紧致差分格式的优化和初步应用
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摘要
高精度紧致差分格式作为数值计算的重要研究问题之一,在很多科学计算领域中占有重要地位.而且,随着工程问题的日趋复杂化,对数值格式的要求越来越高,低阶精度差分格式已经不能完全满足求解问题的要求,因此,有必要构造高分辨率、高精度的差分格式以满足科学和工程计算的需要.
本文的研究内容可以分为以下三个方面：
第一：依据差分格式的伪波数应该在尽可能大的波数范围内接近物理波数的思想,构造了满足四阶精度且具有高分辨率的三对角四阶紧致差分格式.一方面,它可以与近些年发展的求解(循环)三对角方程组的高效算法相结合,以更高的分辨率、更小的计算量来计算偏微分方程的一阶导数；另一方面,与传统格式相比,该格式的最大精确求解波数可以达到2.5761,大于传统格式的1.13097.因此,优化格式更适合模拟小尺度波动.
第二：通过比较我们建立的四阶最优紧致差分格式,以及传统的六阶和八阶紧致差分格式,来研究精度和分辨率之间的关系.一般而言,高精度的差分格式具有较高的分辨率,但是,数值格式精度和分辨率是两个不同的概念,通过差分格式对实际算例模拟的比较,我们发现：针对小尺度波动问题,精度低的四阶格式反而比六阶和八阶的分辨率高.因此,在解决实际问题时,需要选择具有合适的精度和分辨率的数值格式.
第三：与传统的四阶紧致差分格式进行对比,说明在实际应用时,我们建立的四阶紧致差分格式具有高分辨率,更适合求解小尺度波动问题.数值计算结果也表明：虽然优化格式仍然是四阶精度,但是要比传统四阶紧致差分格式的计算误差小；尤其对于小尺度波动,优化格式的计算误差会更小.对于行波问题,优化格式能够更加准确的模拟波动的传播行为,其优势也更加明显.理论分析和数值算例的比较结果均表明,优化的紧致差分格式更适合求解小尺度波动问题.
The research and application of compact fnite diference scheme with high order ofaccuracy have received much attentions in recent years. This kind of scheme plays impor-tant roles in scientifc computations. As the increasing complexity scientifc computations,the requirement on the numerical scheme is increasing also. Thus, the numerical schemeswith lower order do not fulfll the requirement. As a result, it is necessary to develop highorder and high accuracy numerical scheme.
The content of this thesis can be ascribed to the following three points:
First, based on the idea of diference schemes pseudo-wavenumbers should resolvethe physical wavenumbers as accurate high as possible we proposed an optimal tridiagonalcompact fnite diference scheme with fourth order accuracy and high resolution. On the onehand, the optimal compact scheme can be solved efciently by the algorithms which we havedeveloped recently to solve the (cyclic) tridiagonal equations. Thus, the frst derivation ofpartial diferential equations can be solved efciently by the optimized compact diferencescheme efcienty; On the other hand, the maximum of pseudo-wavenumber can be resolvedby the optimal scheme is2.5761, and larger than the pseudo-wavenumber of1.13097,which can be resolved by the traditional fourth order compact schemes. Therefore, theoptimized compact diference scheme is more appropriate to resolve small scale waves influid dynamics and aero-acoustic dynamics.
Second, comparisons among the optimized fourth order compact fnite diference scheme, sixth and eighth order compact diference are performed. The comparative results showedthat the optimized fourth order scheme with low accuracy has higher resolution than thatof the sixth and eight order schemes. Therefore, we have to bear in mind that the accuracyand the resolution are two diferent concepts and should not be replaced with each other.In contrast, one should choose a scheme with appropriate order of accuracy and resolutionaccording to the problem needed to be solved.
Third, compared with traditional fourth-order compact fnite diference scheme, theoptimized fourth order compact fnite diference scheme has high resolution and is moresuitable to resolve small scale fluctuation problems in practical applications. Numerical ex-periments illustrated that: although the accuracy of the optimized scheme is fourth-order,it has a smaller error than that of the traditional fourth-order compact fnite diference,especially for the small scale fluctuations. Our numerical experiments also showed theadvantages of the optimized fourth-order compact fnite diference scheme in simulatingthe wave propagation problem. Thus, according to the theoretical analysis and numericalexperiment results, we conclude that optimized compact fnite diference scheme is moreappropriate to resolve small scale fluctuations problems.
