两类非线性发展方程的分析与计算方法研究
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摘要
非线性发展方程在力学领域、生物医学工程领域、控制工程领域等理论研究和工程应用中都有着十分重要的意义。材料力学中的粘性振动和生物医学中的神经传播的模型可归结为一类非线性发展方程——非线性拟双曲方程,对该类方程解的存在性和进行数值求解方法的研究,是材料力学和生物医学研究领域的重要课题。但其高维情形的数值方法和理论分析难度较大。波浪是海洋中最常见的物理现象,其由深海向近岸传播过程中所产生的各种非线性效应对近岸人类生产生活和近岸地貌变化有着较大影响。而近岸浅水区波浪非线性较强且形式复杂,导致数值模拟难度增大。Boussinesq类方程是目前常用的一类浅水非线性发展方程,对该类方程进行理论分析和计算方法研究,为进一步保障近海生产作业、进行海岸工程设计和施工等有重要意义。本文对非线性拟双曲方程,以及在海岸工程中得到广泛应用的Boussinesq类浅水波浪模型这两类非线性发展方程进行了理论分析与计算方法的研究。
针对材料力学中的粘性振动及神经传播过程中涉及的一类非线性拟双曲方程,目前计算方法的研究一般局限于一维问题,其高维非线性情形的理论分析和数值求解难度较大。本文基于有限元配置法,对该类非线性发展方程的二维初边值问题,采用分片双三次Hermite插值多项式空间作为求解的逼近函数空间,建立了半离散和全离散格式,并对两种格式证明了数值解的存在唯一性;应用微分方程先验估计的理论和技巧得到L2最佳阶误差估计。数值实验结果表明,在保证整体误差估计和不增加计算量的前提下,本方法比传统有限元方法有更高的逼近精度,进一步扩展了配置法的应用范围。
基于交替方向有限元法和变网格法,本文提出了针对上述非线性发展方程三维形式的交替方向变网格有限元方法。基于对二维方程的分析,对其三维方程构造了交替方向变网格有限元格式;通过在原方程中加入扰动项,将高维问题化为一系列简单的一维问题逐次求解;应用常规恒等变换的技巧,使得方程能够进行交替方向计算,且可以并行进行。对该格式的误差分析表明,该格式形式简便、稳定,易于并行计算,适用于大规模科学工程计算。
现有Boussinesq类方程,在提高其非线性的同时,通常方程的最高阶导数的阶数也会随之增加,很多情况下都保留了高于三阶的导数项,这给数值计算带来很大困难。通过重新假定波高水深比和水深波长比的关系,本文建立了不增加方程最高阶导数前提下提高精度的新型二维Boussinesq方程。
对近岸浅水区波浪的分析和计算求解中,求解区域范围通常比较大,需要消耗较大的计算量和计算时间。对经典的Boussinesq方程,本文提出了基于区域分解的Boussinesq方程并行计算方法。基于区域分解方法和重叠度概念,提出了用于并行求解的重叠区域分解方法;结合子区域校正算法,采用单位分解函数合理分配重叠区域上的校正量,构造了一类针对Boussinesq方程的新型并行有限差分算法;通过对孤立波进行算例分析,讨论了子区域重叠度和迭代次数对算法的影响。实验结果表明,该并行计算方法可以有效地提高计算效率,对于单位分解函数集取线性函数时,计算中只需要较小的重叠度和较少的迭代次数即可满足实际需要。
Nonlinear evolution equations play an important role in mechanics, biomedical engineering, control engineering and other fields of theoretical researches and engineering applications. The models of viscous vibration in material mechanics and neural transmission in the biomedical can be reduced to a class of nonlinear evolution equations——Nonlinear pseudo-hyperbolic equation. Thus it is an important subject in the areas of material mechanics and biomedical to study the existence of the solutions and the numerical methods for these equations. However, it is not easy to obtain the numerical methods or the theoretical analysis for the multi-dimensional cases. Waves are the most common physical phenomena in oceans, and a variety of nonlinear effects during the travelling from the deep sea to the shore have a great impact on the production of human life and near-shore coastal landscape change. The strong nonlinearity as well as the complex forms of the waves in the offshore shallow water increases the difficulty in numerical simulation. Nowadays the Boussinesq type equations are commonly used nonlinear evolution equations for shallow water. The study of the class of equations theoretically and numerically are of great significance in the further protection of the offshore production operations, design and construction of coastal projects and so forth. In this paper, the proposed nonlinear hyperbolic equations and the widely used Boussinesq equations in coastal engineering have been theoretical analyzed, and some numerical methods have also been researched.
