非结构网格上求解二维浅水波方程的最小二乘有限体积法
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摘要
浅水波方程在海洋工程、环境问题、生态学等许多领域有着广泛的应用,众所周知,浅水波方程的解很难求得,而且即使在初值很光滑的情况下其解也会出现间断。本文主要讨论了非结构网格上解浅水波方程的最小二乘有限体积方法。
本文共分五章:
第一章简要介绍了浅水波方程的物理背景、数学模型及其数值解法的研究发展。
第二章介绍了结构网格上和非结构网格上解浅水波方程的数值方法,详细介绍了具有代表性的求解浅水波方程的Godunov型有限体积格式和复合有限体积格式。
第三章介绍了双曲型守恒律有限体积法的框架、网格和控制体积、TVD型Runge-Kutta时间离散方法、极值原理以及限制器形式。
第四章构造出了非结构网格上解浅水波方程的非振荡有限体积方法。借助最小二乘的思想,得到了单元上的一个线性插值多项式,并利用极值原理的思想,在保证其解不出现新的极值的情况下,构造出了非结构网格上解浅水波方程的非振荡数值格式。该数值方法避免了模板的选取,计算量要比ENO方法小得多,而且仍具有高分辨率。
第五章利用最小二乘思想,通过一次扩张模板来提高数值方法的精度,发展了在非结构网格上解二维浅水波方程的高精度有限体积法。构造数值算法过程中利用极值原理的思想,来保证解不出现新的极值,构造出解浅水波方程的三阶数值格式。该格式具有高分辨率,准确地模拟了间断现象。
最后总结了四、五两章讨论的数值方法的优缺点,提出了今后工作的发展方向。
The Shallow Water Equations arises in many applications such as oceanic, environment and bionomics. It is well known that the solutions of the Shallow Water Equations can be discontinuous even when the initial condition is smooth. This thesis is concerned with the least square finite volume method for Shallow Water Equations on unstructured meshes.
The thesis consists of five chapters.
In chapter 1, we briefly review the physical background, mathematical model and numerical methods for the Shallow Water Equations.
In chapter 2, we present the numerical schemes for resolving Shallow Water Equations on structured and unstructured meshes, such as Godunov and composite schemes.
In chapter 3, we present frame of the finite volume methods for 2-D hyperbolic conservation laws. Also we introduce control volume, TVD Runge-Kutta time discretization, maximum principle and limiters.
In chapter 4, we present the non-oscillatory finite volume method for the Shallow Water Equations on unstructured meshes. We get the linear interpolation using the least square method and ensure that the solution will not produce the new extremum using the maximum principle. Finally we construct the non-oscillatory numerical scheme. The method avoids choosing template, needs less computation and has high resolution of the discontinue.
In chapter 5, we present the third-order finite volume method for the Shallow Water Equations on unstructured meshes. We get the secondary interpolation using the least square method by enlarging the template once and ensure that the solution will not produce the new extremum using the maximum principle. Finally we construct the third-order high resolution numerical scheme. It can capture discontinuous well.
At last, we summarize the advantages and disadvantages in chapter4, 5, and show our task in the future.
本文共分五章:
第一章简要介绍了浅水波方程的物理背景、数学模型及其数值解法的研究发展。
第二章介绍了结构网格上和非结构网格上解浅水波方程的数值方法,详细介绍了具有代表性的求解浅水波方程的Godunov型有限体积格式和复合有限体积格式。
第三章介绍了双曲型守恒律有限体积法的框架、网格和控制体积、TVD型Runge-Kutta时间离散方法、极值原理以及限制器形式。
第四章构造出了非结构网格上解浅水波方程的非振荡有限体积方法。借助最小二乘的思想,得到了单元上的一个线性插值多项式,并利用极值原理的思想,在保证其解不出现新的极值的情况下,构造出了非结构网格上解浅水波方程的非振荡数值格式。该数值方法避免了模板的选取,计算量要比ENO方法小得多,而且仍具有高分辨率。
第五章利用最小二乘思想,通过一次扩张模板来提高数值方法的精度,发展了在非结构网格上解二维浅水波方程的高精度有限体积法。构造数值算法过程中利用极值原理的思想,来保证解不出现新的极值,构造出解浅水波方程的三阶数值格式。该格式具有高分辨率,准确地模拟了间断现象。
最后总结了四、五两章讨论的数值方法的优缺点,提出了今后工作的发展方向。
The Shallow Water Equations arises in many applications such as oceanic, environment and bionomics. It is well known that the solutions of the Shallow Water Equations can be discontinuous even when the initial condition is smooth. This thesis is concerned with the least square finite volume method for Shallow Water Equations on unstructured meshes.
