论文标题
精确的解决方案和多维戈多诺夫方案
Exact solution and the multidimensional Godunov scheme for the acoustic equations
论文作者
论文摘要
作为Euler方程的线性化得出的声学方程是多维溶液研究的有价值系统。此外，它们具有类似于Euler方程的低马赫数极限。为了在此限制中理解多维Godunov方案的行为，首先得出了在三个空间维度中相应的Cauchy问题的精确解决方案。讨论了在二维中四季度里曼问题的精确解中的对数奇异性的出现。然后，将溶液公式用于在二维中获得多维戈多诺夫有限体积方案。证明它优于尺寸分开的上风/ROE方案，它的稳定性和解决多维Riemann问题的能力。从实验和理论上显示的是，尽管考虑到多维信息，但仍无法解决低的马赫数限制。
The acoustic equations derived as a linearization of the Euler equations are a valuable system for studies of multi-dimensional solutions. Additionally they possess a low Mach number limit analogous to that of the Euler equations. Aiming at understanding the behaviour of the multi-dimensional Godunov scheme in this limit, first the exact solution of the corresponding Cauchy problem in three spatial dimensions is derived. The appearance of logarithmic singularities in the exact solution of the 4-quadrant Riemann Problem in two dimensions is discussed. The solution formulae are then used to obtain the multidimensional Godunov finite volume scheme in two dimensions. It is shown to be superior to the dimensionally split upwind/Roe scheme concerning its domain of stability and ability to resolve multi-dimensional Riemann problems. It is shown experimentally and theoretically that despite taking into account multi-dimensional information it is, however, not able to resolve the low Mach number limit.
