论文标题
相位场晶体方程的线性自适应二阶向后分化配方方案
A linear adaptive second-order backward differentiation formulation scheme for the phase field crystal equation
论文作者
论文摘要
在本文中，我们介绍并分析了一个线性完全离散的二阶方案，该方案具有相位场晶体方程的可变时间步长。更准确地说，我们基于二阶向后分化公式（BDF2）构建线性自适应时间踏脚方案，并使用傅立叶光谱方法进行空间离散化。标量辅助变量方法用于处理非线性术语，在该项中，我们仅采用一阶方法来近似辅助变量。这种处理对于拟议自适应BDF2方案的无条件能量稳定性的推导非常重要。但是，我们首次发现该策略不会通过设置正常常数$ c_ {0} $足够大，以至于$ c_ {0} \ geq 1/\dt。$通过适应性BDF2方案的能量稳定性建立了相关的bdf2方案，可以通过设置正常$ c_ {0} $来影响未知阶段函数$ ϕ^{n} $的二阶准确性。 $γ_{n+1}：= \ dt_ {n+1}/\ dt_ {n} \ leq 4.8645 $。此外，通过使用均匀的$ h^{2} $界的数值解决方案的界限，在非均匀网格上衍生出$ ϕ^{n} $的第二阶级准确性的严格误差估计值。最后，进行了一些数值实验以验证理论结果并证明了完全离散的自适应BDF2方案的效率。
In this paper, we present and analyze a linear fully discrete second order scheme with variable time steps for the phase field crystal equation. More precisely, we construct a linear adaptive time stepping scheme based on the second order backward differentiation formulation (BDF2) and use the Fourier spectral method for the spatial discretization. The scalar auxiliary variable approach is employed to deal with the nonlinear term, in which we only adopt a first order method to approximate the auxiliary variable. This treatment is extremely important in the derivation of the unconditional energy stability of the proposed adaptive BDF2 scheme. However, we find for the first time that this strategy will not affect the second order accuracy of the unknown phase function $ϕ^{n}$ by setting the positive constant $C_{0}$ large enough such that $C_{0}\geq 1/\Dt.$ The energy stability of the adaptive BDF2 scheme is established with a mild constraint on the adjacent time step radio $γ_{n+1}:=\Dt_{n+1}/\Dt_{n}\leq 4.8645$. Furthermore, a rigorous error estimate of the second order accuracy of $ϕ^{n}$ is derived for the proposed scheme on the nonuniform mesh by using the uniform $H^{2}$ bound of the numerical solutions. Finally, some numerical experiments are carried out to validate the theoretical results and demonstrate the efficiency of the fully discrete adaptive BDF2 scheme.
