一种无数值积分的间断Galerkin有限元方法
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摘要
在使用间断Galerkin有限元方法的计算过程中,需要构造相应的积分表达式作为数值求解的出发点,继而会引入体积分和面积分。对于这些积分项的值,一般需要通过数值积分的方法获得。当需要使用高阶间断Galerkin有限元方法时,数值积分计算精度的要求会相应地增加,其所需的计算量将变得很大,而数值积分的计算量又在很大程度上决定了间断Galerkin有限元方法的计算效率。针对这一问题,通过建立Lagrange插值多项式基函数和Jacobi正交多项式基函数的一定关系,构造了一种无数值积分的间断Galerkin有限元方法显式半离散格式,并对不同条件下的线性和非线性一维、二维守恒律进行了直接数值模拟,得到了理想的数值结果。该方法不再需要通过数值积分来计算每个单元的积分项,而且有效地达到了间断Galerkin有限元方法的高阶精度,其对于构造更为高效的高阶间断Galerkin有限元计算方法具有非常显著的意义。
In the calculation process of using discontinuous Galerkin finite element method,the corresponding integral expression needs to be constructed as the starting point of numerical solving method,then the volume integral and surface integral are introduced.Under normal circumstances,the value of these integral items is acquired by using numerical integration method.When high-order discontinuous Galerkin finite element method needs to be used,the demand for numerical integration calculation accuracy increases accordingly.This demand results in large calculation amount,and the numerical integration calculation amount largely determines the computational efficiency of discontinuous Galerkin finite element method.In order to solve this problem,an explicit semi-discretization of quadrature-free discontinuous Galerkin finite element method was structured by establishing the relationship between Lagrange interpolation polynomial basis function and Jacobi orthogonal polynomial basis function.An explicit semidiscretization was applied to the direct numerical simulation of linear and nonlinear onedimensional,two-dimensional conservation laws with different condition,and ideal numerical results were obtained.Using this method,there's no need to calculate the integral items of each element by numerical integration,and the high precision of discontinuous Galerkin finite element method can be achieved effectively.The method has very significant meaning for structuring high efficient high-order discontinuous Galerkin finite element method.
In the calculation process of using discontinuous Galerkin finite element method,the corresponding integral expression needs to be constructed as the starting point of numerical solving method,then the volume integral and surface integral are introduced.Under normal circumstances,the value of these integral items is acquired by using numerical integration method.When high-order discontinuous Galerkin finite element method needs to be used,the demand for numerical integration calculation accuracy increases accordingly.This demand results in large calculation amount,and the numerical integration calculation amount largely determines the computational efficiency of discontinuous Galerkin finite element method.In order to solve this problem,an explicit semi-discretization of quadrature-free discontinuous Galerkin finite element method was structured by establishing the relationship between Lagrange interpolation polynomial basis function and Jacobi orthogonal polynomial basis function.An explicit semidiscretization was applied to the direct numerical simulation of linear and nonlinear onedimensional,two-dimensional conservation laws with different condition,and ideal numerical results were obtained.Using this method,there's no need to calculate the integral items of each element by numerical integration,and the high precision of discontinuous Galerkin finite element method can be achieved effectively.The method has very significant meaning for structuring high efficient high-order discontinuous Galerkin finite element method.
引文
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[21]AN J.The solving method and algorithm implementation of the zeros of derivative of Jacobi polynomial[J].Journal of Guizhou Normal University,2014,32(4):42-45.(in Chinese)安静.Jacobi多项式及其导数零点的求解方法及算法实现[J].贵州师范大学学报:自然科学版,2014,32(4):42-45.
[22]Gordon W J,Hall C A.Construction of curvilinear co-ordinate systems and applications to mesh generation[J].International Journal for Numerical Methods in Engineering,1973,7(4):461-477.
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[24]Tu S,Aliabadi S.A slope limiting procedure in discontinuous Galerkin finite element method for gasdynamics applications[J].International Journal of Numerical Analysis and Modeling,2005,2(2):163-178.
[25]Guo Y H,Yang Y,Liu Y O.An original designed limiter for discontinuous Galerkin method[J].Acta Aerodynamica Sinica,2014,32(4):462-467.(in Chinese)郭永恒,杨永,刘逸鸥.一种基于间断Galerkin格式的新型限制器设计[J].空气动力学学报,2014,32(4):462-467.
[2] Lesaint P,Raviart P A.On a finite element method for solving the neutron transport equation[J].Mathematical Aspects of Finite Elements in Partial Differential Equations,1974(33):89-123.
[3] Wellford L C,Oden J T.A theory of discontinuous finite element galerkin approximations of shock waves in nonlinear elastic solids part 1:Variational theory[J].Computer Methods in Applied Mechanics and Engineering,1976,8(1):1-16.
