### Abstract

This article develops a new two-level three-point implicit finite difference scheme of order 2 in time and 4 in space based on arithmetic average discretization for the solution of nonlinear parabolic equation ε u _{xx} = f(x, t, u, u_{x}, u_{t}), 0 <1 <1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where ε > 0 is a small positive constant. We also propose a new explicit difference scheme of order 2 in time and 4 in space for the estimates of (∂u/∂x). The main objective is the proposed formulas are directly applicable to both singular and nonsingular problems. We do not require any fictitious points outside the solution region and any special technique to handle the singular problems. Stability analysis of a model problem is discussed. Numerical results are provided to validate the usefulness of the proposed formulas.

Original language | English |
---|---|

Pages (from-to) | 640-651 |

Number of pages | 12 |

Journal | Numerical Methods for Partial Differential Equations |

Volume | 23 |

Issue number | 3 |

DOIs | |

Publication status | Published - May 2007 |

### Fingerprint

### Keywords

- Arithmetic average discretization
- Burgers' equation
- Diffusion-convection equation
- Implicit scheme
- Nonlinear parabolic equation
- RMS errors
- Singular problem

### ASJC Scopus subject areas

- Applied Mathematics
- Analysis
- Computational Mathematics

### Cite this

*Numerical Methods for Partial Differential Equations*,

*23*(3), 640-651. https://doi.org/10.1002/num.20195

**An O(k2 + kh2 + h4) arithmetic average discretization for the solution of 1-D nonlinear parabolic equations.** / Mohanty, R. K.; Karaa, Samir; Arora, Urvashi.

Research output: Contribution to journal › Article

*Numerical Methods for Partial Differential Equations*, vol. 23, no. 3, pp. 640-651. https://doi.org/10.1002/num.20195

}

TY - JOUR

T1 - An O(k2 + kh2 + h4) arithmetic average discretization for the solution of 1-D nonlinear parabolic equations

AU - Mohanty, R. K.

AU - Karaa, Samir

AU - Arora, Urvashi

PY - 2007/5

Y1 - 2007/5

N2 - This article develops a new two-level three-point implicit finite difference scheme of order 2 in time and 4 in space based on arithmetic average discretization for the solution of nonlinear parabolic equation ε u xx = f(x, t, u, ux, ut), 0 <1 <1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where ε > 0 is a small positive constant. We also propose a new explicit difference scheme of order 2 in time and 4 in space for the estimates of (∂u/∂x). The main objective is the proposed formulas are directly applicable to both singular and nonsingular problems. We do not require any fictitious points outside the solution region and any special technique to handle the singular problems. Stability analysis of a model problem is discussed. Numerical results are provided to validate the usefulness of the proposed formulas.

AB - This article develops a new two-level three-point implicit finite difference scheme of order 2 in time and 4 in space based on arithmetic average discretization for the solution of nonlinear parabolic equation ε u xx = f(x, t, u, ux, ut), 0 <1 <1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where ε > 0 is a small positive constant. We also propose a new explicit difference scheme of order 2 in time and 4 in space for the estimates of (∂u/∂x). The main objective is the proposed formulas are directly applicable to both singular and nonsingular problems. We do not require any fictitious points outside the solution region and any special technique to handle the singular problems. Stability analysis of a model problem is discussed. Numerical results are provided to validate the usefulness of the proposed formulas.

KW - Arithmetic average discretization

KW - Burgers' equation

KW - Diffusion-convection equation

KW - Implicit scheme

KW - Nonlinear parabolic equation

KW - RMS errors

KW - Singular problem

UR - http://www.scopus.com/inward/record.url?scp=34248146960&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34248146960&partnerID=8YFLogxK

U2 - 10.1002/num.20195

DO - 10.1002/num.20195

M3 - Article

VL - 23

SP - 640

EP - 651

JO - Numerical Methods for Partial Differential Equations

JF - Numerical Methods for Partial Differential Equations

SN - 0749-159X

IS - 3

ER -