2017/01/30 by Adi Ditkowski, Sigal Gottlieb, Ditkowski, Adi +1
Computer Science · Engineering · Mathematics · #65L05 #65L06 #65L70 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods for differential equations #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1701.08568
openalex publication_date 2017/01/30 · openalex created_date 2022/10/02 · openalex updated_date 2026/08/01
The commonly used one step methods and linear multi-step methods all have a\nglobal error that is of the same order as the local truncation error (as\ndefined in\n citegustafsson1995time,quarteroni2010numerical,AllenIsaacson,IsaacsonKeller,Sewell).\nIn fact, this is true of the entire class of general linear methods. In\npractice, this means that the order of the method is typically defined solely\nby the order conditions which are derived by studying the local truncation\nerror. In this work, we investigate the interplay between the local truncation\nerror and the global error, and develop a methodology which defines the\nconstruction of explicit em error inhibiting block one-step methods\n(alternatively written as explicit general linear methods citebutcher1993a).\nThese em error inhibiting schemes are constructed so that the accumulation\nof the local truncation error over time is controlled, which results in a\nglobal error that is one order higher than the local truncation error. In this\nwork, we delineate how to carefully choose the coefficient matrices so that the\ngrowth of the local truncation error is inhibited. We then use this theoretical\nunderstanding to construct several methods that have higher order global error\nthan local truncation error, and demonstrate their enhanced order of accuracy\non test cases. These methods demonstrate that the error inhibiting concept is\nrealizable. Future work will further develop new error inhibiting methods and\nwill analyze the computational efficiency and linear stability properties of\nthese methods.\n