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 solve...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.展开更多
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 fir...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 differ- entials, 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（h2） on uniform point distributions.展开更多
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 explicit expressions and accuracy of th...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 explicit 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 difference 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 neighboring 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.展开更多
基金supported in part by the National Natural Science Foundation of China under(Grant No.10871029)foundation of LCP.
文摘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.
基金This project was supported by the National Natural Science Foundation of China （11371066, 11372050）, and the Foundation of National Key Laboratory of Science and Technology Computation Physics.
文摘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 differ- entials, 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（h2） on uniform point distributions.
基金supported by the National Natural Science Foundation of China(11671049)the Foundation of LCP,and the CAEP Foundation(CX2019026).
文摘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 explicit 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 difference 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 neighboring 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.
