### 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 language | English |
---|---|

Pages (from-to) | 1722-1738 |

Number of pages | 17 |

Journal | Computers and Mathematics with Applications |

Volume | 77 |

Issue number | 6 |

DOIs | |

Publication status | Published - 15 Mar 2019 |

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### Keywords

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

### Cite this

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*Computers and Mathematics with Applications*, vol. 77, no. 6, pp. 1722-1738. https://doi.org/10.1016/j.camwa.2018.09.042

**A modified L-scheme to solve nonlinear diffusion problems.** / Mitra, K. (Corresponding author); Pop, I.S.

Research output: Contribution to journal › Article › Academic › peer-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.

KW - Linearization

KW - Newton, Picard, L-scheme

KW - Nonlinear diffusion problem

KW - Richards equation

KW - Stability

KW - Unconditional convergence

UR - http://www.scopus.com/inward/record.url?scp=85054571368&partnerID=8YFLogxK

U2 - 10.1016/j.camwa.2018.09.042

DO - 10.1016/j.camwa.2018.09.042

M3 - Article

AN - SCOPUS:85054571368

VL - 77

SP - 1722

EP - 1738

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 6

ER -