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摘 要:針对二阶显式TVD RungeKutta局部间断Galerkin方法求解热传导方程的稳定性问题,在方程的解是充分光滑的情况下,通过有限元分析的方法,在理论上严格的证明了对于任意非均匀正则网格和k次分段多项式间断有限元空间,当CourantFriedrichsLewy (CFL)条件取为τ≤λμ-2h2时,算法是L2稳定的,其中τ,h分别是时间步长和空间步长,μ,λ是与h,τ无关的常数。
关键词:
RungKutta法;局部间断Galerkin方法;稳定性分析;热传导方程;L2稳定
DOI:10.15938/j.jhust.2017.06.020
中图分类号: O29
文献标志码: A
文章编号: 1007-2683(2017)06-0109-04
Abstract:To analyze the stability of the local discontinuous Garlerkin method for heat equation, where the time discretization is the explicit TVD RungeKutta method. For the sufficiently smooth solution case, when the finite element space is the kth order piecewise polynomial space on the regular meshes, we use the finite element analysis technique to proof the L2norm stability for hear equation under the CFL condition τ≤λμ-2h2, where τ,h are the time step and the length of the element respectively, and μ,λ are constants independent of h,τ.
Keywords:RungeKutta;finite element;stability analysis;partial differential equations;L2norm stability
0 引 言
间断有限元是一类有限元空间取为间断多项式空间的有限元方法,具有易于实现hp自适应性和灵活处理复杂计算区域等优点。 该方法由Reed和Hill于1973年在求解稳态的中子运输方程时提出[1]。上世纪80年代末90年代初,Cockburn和Shu针对非线性发展型双曲守恒律方程提出了TVD RungeKutta间断有限元方法,详细的讨论了方程组以及多维问题[2-6]。 1998年,根据Bassi和Rebay对于粘性NavierStorkes 方程成功的计算结果[7],Cockburn和Shu又把这个方法推广到了求解对流扩散方程,提出了局部间断有限元思想[8]。 2002年,Yan和Shu针对含有高阶空间导数的偏微分方程给出了局部间断有限元算法[9]。 更多关于间断有限元和局部间断有限元的研究现状可以查阅综述性文献[10-15]和专著[16] 。注意到,间断有限元法只用于空间离散,在时间离散方面,对于热传导方程,可以采用显式的时间离散格式[17],而高阶问题则需要效率更高的隐式或半隐式格式[18]。 同时,随着问题的深入,近年来关于间断有限元和局部间断有限元方法的稳定性问题的研究也逐步展开。 2004年,Zhang 和Shu 首次给出了非线性双曲守恒律方程的二阶显式TVD RungeKutta 间断有限元方法的稳定性分析[19]。2010年,Zhang和Shu讨论了三阶显式TVD RungeKutta 间断有限元解线性双曲守恒律方程的L2稳定性问题[20]。 2015年,Wang和Shu又讨论了半隐式的RungeKutta 局部间断有限元解非线性对流扩散方程时的稳定性问题[21]。 由于全离散格式的复杂性,目前关于稳定性分析的研究成果并不多。
3 结 论
本文证明了在时间和空间步长满足τ≤(6-42)μ-2h2时,二阶显示TVD RungeKutta 局部间断Galerkin方法是L2稳定的。 在将来的工作中,我们会讨论隐式或半隐式的时间离散方法结合局部间断有限元法的稳定性分析。
参 考 文 献:
[1] REED W H, HILL T R. Triangular Mesh Methods for the Neutron Transport Equation[R].Los Alamos Scientific Lab. N. Mex.(USA), 1973.
[2] COCKBURN B, SHU Chiwang. The RungeKutta Local Projection DiscontinuousGalerkin Finite Element Method for Scalar Conservation Laws[J].RAIROModélisation Mathématique et Analyse Numérique 1991,25(3): 337-361.
[3] COCKBURN B, SHU C W. The RungeKutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws.II:General Framework[J]. Mathematics of Computation, 1989, 52(186): 411-435. [4] COCKBURN B, LIN S Y, SHU C W. The RungeKutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws III: Onedimensional Systems[J].Journal of Computational Physics, 1989, 84(1):90-113.
[5] COCKBURN B, HOU S, SHU C W. The RungeKutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws. IV:The Multidimensional Case[J]. Mathematics of Computation, 1990, 54(190): 545-581.
[6] COCKBURN B, SHU C W. The RungeKutta Discontinuous Galerkin Method for Conservation Laws V: Multidimensional Systems[J]. Journal of Computational Physics, 1998, 141(2): 199-224.
[7] BASSI F, REBAY S. A Highorder Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible NavierStokes Equations[J].Journal of Computational Physics, 1997, 131(2): 267-279.
[8] COCKBURN B, SHU CW. The Local Discontinuous Galerkin Method for Timedependent Convectiondiffusion Systems[J]. SIAM J Numer Anal, 1998(35): 2440-2463.
