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

Jan Nordström; Fredrik Laurén; Oskar Ålund; Jan Nordström; Fredrik Laurén; Oskar Ålund

doi:10.3934/math.20241132

AIMS Mathematics

2024, Volume 9, Issue 9: 23291-23312. doi: 10.3934/math.20241132

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

1.
Department of Mathematics, Applied Mathematics, Linköping University, SE-581 83, Linköping, Sweden
2.
Department of Mathematics and Applied Mathematics, University of Johannesburg, P.O. Box 524, Auckland Park 2006, South Africa

Received: 10 June 2024 Revised: 15 July 2024 Accepted: 23 July 2024 Published: 01 August 2024
MSC : 65M06, 65M22

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.
- nonlinear initial boundary value problems,
- Jacobian,
- Newton's method,
- incompressible Navier-Stokes equations,
- summation-by-parts,
- weak boundary conditions
Citation: Jan Nordström, Fredrik Laurén, Oskar Ålund. An explicit Jacobian for Newton's method applied to nonlinear initial boundary value problems in summation-by-parts form[J]. AIMS Mathematics, 2024, 9(9): 23291-23312. doi: 10.3934/math.20241132

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Abstract

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.

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