact solution exists is given.     

A tau method approach for the diffusion equation with nonlocal boundary conditions

An approximate method for solving the diffusion equation with nonlocal boundary conditions is proposed. The method is based upon constructing the double shifted Legendre series to approximate the required solution using Legendre tau method. The differential and integral expressions which arise in the diffusion equation with nonlocal boundary conditions are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. Numerical examples are included to demonstrate the validity and applicability of the method and a comparison is made with existing results.     

Acceleration techniques for cubic interpolation MIP volume rendering

Multimedia Tools and Applications - Maximum intensity projection (MIP) is a volume visualization technique that is important in modern medical imaging systems. We propose a method to accelerate...     

Designing planar cubic B-spline curves with monotonic curvature for curve interpolation

Computational Visual Media -     

Development of a mixed control volume – Finite element method for the advection–diffusion equation with spectral convergence

In this paper we attack the problem of devising a finite volume method for computational fluid dynamics and related phenomena which can deal with complex geometries while attaining high-orders of accuracy and spectral convergence at a reasonable computational cost. As a first step towards this end, we propose a control volume finite element method for the solution of the advection–diffusion equation. The numerical method and its implementation are carefully tested in the paper where h- and p-convergence are checked by comparing numerical results against analytical solutions in several relevant test-cases. The numerical efficiency of a selected set of operations implemented is estimated by operation counts, ill-conditioning of coefficient matrices is avoided by using an appropriate distribution of interpolation points and control-volume edges.     

求解二维对流扩散方程的投影迭代法

鉴于目前流行的求解大型稀疏代数方程组的投影迭代法中,为提高迭代效率,在迭代前通常需要对稀疏矩阵进行预处理,改善迭代矩阵的条件数,从而减少迭代次数,这使得发展稀疏矩阵的存储技术变得尤为关键。基于二维对流扩散方程的四阶紧致差分格式,将其转化为代数方程组,得到其三对角块形式的系数矩阵,利用稀疏矩阵存储技术和预条件迭代法进行求解,并与传统的中心差分格式所得数值解进行比较,充分说明了方法的高效性和可靠性。     

An interactive procedure for shape preserving cubic spline interpolation

An interactive system is presented which allows flexible shape preserving cubic spline interpolation. The corresponding parameter-depending minimization problem is considered, but similar results are then obtained by means of an interactive adjustment procedure for which the total amount in arithmetic operation is only 0(n).     

Uniform convergence of interpolation by cubic splines

For a sequence of meshes on [0, 1] sufficient conditions are given to obtain uniform convergence of cubic spline interpolants for continous functions respectively for the third derivatives of cubic spline interpolants for functions fromC ³ [0, 1].     

Anisotropic diffusion equation with a new diffusion coefficient for image denoising

This paper presents a new anisotropic diffusion model which is based on a new diffusion coefficient for image denoising. In the proposed model, a new diffusion coefficient and a method of automatically set gradient threshold parameter are introduced into an anisotropic diffusion model, which weakens the staircasing effect and preserves fine edges in a processed image. Comparative experiments show that the new model achieves the more satisfied denoising results than the other existing models.     

An improved EEMD method based on the adjustable cubic trigonometric cardinal spline interpolation

The empirical mode decomposition (EMD) has recently emerged as an efficient tool to adaptively decompose non-stationary signals for nonlinear systems, which has a wide range of applications such as automatic control, mechanical engineering and medicine and biology. A noise-assisted variant of EMD named ensemble empirical mode decomposition (EEMD) have been proposed to alleviate the mode mixing phenomenon. In this paper, we proposed an improved EEMD method, namely cardinal spline interpolation based EEMD (C-EEMD), by optimizing the sifting procedure. Specifically, we employ the adjustable cubic trigonometric cardinal spline interpolation (CTCSI) to accurately represent free curves, other than the original one used in the traditional EEMD. The new interpolation approach can be used to build the mean curve in a more precise way. By virtue of CTCSI, we can therefore obtain the mean value curve from midpoints of the local maxima and minima by just one interpolation operations, which saves almost half the computational cost. Extensive experimental results on synthetic data and real EMI signals clearly demonstrate the superiority of the proposed method, compared to the state-of-the-arts.     

Branching object generation and animation system with cubic hermite interpolation

This paper presents geometric modelling methods for branching object generation and animation with cubic Hermite curves. The procedure for creating a branching object involves two steps. First, the skeleton of an object is constructed. The skeleton consists of geometrical and topological information about the segments of the object. Secondly, cubic Hermite curves are used to deform and join consecutive segments in the skeleton in order to form the body of the object with C¹ continuity. Cubic Hermite curves are also used to deform partial segments which allow an object to grow with desirable continuity in space and time. The techniques are easily implemented on any computer and allow the user to create a branching object using objects which have diverse shapes as segments.     

A mollification regularization method for unknown source in time-fractional diffusion equation

In the present paper, we consider an inverse source problem for a fractional diffusion equation. This problem is ill-posed, i.e. the solution (if it exists) does not depend continuously on the data. We give the mollification regularization method to solve this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Moreover, a new a posteriori parameter choice rule is also proposed and a good error estimate is also obtained. Numerical examples are presented to illustrate the validity and effectiveness of this method.     

Hermite interpolation by triangular cubic patches with small Willmore energy

In this paper, a construction of a cubic Bézier spline surface that interpolates prescribed spatial points and the corresponding normal directions of tangent planes is proposed. Boundary curves of each triangular patch minimize the approximated strain energy. A comparison of optimal boundary curves is given. The interpolant minimizes Willmore energy functional. Some numerical examples and applications of the interpolation scheme are presented: surface approximation, hole filling and condensation of parameters.     

An algorithm for computing shape-preserving cubic spline interpolation to data

We present an algorithm for the construction of shape-preserving cubic splines interpolating a set of data point. The method is based upon some existence properties recently developed. Graphical examples are given.