In this paper we consider a linearized variable-time-step two-step backward differentiation formula (BDF2) scheme for solving nonlinear parabolic equations. The scheme is constructed by using the variable time-step BDF2 for the linear term and a Newton linearized method for the nonlinear term in time combining with a Galerkin finite element method (FEM) in space. We prove the unconditionally optimal error estimate of the proposed scheme under mild restrictions on the ratio of adjacent time-steps, i.e. $0<r_k < r_{\max} \approx 4.8645$ and on the maximum time step. The proof involves the discrete orthogonal convolution (DOC) and discrete complementary convolution (DCC) kernels, and the error splitting approach. In addition, our analysis also shows that the first level solution $u^1$ obtained by BDF1 (i.e. backward Euler scheme) does not cause the loss of global accuracy of second order. Numerical examples are provided to demonstrate our theoretical results.
翻译:在本文中,我们考虑了用于解决非线性抛物线式方程式的线性可变时间步骤双步后向差异公式(BDF2)方案。这个方案是使用线性术语的可变时间步骤 BDF2 和非线性术语的牛顿线性方法以及空间的加列尔金有限元素元件法(FEM)来构建的。我们的分析还表明,在对相邻时间步骤的比例,即0.<r_k < r ⁇ max}\ approx 4.8645美元和最大时间步骤的轻度限制下,拟议办法的无条件最佳误差估计值。证据涉及离散的或分层共和离子(DCC)和离散的互补共振内核,以及差分解法。此外,我们的分析还表明,BDFF1(即落后的Euler计划)获得的第一级解决方案($%1)不会导致全球第二顺序的准确性损失。提供了数字示例,以证明我们的理论结果。