Calculations on Nonlinear evolution equations of viscous vibration in material mechanics and neural transmission in the biomedical limited to one-dimensional problem. The numerical methods or do the theoretical analysis for the multi-dimensional cases are difficult.Taking piecewise bicubic Hermite interpolation polynomial space as the solution space, the two-dimensional initial-boundary value problem of the nonlinear hyperbolic equations was solved both in semi-discrete and fully discrete schemes utilizing the finite element configuration method. Furthermore, the existence and uniqueness of the numerical solution for the two schemes have been proved. The theory and techniques of differential equations priori estimate were applied to get the best L2 error estimation. Under the premise that the overall error does not increase as well as the computation, the numerical results showed that the method has higher approximation resolution than the traditional finite element method and expand the scope of application for the configuration method.
Based on alternating direction method and variable grid method, this paper presented a three-dimensional alternating direction variable grid finite element method for the nonlinear evolution equations, and then using this method, a numerical scheme for the three dimension equations was proposed drawing on the analysis of the two dimension equations. Adding the original equation by perturbation, the high dimensional problem is translated into a series of simple one-dimensional problem. Using conventional identical transformation techniques, the equations can be calculated in alternating directions and be paralleled. The error analysis showed that the scheme has a simple form and is stable, moreover, the scheme is easy to implement parallel computing and is capable for large-scale scientific engineering computing.
Usually the order of the highest order derivative in equations will increase as the nonlinearity of the existing Boussinesq type equations increases. In many cases derivatives higher than third order are retained, which brings to the numerical simulation great difficulties. By re-assuming the relationship between the ratio of wave height and water depth and the ratio of water depth and wavelength, the paper developed a new type two-dimensional Boussinesq equation which improves accuracy without increasing the order of the highest order derivative.
Since the range of solving region is usually very large, the analysis and calculation of the waves in the offshore shallow water need consume a large amount of computation and time. Based on region division, this paper developed a parallel computing method for the classical Boussinesq equation. A overlapping region division method was proposed for parallel using region division method and the concept of overlap. Combining subspace correction algorithm and adopting the unit decomposition function to rationally allocate the correction on the overlap regions, a new type of parallel finite difference algorithm for Boussinesq equation was constructed. The sub-region overlap and the number of iterations of the algorithm were discussed by the numerical analysis of solitary waves. The numerical results showed that the parallel computing method can improve computational efficiency. And when the unity partition function set takes the linear function, the calculation requires only small overlap and fewer iterations for the actual needs.
针对材料力学中的粘性振动及神经传播过程中涉及的一类非线性拟双曲方程,目前计算方法的研究一般局限于一维问题,其高维非线性情形的理论分析和数值求解难度较大。本文基于有限元配置法,对该类非线性发展方程的二维初边值问题,采用分片双三次Hermite插值多项式空间作为求解的逼近函数空间,建立了半离散和全离散格式,并对两种格式证明了数值解的存在唯一性;应用微分方程先验估计的理论和技巧得到L2最佳阶误差估计。数值实验结果表明,在保证整体误差估计和不增加计算量的前提下,本方法比传统有限元方法有更高的逼近精度,进一步扩展了配置法的应用范围。
基于交替方向有限元法和变网格法,本文提出了针对上述非线性发展方程三维形式的交替方向变网格有限元方法。基于对二维方程的分析,对其三维方程构造了交替方向变网格有限元格式;通过在原方程中加入扰动项,将高维问题化为一系列简单的一维问题逐次求解;应用常规恒等变换的技巧,使得方程能够进行交替方向计算,且可以并行进行。对该格式的误差分析表明,该格式形式简便、稳定,易于并行计算,适用于大规模科学工程计算。
现有Boussinesq类方程,在提高其非线性的同时,通常方程的最高阶导数的阶数也会随之增加,很多情况下都保留了高于三阶的导数项,这给数值计算带来很大困难。通过重新假定波高水深比和水深波长比的关系,本文建立了不增加方程最高阶导数前提下提高精度的新型二维Boussinesq方程。
对近岸浅水区波浪的分析和计算求解中,求解区域范围通常比较大,需要消耗较大的计算量和计算时间。对经典的Boussinesq方程,本文提出了基于区域分解的Boussinesq方程并行计算方法。基于区域分解方法和重叠度概念,提出了用于并行求解的重叠区域分解方法;结合子区域校正算法,采用单位分解函数合理分配重叠区域上的校正量,构造了一类针对Boussinesq方程的新型并行有限差分算法;通过对孤立波进行算例分析,讨论了子区域重叠度和迭代次数对算法的影响。实验结果表明,该并行计算方法可以有效地提高计算效率,对于单位分解函数集取线性函数时,计算中只需要较小的重叠度和较少的迭代次数即可满足实际需要。
