【作者】 刘建康；

【导师】 郑洲顺；

【作者基本信息】 中南大学 ， 计算数学， 2012， 博士

【摘要】 界面问题是自然界中一种常见的现象,对界面问题的数值方法研究在工业、生物、军事等方面有着重要的理论意义和实际应用价值,近些年一直受到学者们的广泛关注,也成为计算数学研究的前沿问题之一首先,本文对界面问题数值方法的研究现状进行了综述.基于IIM方法,通过对传统的LOD差分格式在非正则点处的差分方程右端加上由跳跃条件确定的修正项,使得在非正则点处的局部截断误差为O(h),而正则点处的局部截断误差仍为O(h2),构造了二维和三维热传导界面问题的LOD-IIM差分格式,所得差分格式在时间和空间方向以无穷范数均二阶收敛,并用三个数值实验验证了格式的收敛阶和稳定性.其次,利用IIM方法的思想,通过对高阶紧致ADI差分格式在非正则点处的差分方程右端加上由跳跃条件确定的修正项,使得在非正则点处的局部截断误差为O(h3),而正则点处的局部截断误差仍为O(h4),构造了二维热传导界面问题的高阶紧致ADI差分格式,所得的差分格式以无穷范数在时间和空间方向分别为二阶和四阶收敛且无条件稳定.基于固定界面的热传导问题的解关于时间的连续性,采用Richardson外推方法将时间方向的精度提高到四阶,从而得到二维热传导界面问题在时间和空间方向均四阶收敛的差分格式.基于所构造的差分格式,对两个分片光滑的数值算例进行求解,验证了格式的收敛阶和无条件稳定性.然后,采用Adams-Bashforth方法处理非线性对流项,通过对传统的ADI差分格式在非正则点处的差分方程右端加上由跳跃条件确定的修正项,使得在非正则点的局部截断误差为O(h),而正则点处的局部截断误差仍为O(h2),构造了二维非线性对流扩散界面问题的ADI-IIM格式,所得格式在时间和空间方向以无穷范数均二阶收敛且无条件稳定.利用修正项关于时间的连续性,将当前时间层的修正项用前两个时间层上的修正项通过Adams-Bashforth方法逼近得到.对三个分片光滑的数值算例进行求解,验证了格式的收敛阶和稳定性以及通过Adams-Bashfoth方法逼近得到的修正项的可靠性.最后,首次将Bernstein多项式用于椭圆界面问题数值方法的研究.在每个子区域上将方程的解表示为Bernstein多项式的线性组合,采用Galerkin方法或者配置方法计算Bernstein多项式的展开系数,得到一维椭圆界面问题和二维直线界面椭圆界面问题的高精度数值方法,数值实验表明该方法具有谱精度.对于二维曲线界面的椭圆界面问题,先通过一个非线性变换将其转化为直线界面问题并求解,再通过逆变换得到曲线界面问题的解.数值实验表明,该方法的逼近精度依赖于界面曲线的复杂程度.更多还原

【Abstract】 Interface problems are ubiquitous in the nature. The research on numeri-cal methods for interface problems has great theoretical significance and actual application value in aspects of industry, biology and military. In recent years, authors have paid close attention to it and it has become one of the frontier prob-lems of computational mathematics.Firstly, the thesis reviews the numerical methods for interfaces problems. Based on IIM, the LOD-IIM difference schemes for2D and3D heat interface problems are constructed by adding corrections at the right-hand side of differ-ence equations of traditional LOD schemes at irregular points so that the local truncation errors at irregular points are of O(h) and of O(h2) at regular points. Thus the LOD-IIM converges at the speed of two-order in both time and space in maximum norm. Three examples verify the convergence and unconditional stability.Secondly, under the strategy of IIM, the HOC-ADI-IIM difference scheme is constructed for2D heat interface problems by adding corrections at the right-hand side of difference equations of traditional HOC-ADI scheme at irregular points so that the local truncation errors at irregular points are of O(h3) and of O(h4) at regular points. Thus the HOC-ADI-IIM scheme is unconditionally sta-ble and convergent two-order in time and four-order in space in maximum norm, respectively. Since the interface is fixed, the primary variables are continuous with time. The convergence order in time can be improved to be of four-order by Richardson extrapolation method. Two examples show that the scheme is unconditionally stable and convergent.Then, by dealing the nonlinear convection term with Adams-Bashforth me-thod, the ADI-IIM scheme is constructed for2D nonlinear convection diffusion interface problems by adding corrections at the right-hand side of difference equations of traditional ADI scheme at irregular points so that the local trunca- tion errors at irregular points are of O(h) and of O(h2) at regular points. The proposed ADI-IIM scheme is two-order convergent in maximum norm in both time and space. Since the corrections are continuous with time, the corrections at present time level can be approximated by Adams-Bashfoth method using that from the two previous time levels. Three examples show that the proposed scheme is convergent and unconditionally stable and the approximated correc-tions are reliable.Finally, the Bernstein polynomials are introduced to investigate numerical methods for elliptic interface problems for the first time. The solution on each subdomain is approximated by the linear combination of Bernstein polynomials. The expansion coefficients of Bernstein polynomials are computed by Galerkin methods or collocation methods. Thus the high accuracy numerical methods are obtained for1D elliptic interface problems and2D elliptic problems with rec-tilinear interfaces. Numerical examples shows that the proposed methods have spectral accuracy. For2D elliptic interface problems with curvilinear interfaces, one can firstly transform it into rectilinear interface problems by a nonlinear transformation and solve it. Then the numerical solutions for curvilinear inter-face problems can be obtained by the inverse transformation. Numerical exper-iments show that the accuracy of proposed method relies on the complexity of interface curve.更多还原