Newton-multigrid method for nonlinear silicon problem with relaxing boundary conditions

Articles in Press

Document Type : Research Paper

Authors

1 Catarinense Federal Institute, BR-280, Araquari, 89245-000, Santa Catarina, Brazil.

2 Department of Mechanical Engineering, Federal University of Paran\'{a}, Polytechnic Center, Curitiba, 81530-000, Paran\'{a}, Brazil.

3 Department of Mathematics, State University of the Central West, Irati, 84505-677, Paran\'{a}, Brazil.

4 Graduate Program in Numerical Methods in Engineering, Federal University of Paran\'{a}, Polytechnic Center, Curitiba, 81530-000, Paran\'{a}, Brazil.

10.22034/cmde.2026.67988.3259

Abstract

This paper introduces a Newton-multigrid (Newton-MG) method to efficiently solve a nonlinear heat transfer problem in a homogeneous silicon rod. The numerical model is constructed using the Finite Difference Method (FDM) with a Central Difference Scheme (CDS) for spatial discretization and the Crank-Nicolson method for temporal approximation. Newton's method is applied to linearize the discretized equations and a multigrid Correction Scheme (CS) is integrated to solve the resulting linear system. Computational experiments demonstrate that, regardless of the physical and numerical parameters, the apparent order of discretization error converges to its theoretical asymptotic order. Additionally, the Newton-MG method exhibits rapid convergence, requiring few linearization steps while achieving a favorable convergence factor. The efficiency gain relative to the singlegrid method increases with the degree of nonlinearity in the physical model. Our findings confirm that Newton-MG is a robust and computationally efficient alternative for nonlinear heat conduction problems.

Keywords

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 20 April 2026

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History

  • Receive Date: 03 July 2025
  • Revise Date: 03 January 2026
  • Accept Date: 18 April 2026

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