Research Of Computational Methods And Analyses For Two Types Of Nonlinear Evolution Equations

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.