基于通量重构形式的高阶算法,在保持间断Galerkin算法局部重构特性和非结构网格中任意高阶精度优点的同时,其计算量大大减小,且具有形式简单、灵活性高等特点。使用显式Runge-Kutta法,隐式非线性LU-SGS法,以及使用无矩阵预处理的广义极小残值法（generalized minimal residual,GMRES）进行求解,并使用p型多重网格在低阶次上光顺低频误差以加快求解。一至四阶精度结果显示使用p型多重网格对显式Runge-Kutta求解以及LU-SGS均具有明显的加速效果,而基于无矩阵预处理的GMRES解法具有更好的稳定性和更快的求解速度。本文提出的基于Gauss-Seidel迭代的无矩阵预处理方法,具有高效和稳定的特征,存储量大大小于ILU预处理。
Compare to discontinuous Galerkin method, high order flux-reconstruction method is faster, simpler, more flexible and also can achieve arbitrary order through local reconstruction on unstructured meshes. Explicit Runge-Kutta（RK）, implicit nonlinear LU-SGS and GMRES with matrix-free preconditioning are presented in this paper and p-multigrid is implemented to smooth low frequency error. Results up to fourth order show that p-multigrid can help to accelerate convergence speed for both RK and LU-SGS; GMRES is more robust and efficient. The matrix-free preconditioning method based on Gauss-Seidel iterative is proved efficient, stable and need much lower memory than ILU.
参考文献
| [1] | Wang, Z.J.;Gao, H. .A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids[J].Journal of Computational Physics,2009(21):8161-8186. |
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