In this paper, we present a numerical strategy to check the strong stability (or GKS-stability) of one-step explicit finite difference schemes for the one-dimensional advection equation with an inflow boundary condition. The strong stability is studied using the Kreiss-Lopatinskii theory. We introduce a new tool, the intrinsic Kreiss-Lopatinskii determinant, which possesses the same regularity as the vector bundle of discrete stable solutions. By applying standard results of complex analysis to this determinant, we are able to relate the strong stability of numerical schemes to the computation of a winding number, which is robust and cheap. The study is illustrated with the O3 scheme and the fifth-order Lax-Wendroff (LW5) scheme together with a reconstruction procedure at the boundary.
翻译:在本文中,我们提出了一个数字战略,以检查单维对流方程式的单维对流边界条件的一步明确的有限差异方案是否牢固稳定(或GKS稳定性);使用Kreiss-Lopatinskii理论研究了强稳定性;我们引入了一种新的工具,即固有的Kreiss-Lopatinskii决定因素,它与离散稳定解决方案的矢量捆绑具有同样的规律性;通过对这一决定因素应用复杂的分析标准结果,我们可以将数字方案的强稳定性与一个稳健和廉价的通风数字的计算联系起来;该研究用O3方案和第五顺序Lax-Wendroff(LW5)方案以及边界的重建程序加以说明。