In this paper, we propose a novel computational implicit method, which we call Isogeometric Residual Minimization (iGRM) with direction splitting. The method mixes the benefits resulting from isogeometric analysis, implicit dynamics, residual minimization, and alternating direction solver. We utilize tensor product B-spline basis functions in space, implicit second order time integration schemes, residual minimization in every time step, and we exploit Kronecker product structure of the matrix to employ linear computational cost alternating direction solver. We implement an implicit time integration scheme and apply, for each space-direction, a stabilized mixed method based on residual minimization. We show that the resulting system of linear equations has a Kronecker product structure, which results in a linear computational cost of the direct solver, even using implicit time integration schemes together with the stabilized mixed formulation. We test our method on three advection-diffusion computational examples, including model ``membrane'' problem, the circular wind problem, and the simulations modeling pollution propagating from a chimney.
翻译:在本文中,我们提出了一个新型的计算隐含方法,我们称之为Isogeology 残留最小化(iGRM),其方向分解。该方法将等离子分析、隐含动态、剩余最小化和交替方向求解器产生的惠益混合在一起。我们在空间使用高压产品B-Sspline基功能,隐含第二顺序时间整合计划,将剩余时间在每一步中最小化,并利用矩阵的克龙克尔产品结构来使用线性计算成本交替方向解答器。我们实施了隐含的时间整合计划,并对每个空间方向应用一种基于剩余最小化的稳定的混合方法。我们显示,由此形成的线性方程式系统有一个Kronecker产品结构,这导致直接解答器的线性计算成本,即使使用隐含时间整合计划以及稳定的混合配方法。我们用三个倾斜-扩散计算模型测试我们的方法,包括“embrane”问题、循环风问题和模拟从烟囱中模拟污染扩散。