For the five-point discrete formulae of directional derivatives in the finite point method,overcoming the challenge resulted from scattered point sets and making full use of the ex-plicit expressions and accuracy of the formulae,this paper obtains a number of theoretical results:(1) a concise expression with definite meaning of the complicated directional differ-ence coefficient matrix is presented,which characterizes the correlation between coefficients and the connection between coefficients and scattered geometric characteristics;(2) various expressions of the discriminant function for the solvability of numerical differentials along with the estimation of its lower bound are given,which are the bases for selecting neigh-boring points and making analysis;(3) the estimations of combinatorial elements and of each element in the directional difference coefficient matrix are put out,which exclude the existence of singularity.Finally,the theoretical analysis results are verified by numerical calculations.The results of this paper have strong regularity,which lay the foundation for further research on the finite point method for solving partial differential equations.

A new approach for numerically solving 3-T diffusion equations on 2-D scattered point distributions is developed by the finite point method. In this paper, a new method for selecting neighboring points is designed, which is robust and well reflects variations of gradients of physical quantities. Based on this, a new discretization method is proposed for the diffusion operator, which results in a new scheme with the stencil of minimal size for numerically solving nonlinear diffusion equations. Distinguished from most of meshless methods often involving dozens of neighboring points, this method needs only five neighbors of the point under consideration. Numerical simulations show the good performance of the proposed methodology. AMS subject classifications: 65D25, 65M06, 65M70

This paper makes some mathematical analyses for the finite point method based on directional difference. By virtue of the explicit expressions of numerical formulae using only five neighboring points for computing first-order and second-order directional differentials, a new methodology is presented to discretize the Laplacian operator defined on 2D scattered point distributions. Some sufficient conditions with very weak limitations are obtained, under which the resulted schemes are positive schemes. As a consequence, the discrete maximum principle is proved, and the first order convergent result of O(h) is achieved for the nodal solutions defined on scattered point distributions, which can be raised up to O(h(2)) on uniform point distributions.

In this paper, relations between directional derivatives are considered for smooth functions both in 2D and 3D spaces. These relations are established in the form of linear combinations of directional derivatives with their coefficients having simple form and structural regularity. By them, expressions based on directional derivatives for some typical differential operators are derived. This builds up a solid mathematical foundation for further study on numerical computation by the finite point method based on directional difference.

本文简要回顾非线性抛物型方程差分方法若干研究工作,包括周毓麟先生在该研究方向取得的部分研究成果,并对近年来相关的部分研究进展进行综述,展望拟开展的研究工作.

A new Lagrangian meshfree difference method for two dimensional compressible hydrodynamics problem is presented. The numerical scheme is a strong formulation which directly approximates the differential operators on the scattered points. These discrete points, which are referred to as Lagrangian points, move with the fluid. Each Lagrangian point carries all fluid information, like density, velocity, pressure, energy, but there is no mass associated with these points. A local point cloud of five neighbors is selected around each Lagrangian point at every time step, and this cloud is used to implement the explicit five-point formulas, which numerically approximates first-order directional derivatives to second order accuracy, to gather information about flow derivatives. Because of numerical stability concerns, a variant of the Lapidus-type artificial viscosity has also been introduced. Furthermore, An adaptive strategy for choosing the time step is also provided. Numerical results show the validity and potential interest of this approach.

We consider a five-point positive meshless collocation method for the numerical solutions of transport process described by hyperbolic conservation laws. This positive meshless method uses the five-point scheme approximation for derivatives, and an artificial dissipation term to ensure the positivity of coefficients. The numerical examples confirm the good performance of the present five-point positive meshless scheme. Copyright (C) 2009 John Wiley & Sons, Ltd.

We propose a new way of rewriting the two dimensional Euler equations and derive an original canonical characteristic relation based on the characteristic theory of hyperbolic systems. This relation contains the derivatives strictly along the bicharacteristic directions, and can be viewed as the 2D extension of the characteristic relation in 1D case.

In this paper,the extremum of second-order directional derivatives,i.e.the gradient of first-order derivatives is discussed.Given second-order directional derivatives in three nonparallel directions,or given second-order directional derivatives and mixed directional derivatives in two nonparallel directions,the formulae for the extremum of second-order directional derivatives are derived,and the directions corresponding to maximum and minimum are perpendicular to each other.

This paper presents a finite point method based on directional derivatives for diffusion equation on 2D scattered points. To discretize the diffusion operator at a given point, a six-point stencil is derived by employing explicit numerical formulae of directional derivatives, namely, for the point under consideration, only five neighbor points are involved, the number of which is the smallest for discretizing diffusion operator with first-order accuracy. A method for selecting neighbor point set is proposed, which satisfies the solvability condition of numerical derivatives. Some numerical examples are performed to show the good performance of the proposed method. Keywords—Finite point method, directional derivatives, diffusion equation, method for selecting neighbor point set.

