【作者】 李媛；

【导师】 程爱杰； 刘争光；

【作者基本信息】 山东大学 ， 计算数学， 2024， 硕士

【摘要】 Allen-Cahn方程是用于描述两相界面扩散性质的经典相场模型,可应用于材料学、流体力学、图像处理等领域,具有深远的研究价值。Allen-Cahn方程不仅满足能量耗散律,即能量随时间演化递减,而且具有最大界原理(maximum bound principle,MBP),即在适当的初始/边界条件下,如果初始数据绝对值的最大值以某一特定常数为界,其解绝对值的最大值也始终以这一常数为界。稳定化指数标量辅助变量(stabilized exponential-SAV,sESAV)算法是高效稳定求解Allen-Cahn方程的有效方法,现已给出并分析了一阶和二阶格式,证明了该格式在离散条件下同时保持能量耗散律和MBP。但此算法得到的数值格式保留了根据辅助变量而不是原始变量的“修正”能量耗散律,即不一定满足原PDE模型的能量耗散律。为了解决这一问题,受松弛SAV算法的启发,本文首先提出了松弛sESAV(relaxed-sESAV,R-sESAV)算法,保留sESAV算法所有优点的同时,很大程度上避免了离散情形下原算法不断偏离原始定义能量的问题。基于此算法,针对Allen-Cahn方程证明了一阶、二阶数值格式在离散条件下同时保持能量耗散律和MBP。其次提出了能量最优的sESAV(energy-optimal sESAV,EOP-sESAV)算法,该算法不仅简化了计算,还得到了最优的能量近似,基于此算法,针对Allen-Cahn方程给出了一阶、二阶数值格式保持能量耗散律和MBP的证明。最后还进行了大量的数值试验和比较,以证明所提算法的准确性和有效性。更多还原

【Abstract】 The Allen-Cahn equation is a classical phase field model used to describe the diffusion properties of the two-phase interface,which can be applied to materials science,fluid mechanics,image processing and other fields,and has far-reaching research value.The Allen-Cahn equation not only satisfies the energy dissipation law in the sense that energy decreases with time,and it has the maximum bound principle in the sense that under appropriate initial/boundary conditions,if the maximum of the absolute value of the initial data is bounded by a particular constant,the maximum of the absolute value of its solution is also always bounded by this constant.The stabilized exponential-scalar auxiliary variable(sESAV)approach,which is an effective method for efficiently and stably solving the Allen-Cahn equation,is now given and analyzed in first-and second-order schemes,and it is shown that the scheme maintains both the energy dissipation law and the MBP under discrete conditions.However,the numerical scheme obtained by this approach retains the "modified" energy dissipation law based on the auxiliary variable rather than the original variable in the sense that the approach does not necessarily satisfy the energy dissipation law of the original PDE model.In order to solve this problem,inspired by the relaxed-SAV approach,this paper firstly proposes the relaxation stabilized exponential SAV approach,which retains all the advantages of the sESAV approach and at the same time largely avoids the problem of the original approach constantly deviating from the original defined energy in the discrete case.Based on this approach,it is proved for the Allen-Cahn equation that the first-order and second-order numerical schemes maintain both the energy dissipation law and the MBP under discrete conditions.Secondly,the energy-optimal stabilized exponential SAV approach is proposed,which not only simplifies the computation,but also obtains the optimal energy approximation,based on which the proofs of the first-order and second-order numerical format preserving the energy dissipation law and MBP are given for the Allen-Cahn equation.Several examples have been presented to demonstrate the effectiveness and accuracy of the proposed approaches.更多还原