On spatial discretization of evolutionary differential equations on the periodic domain with a mixed derivative

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:358

Article

Sato, S.¹  Matsuo, T.¹ 

[1] Univ Tokyo, Grad Sch Informat Sci & Technol, Bunkyo Ku, Tokyo, Japan

关键词: Evolutionary partial differential equations;    Mixed derivative;    Spatial discretization;    Tseng generalized inverses;    Average-difference method;   

DOI  :  10.1016/j.cam.2019.03.021

来源: Elsevier

【 摘 要 】

Recently, various evolutionary partial differential equations (PDEs) with a mixed derivative have been emerged and drawn much attention. Nonetheless, their PDE-theoretical and numerical studies are still in their early stage. In this paper, we aim at the unified framework of numerical methods for such PDEs. However, due to the presence of the mixed derivative, we cannot discuss numerical methods without some appropriate reformulation, which is mathematically challenging itself. Therefore, we first propose a novel procedure for the reformulation of target PDEs into a standard form of evolutionary equations. This contribution may become an important basis not only of numerical analysis, but also of PDE-theory. In order to illustrate this point, we establish the global well-posedness of the sine-Gordon equation. After that, we classify and discuss the spatial discretizations based on the proposed reformulation technique. As a result, we show the average-difference method is suitable for the discretization of the mixed derivative. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free