Nonlinear evolution equations play an important role in mechanics, biomedical engineering, control engineering and other fields of theoretical researches and engineering applications. The models of viscous vibration in material mechanics and neural transmission in the biomedical can be reduced to a class of nonlinear evolution equations——Nonlinear pseudo-hyperbolic equation. Thus it is an important subject in the areas of material mechanics and biomedical to study the existence of the solutions and the numerical methods for these equations. However, it is not easy to obtain the numerical methods or the theoretical analysis for the multi-dimensional cases. Waves are the most common physical phenomena in oceans, and a variety of nonlinear effects during the travelling from the deep sea to the shore have a great impact on the production of human life and near-shore coastal landscape change. The strong nonlinearity as well as the complex forms of the waves in the offshore shallow water increases the difficulty in numerical simulation. Nowadays the Boussinesq type equations are commonly used nonlinear evolution equations for shallow water. The study of the class of equations theoretically and numerically are of great significance in the further protection of the offshore production operations, design and construction of coastal projects and so forth. In this paper, the proposed nonlinear hyperbolic equations and the widely used Boussinesq equations in coastal engineering have been theoretical analyzed, and some numerical methods have also been researched.
Calculations on Nonlinear evolution equations of viscous vibration in material mechanics and neural transmission in the biomedical limited to one-dimensional problem. The numerical methods or do the theoretical analysis for the multi-dimensional cases are difficult.Taking piecewise bicubic Hermite interpolation polynomial space as the solution space, the two-dimensional initial-boundary value problem of the nonlinear hyperbolic equations was solved both in semi-discrete and fully discrete schemes utilizing the finite element configuration method. Furthermore, the existence and uniqueness of the numerical solution for the two schemes have been proved. The theory and techniques of differential equations priori estimate were applied to get the best L2 error estimation. Under the premise that the overall error does not increase as well as the computation, the numerical results showed that the method has higher approximation resolution than the traditional finite element method and expand the scope of application for the configuration method.
Based on alternating direction method and variable grid method, this paper presented a three-dimensional alternating direction variable grid finite element method for the nonlinear evolution equations, and then using this method, a numerical scheme for the three dimension equations was proposed drawing on the analysis of the two dimension equations. Adding the original equation by perturbation, the high dimensional problem is translated into a series of simple one-dimensional problem. Using conventional identical transformation techniques, the equations can be calculated in alternating directions and be paralleled. The error analysis showed that the scheme has a simple form and is stable, moreover, the scheme is easy to implement parallel computing and is capable for large-scale scientific engineering computing.
Usually the order of the highest order derivative in equations will increase as the nonlinearity of the existing Boussinesq type equations increases. In many cases derivatives higher than third order are retained, which brings to the numerical simulation great difficulties. By re-assuming the relationship between the ratio of wave height and water depth and the ratio of water depth and wavelength, the paper developed a new type two-dimensional Boussinesq equation which improves accuracy without increasing the order of the highest order derivative.
Since the range of solving region is usually very large, the analysis and calculation of the waves in the offshore shallow water need consume a large amount of computation and time. Based on region division, this paper developed a parallel computing method for the classical Boussinesq equation. A overlapping region division method was proposed for parallel using region division method and the concept of overlap. Combining subspace correction algorithm and adopting the unit decomposition function to rationally allocate the correction on the overlap regions, a new type of parallel finite difference algorithm for Boussinesq equation was constructed. The sub-region overlap and the number of iterations of the algorithm were discussed by the numerical analysis of solitary waves. The numerical results showed that the parallel computing method can improve computational efficiency. And when the unity partition function set takes the linear function, the calculation requires only small overlap and fewer iterations for the actual needs.
引文
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