A Multilevel Block Preconditioner for the HDG Trace System Applied to Incompressible Resistive MHD
A Multilevel Block Preconditioner for the HDG Trace System Applied to Incompressible Resistive MHD
复制标题
DOI：
10.1016/j.cma.2022.115775
复制
发表时间：
2020-12
期刊：
影响因子：
--
通讯作者：
Sriramkrishnan Muralikrishnan;Stephen Shannon;T. Bui-Thanh;J. Shadid
中科院分区：
文献类型：
--
作者：
Sriramkrishnan Muralikrishnan;Stephen Shannon;T. Bui-Thanh;J. Shadid
提出了一种可扩展的块预处理策略，用于不可压缩电阻磁流体动力学（MHD）的高阶杂交间断Galerkin（HDG）离散所产生的迹系统。我们构建了块预条件与最小二乘换向器（BFBT）近似的舒尔补的逆隔离出的压力未知的跟踪系统。剩余的速度，磁场，和拉格朗日乘子未知数形成一个耦合的节点未知块的系统代数多重网格（AMG）是用来近似逆。MHD方程的复杂性以及静态凝聚HDG跟踪系统的代数性质使得系统AMG部分中平滑器的选择对于块预条件子的收敛性和性能至关重要。我们的数值实验表明，GMRES预处理的ILU（0）重叠零作为一个平滑内部系统AMG表现最好的鲁棒性，每次非线性迭代的时间和内存需求。通过几个二维和三维瞬态测试案例，包括高Lundquist数下的孤岛合并问题，我们证明了块预处理器的鲁棒性和并行可扩展性。此外，对于上层块的交替节点块系统求解器的多层近似嵌套剖分的基础上进行了初步研究。在一个2D的岛屿合并问题的多级近似嵌套解剖预处理显示出更好的可扩展性，相对于网格细化比系统AMG，但相对较弱的鲁棒性Lundquist数缩放。在附录B中，我们严格地证明了：（1）非线性MHD系统在小时间内解的唯一性，（2）Picard迭代法对每一向后Euler步的收敛性，（3）整个时间步的收敛性。
We present a scalable block preconditioning strategy for the trace system arising from the high-order hybridized discontinuous Galerkin (HDG) discretization of incompressible resistive magnetohydrodynamics (MHD). We construct the block preconditioner with a least squares commutator (BFBT) approximation for the inverse of the Schur complement that segregates out the pressure unknowns of the trace system. The remaining velocity, magnetic field, and Lagrange multiplier unknowns form a coupled nodal unknown block for which a system algebraic multigrid (AMG) is used to approximate the inverse. The complexity of the MHD equations together with the algebraic nature of the statically condensed HDG trace system makes the choice of smoother in the system AMG part critical for the convergence and performance of the block preconditioner. Our numerical experiments show GMRES preconditioned by ILU(0) of overlap zero as a smoother inside system AMG performs best in terms of robustness, time per nonlinear iteration and memory requirements. With several transient test cases in 2D and 3D including the island coalescence problem at high Lundquist number we demonstrate the robustness and parallel scalability of the block preconditioner. Additionally for the upper block a preliminary study of an alternate nodal block system solver based on a multilevel approximate nested dissection is presented. On a 2D island coalescence problem the multilevel approximate nested dissection preconditioner shows better scalability with respect to mesh refinement than the system AMG, but is relatively less robust with respect to Lundquist number scaling. In the Appendix B, we rigorously show: (1) the uniqueness of the solution of the nonlinear MHD system for small time, (2) the convergence of the Picard iterations for each backward Euler step, and (3) the convergence of the entire time stepping procedure.