A modified L-scheme to solve nonlinear diffusion problems

K. Mitra (Corresponding author), I.S. Pop

Research output: Contribution to journalArticleAcademicpeer-review

2 Citations (Scopus)

Abstract

In this work, we propose a linearization technique for solving nonlinear elliptic partial differential equations that are obtained from the time-discretization of a wide variety of nonlinear parabolic problems. The scheme is inspired by the L-scheme, which gives unconditional convergence of the linear iterations. Here we take advantage of the fact that at a particular time step, the initial guess for the iterations can be taken as the solution of the previous time step. First it is shown for quasilinear equations that have linear diffusivity that the scheme always converges, irrespective of the time step size, the spatial discretization and the degeneracy of the associated functions. Moreover, it is shown that the convergence is linear with convergence rate proportional to the time step size. Next, for the general case it is shown that the scheme converges linearly if the time step size is smaller than a certain threshold which does not depend on the mesh size, and the convergence rate is proportional to the square root of the time step size. Finally numerical results are presented that show that the scheme is at least as fast as the modified Picard scheme, faster than the L-scheme and is more stable than the Newton or the Picard scheme.

Original languageEnglish
Pages (from-to)1722-1738
Number of pages17
JournalComputers and Mathematics with Applications
Volume77
Issue number6
DOIs
Publication statusPublished - 15 Mar 2019

Fingerprint

Nonlinear Diffusion
Diffusion Problem
Linearization
Partial differential equations
Nonlinear Problem
Convergence Rate
Directly proportional
Nonlinear Parabolic Problems
Converge
Iteration
Linearization Techniques
Quasilinear Equations
Elliptic Partial Differential Equations
Guess
Time Discretization
Diffusivity
Degeneracy
Square root
Nonlinear Partial Differential Equations
Linearly

Keywords

  • Linearization
  • Newton, Picard, L-scheme
  • Nonlinear diffusion problem
  • Richards equation
  • Stability
  • Unconditional convergence

Cite this

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title = "A modified L-scheme to solve nonlinear diffusion problems",
abstract = "In this work, we propose a linearization technique for solving nonlinear elliptic partial differential equations that are obtained from the time-discretization of a wide variety of nonlinear parabolic problems. The scheme is inspired by the L-scheme, which gives unconditional convergence of the linear iterations. Here we take advantage of the fact that at a particular time step, the initial guess for the iterations can be taken as the solution of the previous time step. First it is shown for quasilinear equations that have linear diffusivity that the scheme always converges, irrespective of the time step size, the spatial discretization and the degeneracy of the associated functions. Moreover, it is shown that the convergence is linear with convergence rate proportional to the time step size. Next, for the general case it is shown that the scheme converges linearly if the time step size is smaller than a certain threshold which does not depend on the mesh size, and the convergence rate is proportional to the square root of the time step size. Finally numerical results are presented that show that the scheme is at least as fast as the modified Picard scheme, faster than the L-scheme and is more stable than the Newton or the Picard scheme.",
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A modified L-scheme to solve nonlinear diffusion problems. / Mitra, K. (Corresponding author); Pop, I.S.

In: Computers and Mathematics with Applications, Vol. 77, No. 6, 15.03.2019, p. 1722-1738.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - A modified L-scheme to solve nonlinear diffusion problems

AU - Mitra, K.

AU - Pop, I.S.

PY - 2019/3/15

Y1 - 2019/3/15

N2 - In this work, we propose a linearization technique for solving nonlinear elliptic partial differential equations that are obtained from the time-discretization of a wide variety of nonlinear parabolic problems. The scheme is inspired by the L-scheme, which gives unconditional convergence of the linear iterations. Here we take advantage of the fact that at a particular time step, the initial guess for the iterations can be taken as the solution of the previous time step. First it is shown for quasilinear equations that have linear diffusivity that the scheme always converges, irrespective of the time step size, the spatial discretization and the degeneracy of the associated functions. Moreover, it is shown that the convergence is linear with convergence rate proportional to the time step size. Next, for the general case it is shown that the scheme converges linearly if the time step size is smaller than a certain threshold which does not depend on the mesh size, and the convergence rate is proportional to the square root of the time step size. Finally numerical results are presented that show that the scheme is at least as fast as the modified Picard scheme, faster than the L-scheme and is more stable than the Newton or the Picard scheme.

AB - In this work, we propose a linearization technique for solving nonlinear elliptic partial differential equations that are obtained from the time-discretization of a wide variety of nonlinear parabolic problems. The scheme is inspired by the L-scheme, which gives unconditional convergence of the linear iterations. Here we take advantage of the fact that at a particular time step, the initial guess for the iterations can be taken as the solution of the previous time step. First it is shown for quasilinear equations that have linear diffusivity that the scheme always converges, irrespective of the time step size, the spatial discretization and the degeneracy of the associated functions. Moreover, it is shown that the convergence is linear with convergence rate proportional to the time step size. Next, for the general case it is shown that the scheme converges linearly if the time step size is smaller than a certain threshold which does not depend on the mesh size, and the convergence rate is proportional to the square root of the time step size. Finally numerical results are presented that show that the scheme is at least as fast as the modified Picard scheme, faster than the L-scheme and is more stable than the Newton or the Picard scheme.

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KW - Newton, Picard, L-scheme

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KW - Stability

KW - Unconditional convergence

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