The incompressible Euler equations are an important model system in computational fluid dynamics. Fast high-order methods for the solution of this time-dependent system of partial differential equations are of particular interest: due to their exponential convergence in the polynomial degree they can make efficient use of computational resources. To address this challenge we describe a novel timestepping method which combines a hybridised Discontinuous Galerkin method for the spatial discretisation with IMEX timestepping schemes, thus achieving high-order accuracy in both space and time. The computational bottleneck is the solution of a (block-) sparse linear system to compute updates to pressure and velocity at each stage of the IMEX integrator. Following Chorin's projection approach, this update of the velocity and pressure fields is split into two stages. As a result, the hybridised equation for the implicit pressure-velocity problem is reduced to the well-known system which arises in hybridised mixed formulations of the Poisson- or diffusion problem and for which efficient multigrid preconditioners have been developed. Splitting errors can be reduced systematically by embedding this update into a preconditioned Richardson iteration. The accuracy and efficiency of the new method is demonstrated numerically for two time-dependent testcases that have been previously studied in the literature.

翻译：不可压缩欧拉方程是计算流体力学中的重要模型系统。针对该含时偏微分方程组的快速高阶解法尤其受到关注：因其在多项式次数上具有指数收敛性，能高效利用计算资源。为应对这一挑战，本文提出一种新型时间推进方法，将空间离散化的可杂交间断伽辽金方法与IMEX时间推进格式相结合，从而在空间和时间上均实现高阶精度。计算瓶颈在于求解（分块）稀疏线性系统以计算IMEX积分器每阶段的压力和速度更新值。遵循Chorin投影法思想，将速度场与压力场的更新分解为两个阶段。由此，隐式压力-速度问题的可杂交方程被简化为混合型泊松/扩散问题中常见的经典系统，针对该系统已发展出高效的多重网格预处理器。通过将更新过程嵌入预条件Richardson迭代，可系统性地降低分裂误差。通过对文献中两个经典含时测试算例的数值实验，验证了新方法在精度与效率方面的优越性能。