摘要
偏微分方程数值解在计算数学的研究领域占有重要地位,有限差分是主要方法之一.对于半线性发展方程,一种离散方法是使用显式差分格式,计算量小,但条件稳定；另一种方法是使用隐式差分格式,无条件稳定,但在每一个时间层都要解方程组,当处理高维问题的时候,计算量就会变得非常大.
本文考虑二维半线性发展方程的一类线性化交替方向隐式差分方法.首先基于Crank-Nicolson差分离散思想,将半线性方程离散化,然后通过添加扰动项进行算子分解并充分利用非线性源项的导数信息建立在时间和空间方向均具有二阶精度的一类线性化二层无条件稳定的隐式差分格式.
本文第二节针对半线性反应扩散方程提出了一类线性化二层Peaceman-Rachford交替方向差分方法,该方法充分利用了P-R格式的特点,具有格式简洁、易于使用等优点.利用离散能量方法证明了格式在空间和时间方向按照离散L2范数均具有二阶精度.数值例子验证了理论分析的正确性和格式的有效性.
第三节给出了粘性波动方程的P-R交替方向差分方法.粘性波动方程是一类特殊的半线性双曲型方程,首先通过变量替换将方程从形式上降阶,然后使用P-R离散思想将方程离散导出计算格式.证明了格式按照离散L2范数和离散H1范数在时间和空间方向二阶收敛,实际计算表明该格式计算效果良好.
The study of numerical solutions of partial differential equations plays an important rule in computational mathematics. Finite difference is one of the main methods. For the semilinear evolution equations, one discretizing approach is to use an explicit difference scheme which is easy to be computed, but conditionally stable. The other one is the implicit scheme which is unconditionally stable, but at each time level, we must solve linear systems. When we deal with high dimensional problems, the computational cost becomes very large.
In this paper, we construct linearized alternating direction implicit difference methods for two-dimensional semilinear evolution equations. First, based on the Crank-Nicolson difference discretizing idea, we discretize the semi-linear equations. By adding some perturbing terms and making full use of the derivative information of the semilinear source term, we construct a class of linearized two level unconditionally stable implicit difference scheme possessing second order accuracy both in temporal and spatial directions.
In section 2, we propose a class of linearized two level Peaceman-Rachford type alternating direction implicit difference scheme for semilinear reaction-diffusion equations. By making full use of the P-R method, the scheme has the advantages of concise form and easy to use. This section also proves that the scheme has second order accuracy with respect to discreteL2 norm both in space and time directions by using discrete energy method. Numerical examples illustrate the correctness of the theoretical analysis and the effectiveness of the scheme.
The section 3, we presents a P-R type alternating direction difference scheme for viscous wave equations, which are a special class of semi-linear hyperbolic equations. First, we reduce the order of the equations by replacement of variable, then by using P-R discretizing idea, we derive the computational scheme. This section also proves that the scheme has second order accuracy with respect to discrete L2 norm and discrete H1 norm both in spatial and temporal directions. Numerical examples show that the scheme is effective.
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