An explicit Jacobian for Newton's method applied to nonlinear initial boundary value problems in summation-by-parts form

We derived an explicit form of the Jacobian for discrete approximations of a nonlinear initial boundary value problems (IBVPs) in matrix-vector form. The Jacobian is used in Newton's method to solve the corresponding nonlinear system of equations. The technique was exemplified on the incompress...

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Bibliographic Details
Published inAIMS mathematics Vol. 9; no. 9; pp. 23291 - 23312
Main Authors Nordström, Jan, Laurén, Fredrik, Ålund, Oskar
Format Journal Article
LanguageEnglish
Published AIMS Press 01.01.2024
Subjects
incompressible Navier-Stokes equations
Jacobian
Newton's method
nonlinear initial boundary value problems
summation-by-parts
weak boundary conditions
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Summary:We derived an explicit form of the Jacobian for discrete approximations of a nonlinear initial boundary value problems (IBVPs) in matrix-vector form. The Jacobian is used in Newton's method to solve the corresponding nonlinear system of equations. The technique was exemplified on the incompressible Navier-Stokes equations discretized using summation-by-parts (SBP) difference operators and weakly imposed boundary conditions using the simultaneous approximation term (SAT) technique. The convergence rate of the iterations is verified by using the method of manufactured solutions. The methodology in this paper can be used on any numerical discretization of IBVPs in matrix-vector form, and it is particularly straightforward for approximations in SBP-SAT form.
ISSN:2473-6988
2473-6988
DOI:10.3934/math.20241132