本文的研究内容可以分为以下三个方面：
第一：依据差分格式的伪波数应该在尽可能大的波数范围内接近物理波数的思想,构造了满足四阶精度且具有高分辨率的三对角四阶紧致差分格式.一方面,它可以与近些年发展的求解(循环)三对角方程组的高效算法相结合,以更高的分辨率、更小的计算量来计算偏微分方程的一阶导数；另一方面,与传统格式相比,该格式的最大精确求解波数可以达到2.5761,大于传统格式的1.13097.因此,优化格式更适合模拟小尺度波动.
第二：通过比较我们建立的四阶最优紧致差分格式,以及传统的六阶和八阶紧致差分格式,来研究精度和分辨率之间的关系.一般而言,高精度的差分格式具有较高的分辨率,但是,数值格式精度和分辨率是两个不同的概念,通过差分格式对实际算例模拟的比较,我们发现：针对小尺度波动问题,精度低的四阶格式反而比六阶和八阶的分辨率高.因此,在解决实际问题时,需要选择具有合适的精度和分辨率的数值格式.
第三：与传统的四阶紧致差分格式进行对比,说明在实际应用时,我们建立的四阶紧致差分格式具有高分辨率,更适合求解小尺度波动问题.数值计算结果也表明：虽然优化格式仍然是四阶精度,但是要比传统四阶紧致差分格式的计算误差小；尤其对于小尺度波动,优化格式的计算误差会更小.对于行波问题,优化格式能够更加准确的模拟波动的传播行为,其优势也更加明显.理论分析和数值算例的比较结果均表明,优化的紧致差分格式更适合求解小尺度波动问题.
The research and application of compact fnite diference scheme with high order ofaccuracy have received much attentions in recent years. This kind of scheme plays impor-tant roles in scientifc computations. As the increasing complexity scientifc computations,the requirement on the numerical scheme is increasing also. Thus, the numerical schemeswith lower order do not fulfll the requirement. As a result, it is necessary to develop highorder and high accuracy numerical scheme.
The content of this thesis can be ascribed to the following three points:
First, based on the idea of diference schemes pseudo-wavenumbers should resolvethe physical wavenumbers as accurate high as possible we proposed an optimal tridiagonalcompact fnite diference scheme with fourth order accuracy and high resolution. On the onehand, the optimal compact scheme can be solved efciently by the algorithms which we havedeveloped recently to solve the (cyclic) tridiagonal equations. Thus, the frst derivation ofpartial diferential equations can be solved efciently by the optimized compact diferencescheme efcienty; On the other hand, the maximum of pseudo-wavenumber can be resolvedby the optimal scheme is2.5761, and larger than the pseudo-wavenumber of1.13097,which can be resolved by the traditional fourth order compact schemes. Therefore, theoptimized compact diference scheme is more appropriate to resolve small scale waves influid dynamics and aero-acoustic dynamics.
Second, comparisons among the optimized fourth order compact fnite diference scheme, sixth and eighth order compact diference are performed. The comparative results showedthat the optimized fourth order scheme with low accuracy has higher resolution than thatof the sixth and eight order schemes. Therefore, we have to bear in mind that the accuracyand the resolution are two diferent concepts and should not be replaced with each other.In contrast, one should choose a scheme with appropriate order of accuracy and resolutionaccording to the problem needed to be solved.
Third, compared with traditional fourth-order compact fnite diference scheme, theoptimized fourth order compact fnite diference scheme has high resolution and is moresuitable to resolve small scale fluctuation problems in practical applications. Numerical ex-periments illustrated that: although the accuracy of the optimized scheme is fourth-order,it has a smaller error than that of the traditional fourth-order compact fnite diference,especially for the small scale fluctuations. Our numerical experiments also showed theadvantages of the optimized fourth-order compact fnite diference scheme in simulatingthe wave propagation problem. Thus, according to the theoretical analysis and numericalexperiment results, we conclude that optimized compact fnite diference scheme is moreappropriate to resolve small scale fluctuations problems.
