Multi-level methods with high order in time and space for the numerical simulation of incompressible flows

与现有的数值方法相比，高阶方法具有更高的收敛速度，因此为同时优化流动模拟的速度和精度提供了很大的潜力。为了利用这一优势，空间和时间的顺序必须相互协调。然而，尽管存在强大的高阶空间离散方法，但对于不可压缩流，实际的时间积分方案只能达到2到4阶。因此，时间离散化成为限制因素。它将方法的总体顺序限制在传统有限元方法和有限体积方法已经达到的值上。该项目旨在突破这一障碍。这将通过开发半隐式多级方法来实现，该方法可以对空间和时间上任意顺序的不可压缩流动进行数值模拟。其主要思想是将光谱延迟校正（SDC）方法与定制分裂方案相结合。在Stokes方程的初步研究中，证明了该方法克服常规方法的顺序障碍的能力。目前的项目将大大扩展这种方法，并为包括可变粘度在内的不可压缩Navier-Stokes问题开发一种SDC。主要里程碑是：1)生成和分析了高稳定性和低迭代次数的SDC新方法；2)利用多层SDC实现了阶无关的收敛速度；3)通过时空多重网格求解离散方程的代价最小化；4)将变系数推广到包括扩展到大涡模拟。对于空间离散化，本项目重点研究节点不连续伽辽金方法。通过这种方法，从4到32的订单在空间和时间上都是有针对性的。该方案为变黏度不可压缩Navier-Stokes问题提供了第一个任意高阶半隐式时间积分方法。它与求解离散方程的高效多级方法相结合，为高阶方法在流体力学中的应用开辟了广阔的前景。在可能的后续项目中推广这些技术有望解决更复杂的问题，例如具有弯曲边界或多相流的配置。

High-order methods achieve a higher convergence rate than established numerical methods and, hence, offer a large potential for the simultaneous optimization of the speed and accuracy of flow simulations. To exploit this advantage, the orders in space and time must be attuned to each other. However, while powerful high-order methods exist for spatial discretization, practical time integration schemes reach only order 2 up to 4 for incompressible flows. As a consequence, the time discretization becomes the limiting factor. It constrains the overall order of the method to values that are already achieved with conventional finite element methods and finite volume methods. The project intends to break through this barrier. This will be accomplished by developing semi-implicit multi-level methods which enable the numerical simulation of incompressible flows with arbitrary order in space and time. The main idea consists in combining the spectral deferred correction (SDC) method with tailored splitting schemes. The ability to overcome the order barrier of conventional methods with this approach was demonstrated in preliminary studies at the example of the Stokes equations. The present project will extend this method substantially and develop an SDC for incompressible Navier-Stokes problems including a variable viscosity. The principal milestones are:1) generation and analysis of new SDC methods with high stability and low iteration count,2) achievement of an order-independent convergence rate by means of multilevel SDC, 3) minimization of the cost for solving the discrete equations via space-time multigrid,4) generalization to varying coefficients including the extension to large-eddy simulation. For spatial discretization, the project focuses on nodal discontinuous Galerkin methods. With this approach orders from 4 to 32 are targeted - in space as well as in time.As a result, the project provides the first semi-implicit time integration method of arbitrarily high order for incompressible Navier-Stokes problems with variable viscosity. Combined with the highly efficient multi-level methods for solving the discrete equations, it opens an attractive perspective for the application of high-order methods in fluid mechanics. The generalization of these techniques in possible follow-up projects promises an approach to far more complex problems, such as configurations with curved boundaries or multi-phase flows.