2019/03/27 by Gianluca Frasca-Caccia, Frasca-Caccia, Gianluca, Peter E. Hydon +1
Mathematics · Physics and Astronomy · #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Waves and Solitons #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1903.11491
openalex publication_date 2019/03/27 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
Finite difference schemes that preserve two conservation laws of a given\npartial differential equation can be found directly by a recently-developed\nsymbolic approach. Until now, this has been used only for equations with\nquadratic nonlinearity. In principle, a simplified version of the direct\napproach also works for equations with polynomial nonlinearity of higher\ndegree. For the modified Korteweg-de Vries equation, whose nonlinear term is\ncubic, this approach yields several new families of second-order accurate\nschemes that preserve mass and either energy or momentum. Two of these families\ncontain Average Vector Field schemes of the type developed by Quispel and\ncoworkers. Numerical tests show that each family includes schemes that are\nhighly accurate compared to other mass-preserving methods that can be found in\nthe literature.\n