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

高阶方法实现了更高的收敛速度比既定的数值方法，因此，提供了一个很大的潜力，同时优化的速度和精度的流动模拟。为了利用这一优势，空间和时间的秩序必须相互协调。然而，虽然强大的高阶方法存在空间离散化，实际的时间积分方案达到只有2至4阶不可压缩流。因此，时间离散化成为限制因素。它约束的整体顺序的方法已经实现了与传统的有限元方法和有限体积法的值。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，该方法能够对空间和时间中任意阶的不可压缩流进行数值模拟。其主要思想是将光谱延迟校正（SDC）方法与定制的分裂方案相结合。在初步研究中，以斯托克斯方程为例，证明了这种方法克服传统方法的阶障碍的能力。本计画将大幅延伸此方法，并发展一个求解不可压缩Navier-Stokes方程的SDC。主要的里程碑是：1）具有高稳定性和低迭代次数的新的SDC方法的产生和分析，2）通过多级SDC实现与阶无关的收敛速度，3）通过时空多重网格最小化求解离散方程的成本，4）推广到变系数包括扩展到大涡模拟。对于空间离散化，该项目侧重于节点间断伽辽金方法。该方法的目标阶数从4到32--在空间和时间上都是如此。因此，该项目为具有变粘性的不可压缩Navier-Stokes方程提供了第一个任意高阶的半隐式时间积分方法。结合高效的多层方法求解离散方程，为高阶方法在流体力学中的应用开辟了一个诱人的前景。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，如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.