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Adaptive Second-Order Crank--Nicolson Time-Stepping Schemes for Time-Fractional Molecular Beam Epitaxial Growth Models
SIAM Journal on Scientific Computing ( IF 3.1 ) Pub Date : 2020-06-01 , DOI: 10.1137/19m1259675
Bingquan Ji , Hong-lin Liao , Yuezheng Gong , Luming Zhang

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page B738-B760, January 2020.
Adaptive second-order Crank--Nicolson time-stepping methods using the recent scalar auxiliary variable (SAV) approach are developed for the time-fractional molecular beam epitaxial models with Caputo's fractional derivative. Based on the piecewise linear interpolation, the Caputo's derivative is approximated by a novel second-order formula, which is naturally suitable for a general class of nonuniform meshes and essentially preserves the positive semidefinite property of the integral kernel. The resulting Crank--Nicolson SAV time-stepping schemes are unconditionally energy stable on arbitrary nonuniform time meshes. The fast algorithm and adaptive time strategy are employed to speed up the numerical computation. Ample numerical results show that our methods are computationally efficient in multiscale time simulations and appropriate for accurately resolving the intrinsically initial singularity of the solution and for efficiently capturing the fast dynamics away from the initial time.


中文翻译:

分数阶分子束外延生长模型的自适应二阶Crank-Nicolson时间步长方案

SIAM科学计算杂志,第42卷,第3期,第B738-B760页,2020年1月。
针对具有Caputo分数导数的时间分数分子束外延模型,开发了使用最新标量辅助变量(SAV)方法的自适应二阶Crank-Nicolson时间步进方法。基于分段线性插值,Caputo的导数由一个新颖的二阶公式逼近,该公式自然适用于一般类别的非均匀网格,并基本上保留了积分核的正半定性。由此产生的Crank-Nicolson SAV时间步长方案在任意非均匀时间网格上都是无条件能量稳定的。采用快速算法和自适应时间策略来加速数值计算。
更新日期:2020-06-01
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