论文标题
线性和半线性抛物线方程的变量级尺寸BDF2方法的稳定性和误差估计值
Stability and error estimates for the variable step-size BDF2 method for linear and semilinear parabolic equations
论文作者
论文摘要
在本文中，通过两步向后分化公式（BDF2）方法（带有可变台阶尺寸）的线性和半连接方程的时间离散量。为问题提供了肯定的答案：$ l^\ infty（0，t; h）$的步长比率的上限是否与线性和半线性抛物线方程的BDF2方法的稳定性与零稳定性的上限相同。 $ l^\ infty（0，t; v）$ - 在更轻松的条件下，在连续的台阶尺寸的比率下，也建立了变量级尺寸BDF2方法的稳定性。基于这些稳定性结果，得出了几种不同规范的错误估计。要使用BDF方法，采用梯形方法和向后的Euler方案来计算起始值。对于后一种选择，在几种规范中从理论和数值上观察到恒定阶梯尺寸BDF2方法的阶递减现象。数值结果还说明了线性和半线性抛物线方程的提议方法的有效性。
In this paper stability and error estimates for time discretizations of linear and semilinear parabolic equations by the two-step backward differentiation formula (BDF2) method with variable step-sizes are derived. An affirmative answer is provided to the question: whether the upper bound of step-size ratios for the $l^\infty(0,T;H)$-stability of the BDF2 method for linear and semilinear parabolic equations is identical with the upper bound for the zero-stability. The $l^\infty(0,T;V)$-stability of the variable step-size BDF2 method is also established under more relaxed condition on the ratios of consecutive step-sizes. Based on these stability results, error estimates in several different norms are derived. To utilize the BDF method the trapezoidal method and the backward Euler scheme are employed to compute the starting value. For the latter choice, order reduction phenomenon of the constant step-size BDF2 method is observed theoretically and numerically in several norms. Numerical results also illustrate the effectiveness of the proposed method for linear and semilinear parabolic equations.