[9] YAN J, SHU CW. Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives[J].J Sci Comput, 2002(17): 27-47.
[10]COCKBURN B, SHU C W. RungeKutta Discontinuous Galerkin Methods for Convectiondominated Problems[J]. Journal of Scientific Computing, 2001, 16(3): 173-261.
[11]SHU C W. A Brief Survey on Discontinuous Galerkin Methods in Computational Fluid Dynamics[J]. Advances in Mechanics, 2013(43): 541-554.
[12]XU Y, SHU C. Local Discontinuous Galerkin Methods for Three Classes of Nonlinear Wave Equations[J]. Journal of Computational Mathematics, 2004,22(2):250-274.
[13]SHU C W. Discontinuous Galerkin Methods: General Approach and Stability[J].Numerical Solutions of Partial Differential Equations, 2009: 149-201.
[14]SHU C W. Discontinuous Galerkin Method for Timedependent Problems: Survey and Recent Developments[M]. Springer International Publishing, 2014: 25-62.
[15]SHU C W. High Order WENO and DG Methods for Timedependent Convectiondominated PDEs: A Brief Survey of Several Recent Developments [J]. Journal of Computational Physics, 2016, 316: 598-613.
[16]HESTHAVEN J S, WARBURTON T. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications[M]. Springer: Science & Business.Media, 2007.
[17]SHU C W, OSHER S. Efficient Implementation of Essentially Nonoscillatory Shockcapturing Schemes[J]. Journal of Computational Physics, 1988, 77(2):439-471.
[18]XIA Y, XU Y, SHU CW. Efficient Time Discretization for Local Discontinuous Galerkin Methods[J]. Discrete and Continuous Dynamical Systems Series B, 2007, 8(3): 677.
[19]ZHANG Q, SHU C W. Error Estimates to Smooth Solutions of RungeKutta Discontinuous Galerkin Methods for Scalar Conservation Laws[J]. SIAM Journal on Numerical Analysis, 2004, 42(2): 641-666.
[20]ZHANG Q, SHU C W. Stability Analysis and a Priori Error Estimates of the Third Order Explicit RungeKutta Discontinuous Galerkin Method for Scalarconservation Laws[J]. SIAM Journal on Numerical Analysis, 2010, 48(3): 1038-1063.
[21]WANG H, SHU CW, ZHANG Q. Stability Analysis and Error Estimates of Local Discontinuous Galerkin Methods with Implicitexplicit Timemarching for Nonlinear Convectiondiffusion Problems[J]. Applied Mathematic and Computation, 2016, 272(2): 237-258.
(編辑:温泽宇)
关键词:
RungKutta法;局部间断Galerkin方法;稳定性分析;热传导方程;L2稳定
DOI:10.15938/j.jhust.2017.06.020
中图分类号: O29
文献标志码: A
文章编号: 1007-2683(2017)06-0109-04
Abstract:To analyze the stability of the local discontinuous Garlerkin method for heat equation, where the time discretization is the explicit TVD RungeKutta method. For the sufficiently smooth solution case, when the finite element space is the kth order piecewise polynomial space on the regular meshes, we use the finite element analysis technique to proof the L2norm stability for hear equation under the CFL condition τ≤λμ-2h2, where τ,h are the time step and the length of the element respectively, and μ,λ are constants independent of h,τ.
Keywords:RungeKutta;finite element;stability analysis;partial differential equations;L2norm stability
0 引 言
间断有限元是一类有限元空间取为间断多项式空间的有限元方法,具有易于实现hp自适应性和灵活处理复杂计算区域等优点。 该方法由Reed和Hill于1973年在求解稳态的中子运输方程时提出[1]。上世纪80年代末90年代初,Cockburn和Shu针对非线性发展型双曲守恒律方程提出了TVD RungeKutta间断有限元方法,详细的讨论了方程组以及多维问题[2-6]。 1998年,根据Bassi和Rebay对于粘性NavierStorkes 方程成功的计算结果[7],Cockburn和Shu又把这个方法推广到了求解对流扩散方程,提出了局部间断有限元思想[8]。 2002年,Yan和Shu针对含有高阶空间导数的偏微分方程给出了局部间断有限元算法[9]。 更多关于间断有限元和局部间断有限元的研究现状可以查阅综述性文献[10-15]和专著[16] 。注意到,间断有限元法只用于空间离散,在时间离散方面,对于热传导方程,可以采用显式的时间离散格式[17],而高阶问题则需要效率更高的隐式或半隐式格式[18]。 同时,随着问题的深入,近年来关于间断有限元和局部间断有限元方法的稳定性问题的研究也逐步展开。 2004年,Zhang 和Shu 首次给出了非线性双曲守恒律方程的二阶显式TVD RungeKutta 间断有限元方法的稳定性分析[19]。2010年,Zhang和Shu讨论了三阶显式TVD RungeKutta 间断有限元解线性双曲守恒律方程的L2稳定性问题[20]。 2015年,Wang和Shu又讨论了半隐式的RungeKutta 局部间断有限元解非线性对流扩散方程时的稳定性问题[21]。 由于全离散格式的复杂性,目前关于稳定性分析的研究成果并不多。
3 结 论
本文证明了在时间和空间步长满足τ≤(6-42)μ-2h2时,二阶显示TVD RungeKutta 局部间断Galerkin方法是L2稳定的。 在将来的工作中,我们会讨论隐式或半隐式的时间离散方法结合局部间断有限元法的稳定性分析。
参 考 文 献:
[1] REED W H, HILL T R. Triangular Mesh Methods for the Neutron Transport Equation[R].Los Alamos Scientific Lab. N. Mex.(USA), 1973.