引文
[1]傅德熏,马延文.计算流体力学[M].北京：高等教育出版社,2002
[2]Smith G D. Numerical solutions of partial differential equations[M]. Oxford:Clarendon Press,1985
[3]Richtmyer R D, Morton K W. Difference methods for initial problems[M]. New York:Inter-science Publishers,1967
[4]王书强,杨顶辉,杨宽德.弹性波方程的紧致差分方法[J].清华大学学报：自然科学版,2002,42(8)：1128-1131
[5]Lele S K. Compact Finite Difference Schemes with Spectral-like Resolution[J]. Journal of Computational Physics,1992,103:16-42
[6]Visbal M R, Gaitonde D V. Computation of aeroacoustic on general geometries using com-pact differencing and filtering schemes [R]. AIAA-99-33696,1999
[7]Visbal M R, Gaitonde D V. On the use of higher order finite difference schemes on curvilinear and deforming meshes [J]. Journal of Computational Physics,2002,181(1):155-185
[8]Tam C K W, Webb J C. Dispersion-relation-preserving finite difference schemes for compu-tational acoustics[J]. Journal of computational Physics,1992,107:262-281
[9]罗柏华.用于波动方程计算的高阶精度紧致差分格式[J].上海大学学报：自然科学版,2009,15(2)：134-141
[10]Kim J W, Lee D J. Optimized compact finite difference schemes with maximum resolution [J]. AIAA Journal,1997,34(5):887-893
[11]Kim J W. Optimized boundary compact finite difference schemes for computational aeroa-coustics [J]. Journal of computational Physics,2007,225(1):995-1019
[12]柳占新,黄其柏,胡溧,等.计算气动声学中的高精度紧致差分格式研究[J].航空动力学报,2009,24(1)：83-90
[13]柳占新,黄其柏,袁骥轩,等.计算气动声学中的紧致滤波格式[J].航空学报,2009,30(3)：403-410
[14]KRISHNAM Mahesh. A family of high order finite difference schemes with good spectral resolution [J]. J Comp ut Phys,1998,145:332-358
[15]王彩华.一维对流扩散方程的一类新型高精度紧致差分格式[J].水动力学研究与进展,2004,19(5)：655-663
[16]陈国谦,杨志峰.对流扩散方程的指数型摄动差分[J].计算物理,1994,11(4)：413-424
[17]陈国谦,高智.对流扩散方程的四阶紧致迎风差分格式[J].计算数学,1992,第3期,345-357
[18]Dennis S C R, Hundson J D. Compact h4finite difference approximations to operators of Navier-Stokes type. J Comput Phys,1989,85(2):390-416
[19]朱庆勇,马延文.数值求解Navier-Stokes方程的迎风紧致差分格式[J].计算力学学报,2000,17(4)：379-384
[20]Ma Y W, Fu D X. Super compact finite difference methods with uniform and nonuniform grid system [A]. Proceeding of6th International Symposium on Computational Fluid Dy-namics[C]. Lake Tahoe, Nevada,1995
[21]Dexun Fu, Yanwen Ma. A high order accurate finite difference scheme for complex flow fields[J]. J of Comput Phys,1997,134:1-15
[22]朱庆勇,马延文.求解双曲型守恒律方程的高精度迎风紧致群速度控制法[J].计算物理,1998,15(5)：531-536
[23]Harten A. High resolution schemes for conservation laws[J]. J Comp Phys,1983,49:357-393
[24]Rainer Kress. Numerical Analysis[M]. Springer-Verlag,1941
[25]石东洋.数值计算方法[M].郑州：郑州大学出版社,2007：103-111
[26]李文强,马民.求解循环三对角方程组的追赶法[J].科技导报2009,27(14)：69-72
[27]Hu F Q, Hussanini M Y, Manthey Y J L. Low-dissipation and low-dispersion runge-kutta schemes for computational acoustics[J]. Journal of Computational Physics,1996,124:177-191
[28]Taegee Min, Jung Yul Yoo, Haecheon Choi. Effect of spatial discretization schemes on nu-merical solutions of viscoelastic fluid flows. J.Non-Newtonian Fluid Mech,100:27-47,2001
[29]Li Xinliang, Ma Yanwen, Fu Dexun. DNS of incompressible turbulent channel flow with upwind compact scheme on non-uniform meshes. CFD J,2000,8(4):536-543
