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.
Adaptive-Growth Randomized Neural Networks for Level-Set Computation of Multivalued Nonlinear First-Order PDEs with Hyperbolic Characteristics
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China（数学与统计学学院，西安交通大学，西安，陕西 710049，中国）；School of Mathematical Sciences & Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China（数学科学学院及自然科学研究所，上海交通大学，上海 200240，中国）；School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China（数学与统计学学院及电力工程多相流国家重点实验室，西安交通大学，西安，陕西 710049，中国）