The thesis consists of five chapters.
In chapter 1, we briefly review the physical background, mathematical model and numerical methods for the Shallow Water Equations.
In chapter 2, we present the numerical schemes for resolving Shallow Water Equations on structured and unstructured meshes, such as Godunov and composite schemes.
In chapter 3, we present frame of the finite volume methods for 2-D hyperbolic conservation laws. Also we introduce control volume, TVD Runge-Kutta time discretization, maximum principle and limiters.
In chapter 4, we present the non-oscillatory finite volume method for the Shallow Water Equations on unstructured meshes. We get the linear interpolation using the least square method and ensure that the solution will not produce the new extremum using the maximum principle. Finally we construct the non-oscillatory numerical scheme. The method avoids choosing template, needs less computation and has high resolution of the discontinue.
In chapter 5, we present the third-order finite volume method for the Shallow Water Equations on unstructured meshes. We get the secondary interpolation using the least square method by enlarging the template once and ensure that the solution will not produce the new extremum using the maximum principle. Finally we construct the third-order high resolution numerical scheme. It can capture discontinuous well.
At last, we summarize the advantages and disadvantages in chapter4, 5, and show our task in the future.
引文
[1] Anastasiou K. and Chan C T. , Solution of the 2D shallow water equations using the finite volume method on unstructured triangular meshes, Inter J Numer Methods in Fluids, 1997, 24, 1225-1245
[2] S. Chippada, C. Dawson, M. F. Martinez and M. Wheeler, A Godunov-Type Finite Volume Method for Systems of shallow water equations, Center for Research on Parallel Computation, 1997, 1-34
[3] Luettich Jr. R. A., Westerink J. J. and Scheffiner N. W., An Advanced Three-Dimensional Circulation Model for Shelves, Coasts and Estuaries, Report 1, U.S. Army Corps of Engineers, Washington, D.C. 20314 (1000), 1991
[4] Lynch, D.R. and Gray W.G., A wave equation model for finite element tidal computations, Comput Fluids, 1979, 7, 207-228
[5] Harten A., High resolution schemes for hyperbolic conservation laws, J Comp Phys, 1983, 49, 357-393
[6] Harten A., On high-order accurate interpolation for non-oscillatory shock-capturing schemes , The IMA Volumes in Mathematics and its Applications.springer- ver -lag, 1986, 2, 71-106
[7] Roe P L., Approximate Riemann solvers parameter vector and difference schemes. J Comp Phys, 1981, 43, 357-372
[8] Bermudez A, bodriguez C and Vilar M A., Solving shallow water equations by a mixed implicit finite element method, IMA J Numer Anal, 1991, 11(1), 79-97
[9] Chipada S, Dawson C N, Martinez M L, et al., Finite element approximations to the system of shallow water equations I: Continuous-time a priori error estimates , SIAM J Num Anal, 1998, 35(2), 692-711
[10]Agoshkov V I, Crochinnikov E, Quarteroni A, et at, Recent developments in the numerical simulation of shalow water equations R: Temporal discretization, Math Models Methods Appl Sci, 1994, 55(4), 533-566
[11]Agoshkov V I, Quarteroni A and Saleri F.Recent developments in the numerical simulation of shallow water equation I: Boundary conditions, Appl Numer Math, 1994, 15(1), 175-200
[12] Gray W G, A finite element study of tidal flow data for the north sea and English channel, Adv Water Res, 1989, 12(1), 143-154
[13] Gray W G, Drolet J and Kinnmark I P E, A simulation of tidal flow in the southern part of the north sea and English charnel, Adv Water Res, 1987, 10(1), 131-137
[14]P.Glaister, Flux difference splitting for open-channel flows, Int.J.for Numerical Methods in Fluids, 1993, 16, 629-654