[4] Delfour M C,Trochu F.Discontinuous finite element methods for the approximation of optimal control problems governed by hereditarydifferentialsystems[M]//SpringerBerlin Heidelberg,1978:256-271.
[5] Chavent G,Salzano G.A finite-element method for the 1-D water flooding problem with gravity[J]. Journal of Computational Physics,1982,45(3):307-344.
[6] Chavent G,Cockburn B.The local projection discontinuousGalerkin finite element method for scalar conservation laws[J].RAIRO-Modélisation Mathématique et Analyse Numérique,1989,23(4):565-592.
[7] Shu C W.Total-variation-diminishing time discretizations[J].SIAM Journal on Scientific and Statistical Computing,1988,9(6):1073-1084.
[8] Shu C W.TVB uniformly high-order schemes for conservation laws[J].Mathematics of Computation,1987,49(179):105-121.
[9] Cockburn B,Shu C W.The Runge-Kutta local projection discontinuous-Galerkin finite element method for scalar conservation laws[J].RAIRO-Modélisation Mathématique et Analyse Numérique,1991,25(3):337-361.
[10]Cockburn B,Shu C W.TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws.II.General framework[J].Mathematics of Computation,1989,52(186):411-435.
[11]Cockburn B,Lin S Y,Shu C W.TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III:one-dimensional systems[J].Journal of Computational Physics,1989,84(1):90-113.
[12]Cockburn B, Hou S,Shu C W.The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws.IV. The multidimensional case[J].Mathematics of Computation,1990,54(190):545-581.
[13]Cockburn B, Shu C W.The Runge-Kutta discontinuous Galerkin method for conservation laws V:multidimensional systems[J].Journal of Computational Physics,1998,141(2):199-224.
[14]Guo Y H, Yang Y,Zhang Q, A high efficient implicit discontinuous Galerkin method[J].Acta Aerodynamica Sinica,2012,30(2):250-253.(in Chinese)郭永恒,杨永,张强.一种高效的隐式间断Galerkin方法研究[J].空气动力学学报,2012,30(2):250-253.
[15]Xia Y D,Wu Y Z,Lv H Q,et al,Parallel computation of a high-order discontinuous Galerkin method on unstructured grids[J].Acta Aerodynamica Sinica,2011,29(05):537-541.(in Chinese)夏轶栋,伍贻兆,吕宏强,等.高阶间断有限元法的并行计算研究[J].空气动力学学报,2011,29(05):537-541.
[16]Atkins H L,Shu C W.Quadrature-free implementation of discontinuous Galerkin method for hyperbolic equations[J].AIAA Journal,1996,36(5):775-782.
[17]Atkins H L.Continued development of the discontinuous Galerkin method for computational aeroacoustic applications[M].American Institute of Aeronautics and Astronautics,1997.
[18]Cockburn B,Shu C W.Runge-Kutta discontinuous Galerkin methods for convection-dominated problems[J].Journal of Scientific Computing,2001,16(3):173-261.
[19] Hesthaven J S,Gottlieb S,Gottlieb D.Spectral methods for time-dependent problems-[M].Cambridge University Press,2007.
[20]Zhao T G,Zhang J H,Some properties of Jacobi orthogonal polynomials[J].Journal of Gansu Normal Colleges,2009,14(5):1-3.(in Chinese)赵廷刚,张建宏.Jacobi正交多项式的一些性质[J].甘肃高师学报,2009,14(5):1-3.
[21]AN J.The solving method and algorithm implementation of the zeros of derivative of Jacobi polynomial[J].Journal of Guizhou Normal University,2014,32(4):42-45.(in Chinese)安静.Jacobi多项式及其导数零点的求解方法及算法实现[J].贵州师范大学学报:自然科学版,2014,32(4):42-45.
[22]Gordon W J,Hall C A.Construction of curvilinear co-ordinate systems and applications to mesh generation[J].International Journal for Numerical Methods in Engineering,1973,7(4):461-477.
[23]LeVeque R J.Finite volume methods for hyperbolic problems[M].Cambridge university press,2002.
[24]Tu S,Aliabadi S.A slope limiting procedure in discontinuous Galerkin finite element method for gasdynamics applications[J].International Journal of Numerical Analysis and Modeling,2005,2(2):163-178.
[25]Guo Y H,Yang Y,Liu Y O.An original designed limiter for discontinuous Galerkin method[J].Acta Aerodynamica Sinica,2014,32(4):462-467.(in Chinese)郭永恒,杨永,刘逸鸥.一种基于间断Galerkin格式的新型限制器设计[J].空气动力学学报,2014,32(4):462-467.