[2] COCKBURN B, SHU Chiwang. The RungeKutta Local Projection DiscontinuousGalerkin Finite Element Method for Scalar Conservation Laws[J].RAIROModélisation Mathématique et Analyse Numérique 1991,25(3): 337-361.
[3] COCKBURN B, SHU C W. The RungeKutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws.II:General Framework[J]. Mathematics of Computation, 1989, 52(186): 411-435. [4] COCKBURN B, LIN S Y, SHU C W. The RungeKutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws III: Onedimensional Systems[J].Journal of Computational Physics, 1989, 84(1):90-113.
[5] COCKBURN B, HOU S, SHU C W. The RungeKutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws. IV:The Multidimensional Case[J]. Mathematics of Computation, 1990, 54(190): 545-581.
[6] COCKBURN B, SHU C W. The RungeKutta Discontinuous Galerkin Method for Conservation Laws V: Multidimensional Systems[J]. Journal of Computational Physics, 1998, 141(2): 199-224.
[7] BASSI F, REBAY S. A Highorder Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible NavierStokes Equations[J].Journal of Computational Physics, 1997, 131(2): 267-279.
[8] COCKBURN B, SHU CW. The Local Discontinuous Galerkin Method for Timedependent Convectiondiffusion Systems[J]. SIAM J Numer Anal, 1998(35): 2440-2463.
[9] YAN J, SHU CW. Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives[J].J Sci Comput, 2002(17): 27-47.
[10]COCKBURN B, SHU C W. RungeKutta Discontinuous Galerkin Methods for Convectiondominated Problems[J]. Journal of Scientific Computing, 2001, 16(3): 173-261.
[11]SHU C W. A Brief Survey on Discontinuous Galerkin Methods in Computational Fluid Dynamics[J]. Advances in Mechanics, 2013(43): 541-554.
[12]XU Y, SHU C. Local Discontinuous Galerkin Methods for Three Classes of Nonlinear Wave Equations[J]. Journal of Computational Mathematics, 2004,22(2):250-274.
[13]SHU C W. Discontinuous Galerkin Methods: General Approach and Stability[J].Numerical Solutions of Partial Differential Equations, 2009: 149-201.
[14]SHU C W. Discontinuous Galerkin Method for Timedependent Problems: Survey and Recent Developments[M]. Springer International Publishing, 2014: 25-62.
[15]SHU C W. High Order WENO and DG Methods for Timedependent Convectiondominated PDEs: A Brief Survey of Several Recent Developments [J]. Journal of Computational Physics, 2016, 316: 598-613.
[16]HESTHAVEN J S, WARBURTON T. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications[M]. Springer: Science & Business.Media, 2007.
[17]SHU C W, OSHER S. Efficient Implementation of Essentially Nonoscillatory Shockcapturing Schemes[J]. Journal of Computational Physics, 1988, 77(2):439-471.
[18]XIA Y, XU Y, SHU CW. Efficient Time Discretization for Local Discontinuous Galerkin Methods[J]. Discrete and Continuous Dynamical Systems Series B, 2007, 8(3): 677.
[19]ZHANG Q, SHU C W. Error Estimates to Smooth Solutions of RungeKutta Discontinuous Galerkin Methods for Scalar Conservation Laws[J]. SIAM Journal on Numerical Analysis, 2004, 42(2): 641-666.
[20]ZHANG Q, SHU C W. Stability Analysis and a Priori Error Estimates of the Third Order Explicit RungeKutta Discontinuous Galerkin Method for Scalarconservation Laws[J]. SIAM Journal on Numerical Analysis, 2010, 48(3): 1038-1063.
[21]WANG H, SHU CW, ZHANG Q. Stability Analysis and Error Estimates of Local Discontinuous Galerkin Methods with Implicitexplicit Timemarching for Nonlinear Convectiondiffusion Problems[J]. Applied Mathematic and Computation, 2016, 272(2): 237-258.
(編辑:温泽宇)