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[19] R.A.Walters and E.J.Barragy, Comparison of h and p finite element approximations of the shallow water equations, Int.J.for Numerical Methods in Fluids, 1997, 24, 61 -79
[20] Dells A I, Skeels C P and Ryrie S C, Evaluation of some approximate Riemann solver for transient open channel flows, Journal of hydraulic engineering, 2000, 38(3), 217-231
[21]Vadym Aizinger and Clint Dawson , A discontinuous Galerkin method for two-dimensional flow and transport in shallow water.Advances in Water Resources, 2002, 25, 67-84
[22] Christopher Zoppou and Stephen Roberts, Numerical Solution Of The Two-Dimensional Unsteady Dam Break, Mathematics Subject Classification, 1991, 1-22
[23]Alcrudo F and Garicia-Navarro P., A high resolution Godunov-type scheme in finite volumes for the 2D shallow water equations.Inter for Number Methods in Fluids, 1995, 20, 743-770
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[25] LeVeque R.J, Numerical Methods for Conservation Laws, Birkhauser Verlag, 1992
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[27]R. A. Walters and G. F. Carey, Analysis of spurious oscillation modes for the shallow-water and Navier-Stokes equations, Comput. Fluids, 1983, 11, 51-68
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[33] Von Neuman J, Richtmyer R D.A method for numerical calculations of hydrodynamical shocks, J Appl Phys, 21(1950), 232-256
[34] Van Leer, Towards the ultimate conservative difference V.A second order sequel to Godunov method, J Comp Phys, 1979, 32, 101-136
[35]Sweby P K,High resolution schemes using flux limiters for hyperbolic conservation laws,SIAM J Num Anal,1984,21,995-1011
[36]Yee H C,Construction of explicit and implicit symmetric TVD schemes and their applications,J CompPhys,1987,68,151-179
[37]Goodman J B and Leveque R J,On the accuracy of stable schemes for 2D scalar conservation laws.Math Comput,1998,45,15-21
[38]Harten A,Engquist B,Osher S and Chakravarthy R.Uniformly high order accurate essentially nonoscillatory schemes Ⅲ.J Comp Phys,1987,71,231-303
[39]HartenA,LaxPD and van Leer B,On upstream differencing and Godunov-type schemes for hyperbolic conservation laws,SIAM Rev,1983,25,35-61
[40]McDonald P W,The Computation of transonic flow though two dimensional gas turbine cascades,ASME,1971,71-89
[41]MacCormack R W,and Paullay A J,Computational efficiency achieved by time splitting of finite difference operators,AIAApaper,1972,72-154
[42]Jameson A and Mavriplis D.Finite volume solution of the two-dimensional Euler equation on a regular triangular mesh,AIAApaper,1985,85(0435)
[43]Mavriplis D,Multigrid solution of the Euler equation on unstructured and adaptive meshes,ICASEReport,1987,87(53),413-430
[44]Mavriplis D and Jameson A,Multigrid solution of two-dimensional Euler equation on unstructured triangular meshes,AIAA paper,1987,87(0353)
[45]Shu C W,Total-variation-diminishing time discretizations,SIAM Journal on Scientific Computing,1988,9,1073-1084
[46]Jameson A,Multigrid Algorithms for Compressible Flow Calculations,In Proceeding of the Second European Conference on Multigrid Methods,1986,1228,166-201
[47]Barth T and Frederickson P,High order solution of the Euler equations on unstructured grids using quadratic reconstruction,AIAA paper,1990,90(0013)
[48]Wang J S and Liu R X,The composite finite volume method on unstructured meshes for 2D shallow water equations,Inter J for Number Methods in Fluids,2001,37,933-949
[49]A.Valiani,V.Caleffi and A.Zanni,Malpasset dam-break Simulation using a Two-Dimen -sional Finite Volume Method,J.Hydr.Engrg.,2002,128(5),460-472
[50]Ji-Wen Wang and Ru-Xun Liu,A comparative study of finite volume methods on unstructured meshes for simulation of 2D shallow water wave problem,Mathematics and Computers in Simulation,2000,53,171-184
[51]潘存鸿,徐昆,三角形网格下求解二维浅水波方程的KFVS格式,水利学报,2006,37(7),858-864
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[53]王如云,浅水波方程的TVD有限差分数值模拟,海洋与湖沼,1991,22(2),115-123
