A Mass-Preserving Two-Step Lagrange?Galerkin Scheme for Convection-Diffusion Problems
A Mass-Preserving Two-Step Lagrange?Galerkin Scheme for Convection-Diffusion Problems
复制标题
对流扩散问题的质量守恒两步拉格朗日伽辽金方案
DOI：
10.1007/s10915-022-01885-w
复制
发表时间：
2022
影响因子：
2.5
通讯作者：
Suzuki Tasuku
中科院分区：
文献类型：
--
作者：
Futai Kouta;Kolbe Niklas;Notsu Hirofumi;Suzuki Tasuku
针对对流扩散问题，提出了一种二阶保质量拉格朗日-伽辽金格式，并在理论框架内证明了该格式具有最优误差估计的收敛性。所引入的方案保持了拉格朗日-伽辽金方法的优点，即对流主导问题的无cfl鲁棒性和离散化后的对称正系数矩阵。此外，如果所涉及的积分计算准确，该方案在离散水平上保持质量。通过引入两个新的关键引理，证明了该方法在时间上的无条件稳定性和二阶误差估计。质量保持特性是通过Rui和Tabata（2010）引入的雅可比矩阵乘法技术实现的，二阶精度是基于多步伽辽金方法的思想，沿着Ewing和Russel（1981）最初引入的特征得到的。对于第一个时间步，采用Rui和Tabata在2010年提出的一阶时间质量保持方案，该方案效率高，且不会造成-和范数的收敛阶损失。对于时间增量，网格大小为多项式次的一致性有限元空间，收敛阶为范数以内，如果采用对偶论证，收敛阶为范数以内。另外证明了-范数的离散版本的误差估计。数值结果证实了理论在一维、二维和三维上的收敛顺序。
A mass-preserving two-step Lagrange–Galerkin scheme of second order in time for convection-diffusion problems is presented, and convergence with optimal error estimates is proved in the framework of-theory. The introduced scheme maintains the advantages of the Lagrange–Galerkin method, i.e., CFL-free robustness for convection-dominated problems and a symmetric and positive coefficient matrix resulting from the discretization. In addition, the scheme conserves the mass on the discrete level if the involved integrals are computed exactly. Unconditional stability and error estimates of second order in time are proved by employing two new key lemmas on the truncation error of the material derivative in conservative form and on a discrete Gronwall inequality for multistep methods. The mass-preserving property is achieved by the Jacobian multiplication technique introduced by Rui and Tabata in 2010, and the accuracy of second order in time is obtained based on the idea of the multistep Galerkin method along characteristics originally introduced by Ewing and Russel in 1981. For the first time step, the mass-preserving scheme of first order in time by Rui and Tabata in 2010 is employed, which is efficient and does not cause any loss of convergence order in the- and-norms. For the time increment, the mesh sizehand a conforming finite element space of polynomial degree, the convergence order is ofin the-norm and ofin the-norm if the duality argument can be employed. Error estimates ofin discrete versions of the- and-norm are additionally proved. Numerical results confirm the theoretical convergence orders in one, two and three dimensions.