[30]彭国伦Fortran95程序设计[M].北京：中国电力出版社,2002.6
[31]李润明,吴晓明.图解Origin8.0科技绘图及数据分析[M].北京：人民邮电出版社,2009,10
[32]韩培友.IDL可视化分析与应用[M].西安：西安工业大学出版社,2006,9
[33]Rai M M, Moin P. Direct simulations of turbulent flow using finite difference schemes [J]. Sci Comput,1991,96(1):15-53
[34]Rai M M, Catski T B, Erlebacher G. Direct simulation of spatially evolving compressible turbulent boundary layers [J]. AIAA,1995,95:583
[35]李珍,张海娥,王涛.基于非等距网格的紧致差分方法与经典差分方法的比较[J].重庆工学院学报(自然科学),2008,22(5)：163-167
[36]张毅锋.高阶精度格式(WCNS)加速收敛和复杂流动数值模拟的应用研究[D].中国空气动力研究与发展中心,2008
[37]李松.高阶精度耗散紧致差分格式的研究与应用[D].中国空气动力研究与发展中心,2007
[38]S.Nagarajan, S.K.Lele, J.H.Ferziger. A robust high-order compactmethod for large eddy simulation [J]. Journal of Computational physics,2003,191:392-419
[39]林东,詹杰民.浅水方程组合型超紧致差分格式[J].计算力学学报,2008,25(6)：791-796
[40]陶文栓.数值传热学[M].2版.西安：西安交通大学出版社,2001：503-505
[41]Kim J, Moin P, Moser, R D. Turbulence statistics in fully-developed channel flow at low Reynolds number. J Fluid Mech,1987,177:133-166
[42]Tam C K W. Fourth computational aeroacoustics workshop on benchmark problems in computational aeroacoustics [R]. Cleveland, USA:NASA Glenn Research Centre,2004,31-32
[43]李文强,刘晓.多项式拟合数值边界格式及其稳定性分析[J].河南师范大学学报：自然科学版,2010,38(1)：16-20
[44]Haedin J C, Risrorcelli J R, Tam C K W. ICASE/LaRC workshop on benchmark problems in computational aeroacoustics[R]. Virginia, USA:Langley Research Centre,1994
[45]傅德熏,马延文.高精度差分格式及多尺度流场特性的数值模拟[J].空气动力学学报,1998,16(1)：24-35
[46]Tam C K W. Recent advances in computational aeroacoustics [J]. Fluid Dynamics Research,2006,38(9):591-615
[2]Smith G D. Numerical solutions of partial differential equations[M]. Oxford:Clarendon Press,1985
[3]Richtmyer R D, Morton K W. Difference methods for initial problems[M]. New York:Inter-science Publishers,1967
[4]王书强,杨顶辉,杨宽德.弹性波方程的紧致差分方法[J].清华大学学报：自然科学版,2002,42(8)：1128-1131
[5]Lele S K. Compact Finite Difference Schemes with Spectral-like Resolution[J]. Journal of Computational Physics,1992,103:16-42
[6]Visbal M R, Gaitonde D V. Computation of aeroacoustic on general geometries using com-pact differencing and filtering schemes [R]. AIAA-99-33696,1999
[7]Visbal M R, Gaitonde D V. On the use of higher order finite difference schemes on curvilinear and deforming meshes [J]. Journal of Computational Physics,2002,181(1):155-185
[8]Tam C K W, Webb J C. Dispersion-relation-preserving finite difference schemes for compu-tational acoustics[J]. Journal of computational Physics,1992,107:262-281
[9]罗柏华.用于波动方程计算的高阶精度紧致差分格式[J].上海大学学报：自然科学版,2009,15(2)：134-141
[10]Kim J W, Lee D J. Optimized compact finite difference schemes with maximum resolution [J]. AIAA Journal,1997,34(5):887-893
[11]Kim J W. Optimized boundary compact finite difference schemes for computational aeroa-coustics [J]. Journal of computational Physics,2007,225(1):995-1019
[12]柳占新,黄其柏,胡溧,等.计算气动声学中的高精度紧致差分格式研究[J].航空动力学报,2009,24(1)：83-90
[13]柳占新,黄其柏,袁骥轩,等.计算气动声学中的紧致滤波格式[J].航空学报,2009,30(3)：403-410
[14]KRISHNAM Mahesh. A family of high order finite difference schemes with good spectral resolution [J]. J Comp ut Phys,1998,145:332-358