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[2] S. Chippada, C. Dawson, M. F. Martinez and M. Wheeler, A Godunov-Type Finite Volume Method for Systems of shallow water equations, Center for Research on Parallel Computation, 1997, 1-34
[3] Luettich Jr. R. A., Westerink J. J. and Scheffiner N. W., An Advanced Three-Dimensional Circulation Model for Shelves, Coasts and Estuaries, Report 1, U.S. Army Corps of Engineers, Washington, D.C. 20314 (1000), 1991
[4] Lynch, D.R. and Gray W.G., A wave equation model for finite element tidal computations, Comput Fluids, 1979, 7, 207-228
[5] Harten A., High resolution schemes for hyperbolic conservation laws, J Comp Phys, 1983, 49, 357-393
[6] Harten A., On high-order accurate interpolation for non-oscillatory shock-capturing schemes , The IMA Volumes in Mathematics and its Applications.springer- ver -lag, 1986, 2, 71-106
[7] Roe P L., Approximate Riemann solvers parameter vector and difference schemes. J Comp Phys, 1981, 43, 357-372
[8] Bermudez A, bodriguez C and Vilar M A., Solving shallow water equations by a mixed implicit finite element method, IMA J Numer Anal, 1991, 11(1), 79-97
[9] Chipada S, Dawson C N, Martinez M L, et al., Finite element approximations to the system of shallow water equations I: Continuous-time a priori error estimates , SIAM J Num Anal, 1998, 35(2), 692-711
[10]Agoshkov V I, Crochinnikov E, Quarteroni A, et at, Recent developments in the numerical simulation of shalow water equations R: Temporal discretization, Math Models Methods Appl Sci, 1994, 55(4), 533-566
[11]Agoshkov V I, Quarteroni A and Saleri F.Recent developments in the numerical simulation of shallow water equation I: Boundary conditions, Appl Numer Math, 1994, 15(1), 175-200
[12] Gray W G, A finite element study of tidal flow data for the north sea and English channel, Adv Water Res, 1989, 12(1), 143-154
[13] Gray W G, Drolet J and Kinnmark I P E, A simulation of tidal flow in the southern part of the north sea and English charnel, Adv Water Res, 1987, 10(1), 131-137
[14]P.Glaister, Flux difference splitting for open-channel flows, Int.J.for Numerical Methods in Fluids, 1993, 16, 629-654
[15]B.T. Nadiga, An adaptive discrete-velocity model for the shallow water equations, J.comp.Phys, 1995, 121, 271-280
[16]B.T. Nadiga, L.Margolin, P.K.Smolarkiewic, Semi-lagrangian shllow water modeling on the cm-5, In A.Ecer, J.Periaux, N.Satofuka, and S.Taylor, esitor, Parallel Computational Fluid Dynamics: Implementations and Results Using Parallel Computaters, Proceedings of the Parallel CFD'95 Conference, Pasadena, CA, U.S.A., 1995, 26(29), 529-536
[17]O.C. Zienkiewicz and P. Ortiz, A split-characteristic based finite element model for the shallow water equations, Int.J.for Numerical Methods in Fluids, 1995, 20, 1061-1080
[18]D.Ambrosi, Approximation of shallow water equations by roe's Riemann solver, Int.J. for Numerical Methods in Fluids, 1995, 20, 157-168
[19] R.A.Walters and E.J.Barragy, Comparison of h and p finite element approximations of the shallow water equations, Int.J.for Numerical Methods in Fluids, 1997, 24, 61 -79
[20] Dells A I, Skeels C P and Ryrie S C, Evaluation of some approximate Riemann solver for transient open channel flows, Journal of hydraulic engineering, 2000, 38(3), 217-231
[21]Vadym Aizinger and Clint Dawson , A discontinuous Galerkin method for two-dimensional flow and transport in shallow water.Advances in Water Resources, 2002, 25, 67-84
[22] Christopher Zoppou and Stephen Roberts, Numerical Solution Of The Two-Dimensional Unsteady Dam Break, Mathematics Subject Classification, 1991, 1-22
[23]Alcrudo F and Garicia-Navarro P., A high resolution Godunov-type scheme in finite volumes for the 2D shallow water equations.Inter for Number Methods in Fluids, 1995, 20, 743-770
[24]Godunov S.K.A difference scheme for numerical computation of discontinuous solution of hydrodynamics equations, Math. Sbornik, 1959, 47, 271-306
