Efficient Solutions of the Euler Equations in a Time-Adaptive Space-Time Framework
Efficient Solutions of the Euler Equations in a Time-Adaptive Space-Time Framework
复制标题
时间自适应时空框架中欧拉方程的有效解
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
D. Mavriplis
中科院分区：
文献类型：
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作者：
K. Mani;D. Mavriplis
采用时空有限体积离散方法求解无粘欧拉方程。该方法固有地考虑了计算网格的变形。传统的隐式非定常离散依赖于在每个时间层添加适当的时间项来求解空间问题，同时在时间上一步一步地向前推进。本文的方法是统一时间和空间维度，并在一个跨越感兴趣的空间和时间域的单一计算网格上操作。本质上，在感兴趣领域的所有空间和时间位置的所有未知数在一次射击中被隐含地解决。这种方法的主要优点是时间步长在空间上的非均匀变化。其目标是通过要求只有某些空间位置在时间上以高分辨率前进，从而减少总自由度。伴随加权残差法被用来识别时空域中需要更高时间分辨率的区域，从而在保持整体解精度的同时只针对必须在时间上缓慢推进的某些空间位置。解中总未知量的减少和隐式耦合在时间上的快速收敛共同构成了一个高效的非定常求解器。以ILU(0)为预条件，用牛顿-克雷洛夫预条件GMRES方法求解实际的非线性问题。虽然该方法收敛速度很快，但即使对于相对较小的问题，对内存的要求也很高。通过将时间域分割成有限数量的片并且在逐片推进的同时隐式地求解片内的所有未知数来缓解该问题。
The inviscid Euler equations are solved using a space-time finite-volume discretization. The method inherently accounts for deforming computational meshes. Traditional implicit unsteady discretizations rely on solving the spatial problem with appropriate temporal terms added in an implicit sense at each time level while advancing forward in time one time-step at a time. The approach in this paper is to unify both time and space dimensions and operate on a single computational mesh that spans both the spatial and temporal domains of interest. In essence all unknowns at all spatial and temporal locations in the domain of interest are solved implicitly in one shot. The primary advantage of this approach is the spatially non-uniform variation of the time-step size. The goal is to reduce the total degrees-of-freedom by requiring that only certain spatial locations advance with high resolution in time. The adjoint weighted residual method is used to identify regions in the space-time domain that require higher temporal resolution, thus targeting only certain spatial locations that have to be advanced slowly in time while maintaining overall solution accuracy. Both the reduction of the total unknowns in the solution and the faster convergence due to implicit coupling in time combine to form a efficient unsteady solver. The actual non-linear problem is solved using the Newton-Krylov preconditioned GMRES method with ILU(0) as the preconditioner. While the method converges rapidly, the memory requirements are high even for relatively small problems. This problem is alleviated by splitting the time domain into a limited number of slabs and solving for all unknowns within a slab implicitly while advancing in time slab-by-slab.