A Lagrangian finite point method for one-dimensional compressible multifluids is presented.The proposed method is a meshfree numerical procedure based on a combination of interior point scheme and interface point tracking algorithm.The discretization of unknown function and its derivatives are defined only by position of the so called Lagrangian points.The interior point formulation is based on Taylor series expansion in continuous regions on both sides of a interface.Unlike most current meshfree method,a point is settled at the interface position initially.State of interface point is updated using Rankine-Hugoniot conditions at interface together with characteristics difference computation.The interface tracking algorithm is the main feature of the method.Numerical tests show that the algorithm is oscillation-free at material interfaces and accuracy of the method is demonstrated.

In this note, we propose a new method to cure numerical shock instability by hybriding different numerical fluxes in the two-dimensional Euler equations. The idea of this method is to combine a "full-wave" Riemann solver and a "less-wave" Riemann solver, which uses a special modified weight based on the difference in velocity vectors. It is also found that such blending does not need to be implemented in all equations of the Euler system. We point out that the proposed method is easily extended to other "full-wave" fluxes that suffer from shock instability. Some benchmark problems are presented to validate the proposed method.

A new approach to the finite point method (FPM) on scattered points in two space dimensions is presented which is based on the directional differential and directional difference. The relations between the multidirectional differentials of each order are derived. Based on these relations, some explicit five-point formulae are obtained for second-order accurate approximation of the first-order directional derivatives and for first-order accurate approximation of the second-order directional derivatives. Solvability conditions for the five-point formulae and the methods for selecting the permissible neighboring point set are discussed. Numerical experiments are presented to demonstrate the performance and convergence of the proposed method.

Relations of the 3D multi-directional derivatives are studied in this paper. These relations are applied to a general second-order linear elliptical operator and the corresponding expression are obtained. These relations and expressions play important roles in the meshless finite point method.

A new upwind isoparametric interpolation finite point (UIIFP) scheme for meshless solver is proposed. This method is developed based on the least-squares procedure and an upwind isoparametric interpolation approximation of the fictitious interface directional flux. The accuracy of UIIFP scheme and 2D test Computation is presented. Copyright (C) 2007 John Wiley & Sons, Ltd.

In this paper, a new approach for approximating energy flux of temperature diffusion equation on unstructured meshes is presented, which is based on different formulae of the finite point method with different accuracies. In addition, a new numerical formula for computing cell nodal temperature is given. Numerical experiments show the good performance and accuracy of our methods. Copyright © 2008 John Wiley & Sons, Ltd.

A class of meshfree methods——finite point method on a set of two-dimensional disordered points is studied.Fundamentals of the method are established by means of directional differentials and directional differences. Formulae relating to multi-directional differentials of each order are given.Based on these formulae and with different numbers of neighboring points,five-point formulae and less-point(two-point,three-point and four-point) formulae are derived,respectively.Solvability conditions of the five-point formulae and permissible set of neighboring points are discussed.Approximate expressions for classical differential operators on a set of disordered points are derived.It is demonstrated with theoretical analysis and numerical experiments that the accuracy of these formulae is improved as the number of neighboring points increases. These approximate formulae lay foundation for constructing computational schemes of partial differential equations on a set of disordered points.They can be applied to computational methods on unstructured meshes to increase accuracy as well.

A new meshless method, called total variation diminishing (TVD) finite point method (TVDFP), is proposed. The TVDFP method is developed on the least-square procedure which uses a global stencil of grid points and the two-dimensional (2D) TVD procedure for the approximation of fictitious interface directional fluxes. We present the accuracy of the TVDFP method and several 2D test computations.

This paper describes the artificial viscosity and Godunov methods in Smoothed Particle Hydrodynamics (SPH) and analyzes the dissipative term which appears in traditional SPH and Godunov method. A traditional artificial viscosity can be mapped to a Riemann solver to keep the dissipative term almost the same, although not equivalence. Due to turning off the viscosity in rarefaction wave region the artificial viscosity method is more accurate than Godunov method. Utilizing some different approximate Riemann solvers directly, we construct some new artificial viscosities correspondingly without adjusting viscosity coefficients. With the help of heat conduction viscosity, these new viscosities give some very good numerical results in numerical experiments.

In transport theory, the convergence of the inner iteration scheme to the spherical neutron transport equation has been an open problem. In this paper, the inner iteration for a positive step function scheme is considered and its convergence in spherical geometry is proved.