This paper presents a kind of new Crank-Nicolson difference scheme for one and two dimensional convection-diffusion equations. It also gives the alternating direction method for two-dimensional problems. Because the c...This paper presents a kind of new Crank-Nicolson difference scheme for one and two dimensional convection-diffusion equations. It also gives the alternating direction method for two-dimensional problems. Because the coefficient matrix formed by the scheme is always diagonally dominant, the scheme can be solved by general iteration method. In this paper, we prove that the new CN scheme for one dimensional problems is convergent with respect to discrete L^2 norm with orderO(△t^2+△th+h^2). We also prove that the new CN scheme for two dimensional problems is stable by discrete Fourier method. Finally, numerical examples show that the method in this paper is very effective for solving convection-diffusion equations.展开更多
黏声波方程常被用于描述地下介质的黏弹性及波的传播现象,频域有限差分(finite difference frequency domain,FDFD)方法是黏声波和黏弹性波波场模拟的常用工具.目前FDFD黏声波模拟常用的二阶五点方法和优化九点方法在一个波长内的网格...黏声波方程常被用于描述地下介质的黏弹性及波的传播现象,频域有限差分(finite difference frequency domain,FDFD)方法是黏声波和黏弹性波波场模拟的常用工具.目前FDFD黏声波模拟常用的二阶五点方法和优化九点方法在一个波长内的网格点数小于4时误差较大.通过令FDFD系数随一个波长内的网格点数自适应从而提高FDFD方法的精度,本文针对黏声波波场模拟发展了一种适用于不同空间采样间隔之比的通用格式自适应系数FDFD方法.同时,为了验证自适应系数FDFD方法对一般黏声波模型的有效性,本文针对三个典型的黏声波模型,分别采用解析解和基于高阶FDFD的参考解验证了所提出方法的有效性.本方法的FDFD格式通过在传统的二阶FDFD格式的基础上引入相关校正项得到,其中校正项按网格点与中心点的距离进行分类选取,同时校正项对应的自适应FDFD系数不仅和空间采样间隔之比相关,还和一个波长内的采样点数相关.所需的自适应FDFD系数可通过声波方程的数值频散关系和查找表高效给出.数值频散分析表明,在空间采样间隔相等或不等的情况下,以相速度误差不超过1%为标准,通用格式自适应系数FDFD方法所需的一个波长内的采样点数均小于2.5.数值模拟实验表明,对于不同的空间采样间隔之比,相对于常用的二阶五点FDFD方法和优化九点FDFD方法,通用格式自适应系数FDFD方法均可在相似的计算量和内存需求下,有效提高黏声波模拟的精度.展开更多
文摘This paper presents a kind of new Crank-Nicolson difference scheme for one and two dimensional convection-diffusion equations. It also gives the alternating direction method for two-dimensional problems. Because the coefficient matrix formed by the scheme is always diagonally dominant, the scheme can be solved by general iteration method. In this paper, we prove that the new CN scheme for one dimensional problems is convergent with respect to discrete L^2 norm with orderO(△t^2+△th+h^2). We also prove that the new CN scheme for two dimensional problems is stable by discrete Fourier method. Finally, numerical examples show that the method in this paper is very effective for solving convection-diffusion equations.
文摘黏声波方程常被用于描述地下介质的黏弹性及波的传播现象,频域有限差分(finite difference frequency domain,FDFD)方法是黏声波和黏弹性波波场模拟的常用工具.目前FDFD黏声波模拟常用的二阶五点方法和优化九点方法在一个波长内的网格点数小于4时误差较大.通过令FDFD系数随一个波长内的网格点数自适应从而提高FDFD方法的精度,本文针对黏声波波场模拟发展了一种适用于不同空间采样间隔之比的通用格式自适应系数FDFD方法.同时,为了验证自适应系数FDFD方法对一般黏声波模型的有效性,本文针对三个典型的黏声波模型,分别采用解析解和基于高阶FDFD的参考解验证了所提出方法的有效性.本方法的FDFD格式通过在传统的二阶FDFD格式的基础上引入相关校正项得到,其中校正项按网格点与中心点的距离进行分类选取,同时校正项对应的自适应FDFD系数不仅和空间采样间隔之比相关,还和一个波长内的采样点数相关.所需的自适应FDFD系数可通过声波方程的数值频散关系和查找表高效给出.数值频散分析表明,在空间采样间隔相等或不等的情况下,以相速度误差不超过1%为标准,通用格式自适应系数FDFD方法所需的一个波长内的采样点数均小于2.5.数值模拟实验表明,对于不同的空间采样间隔之比,相对于常用的二阶五点FDFD方法和优化九点FDFD方法,通用格式自适应系数FDFD方法均可在相似的计算量和内存需求下,有效提高黏声波模拟的精度.