[25] LeVeque R.J, Numerical Methods for Conservation Laws, Birkhauser Verlag, 1992
[26] A.Valiani, V. Caleffi and A. Zanni, Finite volume scheme for 2d shallow-water equations application to the malpasset dam-break, 63-93
[27]R. A. Walters and G. F. Carey, Analysis of spurious oscillation modes for the shallow-water and Navier-Stokes equations, Comput. Fluids, 1983, 11, 51-68
[28]Richard Liska and Burton Wendroff, 2D Shallow Water by Composite Schemes, the CHAMMP program of the US Department of Engergy, 1997, 1-20
[29]TAE HOON YOON and SEOK-KOO KANG, Finite volume model for two-dimensional shallow water flows on unstructured grids, J.Hy.ASCE, 2004, 130(7), 678-688
[30]Toro E., Riemann problems and the WAF method for solving the two-dimensional shallow water equations, Philosophical Trans. Royal Soc, London, U.K., A338, 1992, 43-68
[31]Hu.K, Mingham C.G and Causon D M, A bore-capturing finite volume method for open channel flows, Inter for Number Methods in Fluids, 1998, 28, 1421-1561
[32]TORO EF, Shock-capturing methods for free-surface shallow flows, Chichester: John Wileyh Sons, 2001, 5-165
[33] Von Neuman J, Richtmyer R D.A method for numerical calculations of hydrodynamical shocks, J Appl Phys, 21(1950), 232-256
[34] Van Leer, Towards the ultimate conservative difference V.A second order sequel to Godunov method, J Comp Phys, 1979, 32, 101-136
[35]Sweby P K,High resolution schemes using flux limiters for hyperbolic conservation laws,SIAM J Num Anal,1984,21,995-1011
[36]Yee H C,Construction of explicit and implicit symmetric TVD schemes and their applications,J CompPhys,1987,68,151-179
[37]Goodman J B and Leveque R J,On the accuracy of stable schemes for 2D scalar conservation laws.Math Comput,1998,45,15-21
[38]Harten A,Engquist B,Osher S and Chakravarthy R.Uniformly high order accurate essentially nonoscillatory schemes Ⅲ.J Comp Phys,1987,71,231-303
[39]HartenA,LaxPD and van Leer B,On upstream differencing and Godunov-type schemes for hyperbolic conservation laws,SIAM Rev,1983,25,35-61
[40]McDonald P W,The Computation of transonic flow though two dimensional gas turbine cascades,ASME,1971,71-89
[41]MacCormack R W,and Paullay A J,Computational efficiency achieved by time splitting of finite difference operators,AIAApaper,1972,72-154
[42]Jameson A and Mavriplis D.Finite volume solution of the two-dimensional Euler equation on a regular triangular mesh,AIAApaper,1985,85(0435)
[43]Mavriplis D,Multigrid solution of the Euler equation on unstructured and adaptive meshes,ICASEReport,1987,87(53),413-430
[44]Mavriplis D and Jameson A,Multigrid solution of two-dimensional Euler equation on unstructured triangular meshes,AIAA paper,1987,87(0353)
[45]Shu C W,Total-variation-diminishing time discretizations,SIAM Journal on Scientific Computing,1988,9,1073-1084
[46]Jameson A,Multigrid Algorithms for Compressible Flow Calculations,In Proceeding of the Second European Conference on Multigrid Methods,1986,1228,166-201
[47]Barth T and Frederickson P,High order solution of the Euler equations on unstructured grids using quadratic reconstruction,AIAA paper,1990,90(0013)
[48]Wang J S and Liu R X,The composite finite volume method on unstructured meshes for 2D shallow water equations,Inter J for Number Methods in Fluids,2001,37,933-949
[49]A.Valiani,V.Caleffi and A.Zanni,Malpasset dam-break Simulation using a Two-Dimen -sional Finite Volume Method,J.Hydr.Engrg.,2002,128(5),460-472
[50]Ji-Wen Wang and Ru-Xun Liu,A comparative study of finite volume methods on unstructured meshes for simulation of 2D shallow water wave problem,Mathematics and Computers in Simulation,2000,53,171-184
[51]潘存鸿,徐昆,三角形网格下求解二维浅水波方程的KFVS格式,水利学报,2006,37(7),858-864
[52]付德薰,流体力学数值模拟,北京:国防工业出版社,1993
[53]王如云,浅水波方程的TVD有限差分数值模拟,海洋与湖沼,1991,22(2),115-123
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