论文标题
高阶抛物线方程的二阶半密码方法的质量保守和积极性
Mass-conservative and positivity preserving second-order semi-implicit methods for high-order parabolic equations
论文作者
论文摘要
我们考虑了四阶抛物线方程的一类有限元近似值，可以通过引入辅助变量来写入二阶方程系统。在我们的方法中，我们首先解决了一个变分问题，然后解决了优化问题，以满足解决方案的所需物理特性，例如质量保护，溶液的阳性（非阴性）和能量耗散。此外，我们显示了解决优化问题的解决方案的存在和独特性，并证明该方法会收敛到截断方案\ cite {berger1975}。我们还提出了用于高阶抛物线方程的新的保守截断方法。进行了数值收敛研究，并提出了一系列数值测试，以显示和比较不同方案的效率和鲁棒性。
We consider a class of finite element approximations for fourth-order parabolic equations that can be written as a system of second-order equations by introducing an auxiliary variable. In our approach, we first solve a variational problem and then an optimization problem to satisfy the desired physical properties of the solution such as conservation of mass, positivity (non-negativity) of solution and dissipation of energy. Furthermore, we show existence and uniqueness of the solution to the optimization problem and we prove that the methods converge to the truncation schemes \cite{Berger1975}. We also propose new conservative truncation methods for high-order parabolic equations. A numerical convergence study is performed and a series of numerical tests are presented to show and compare the efficiency and robustness of the different schemes.