[15]王彩华.一维对流扩散方程的一类新型高精度紧致差分格式[J].水动力学研究与进展,2004,19(5)：655-663
[16]陈国谦,杨志峰.对流扩散方程的指数型摄动差分[J].计算物理,1994,11(4)：413-424
[17]陈国谦,高智.对流扩散方程的四阶紧致迎风差分格式[J].计算数学,1992,第3期,345-357
[18]Dennis S C R, Hundson J D. Compact h4finite difference approximations to operators of Navier-Stokes type. J Comput Phys,1989,85(2):390-416
[19]朱庆勇,马延文.数值求解Navier-Stokes方程的迎风紧致差分格式[J].计算力学学报,2000,17(4)：379-384
[20]Ma Y W, Fu D X. Super compact finite difference methods with uniform and nonuniform grid system [A]. Proceeding of6th International Symposium on Computational Fluid Dy-namics[C]. Lake Tahoe, Nevada,1995
[21]Dexun Fu, Yanwen Ma. A high order accurate finite difference scheme for complex flow fields[J]. J of Comput Phys,1997,134:1-15
[22]朱庆勇,马延文.求解双曲型守恒律方程的高精度迎风紧致群速度控制法[J].计算物理,1998,15(5)：531-536
[23]Harten A. High resolution schemes for conservation laws[J]. J Comp Phys,1983,49:357-393
[24]Rainer Kress. Numerical Analysis[M]. Springer-Verlag,1941
[25]石东洋.数值计算方法[M].郑州：郑州大学出版社,2007：103-111
[26]李文强,马民.求解循环三对角方程组的追赶法[J].科技导报2009,27(14)：69-72
[27]Hu F Q, Hussanini M Y, Manthey Y J L. Low-dissipation and low-dispersion runge-kutta schemes for computational acoustics[J]. Journal of Computational Physics,1996,124:177-191
[28]Taegee Min, Jung Yul Yoo, Haecheon Choi. Effect of spatial discretization schemes on nu-merical solutions of viscoelastic fluid flows. J.Non-Newtonian Fluid Mech,100:27-47,2001
[29]Li Xinliang, Ma Yanwen, Fu Dexun. DNS of incompressible turbulent channel flow with upwind compact scheme on non-uniform meshes. CFD J,2000,8(4):536-543
[30]彭国伦Fortran95程序设计[M].北京：中国电力出版社,2002.6
[31]李润明,吴晓明.图解Origin8.0科技绘图及数据分析[M].北京：人民邮电出版社,2009,10
[32]韩培友.IDL可视化分析与应用[M].西安：西安工业大学出版社,2006,9
[33]Rai M M, Moin P. Direct simulations of turbulent flow using finite difference schemes [J]. Sci Comput,1991,96(1):15-53
[34]Rai M M, Catski T B, Erlebacher G. Direct simulation of spatially evolving compressible turbulent boundary layers [J]. AIAA,1995,95:583
[35]李珍,张海娥,王涛.基于非等距网格的紧致差分方法与经典差分方法的比较[J].重庆工学院学报(自然科学),2008,22(5)：163-167
[36]张毅锋.高阶精度格式(WCNS)加速收敛和复杂流动数值模拟的应用研究[D].中国空气动力研究与发展中心,2008
[37]李松.高阶精度耗散紧致差分格式的研究与应用[D].中国空气动力研究与发展中心,2007
[38]S.Nagarajan, S.K.Lele, J.H.Ferziger. A robust high-order compactmethod for large eddy simulation [J]. Journal of Computational physics,2003,191:392-419
[39]林东,詹杰民.浅水方程组合型超紧致差分格式[J].计算力学学报,2008,25(6)：791-796
[40]陶文栓.数值传热学[M].2版.西安：西安交通大学出版社,2001：503-505
[41]Kim J, Moin P, Moser, R D. Turbulence statistics in fully-developed channel flow at low Reynolds number. J Fluid Mech,1987,177:133-166
[42]Tam C K W. Fourth computational aeroacoustics workshop on benchmark problems in computational aeroacoustics [R]. Cleveland, USA:NASA Glenn Research Centre,2004,31-32
[43]李文强,刘晓.多项式拟合数值边界格式及其稳定性分析[J].河南师范大学学报：自然科学版,2010,38(1)：16-20
[44]Haedin J C, Risrorcelli J R, Tam C K W. ICASE/LaRC workshop on benchmark problems in computational aeroacoustics[R]. Virginia, USA:Langley Research Centre,1994
[45]傅德熏,马延文.高精度差分格式及多尺度流场特性的数值模拟[J].空气动力学学报,1998,16(1)：24-35
[46]Tam C K W. Recent advances in computational aeroacoustics [J]. Fluid Dynamics Research,2006,38(9):591-615