### Abstract

We consider the one-dimensional heat conduction equation. The so-called θ-method will be applied to the numerical solution of the problem. Here, the question is the suitable choice of the mesh on which the continuous problem is discretized, that is, the choice of the stepsize of the discretization in both variables. The basic condition comes from the condition of the convergence. Moreover, it is reasonable to choose such a convergent numerical method which is optimal in a certain sense. For any given implicit finite difference method with fixed number of arithmetic operations, we introduce an optimal parameter choice and define the optimal mesh in this sense of values of the stepsizes. We compare our results with the bounds obtained for the preservation of basic qualitative properties. As a result, we obtain the parameter choice for any convergent method being both optimal and preserving the main qualitative properties. Finally, a numerical example is given.

Original language | English |
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Pages (from-to) | 79-85 |

Number of pages | 7 |

Journal | Computers and Mathematics with Applications |

Volume | 38 |

Issue number | 9 |

DOIs | |

Publication status | Published - Nov 1999 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Modelling and Simulation

### Cite this

*Computers and Mathematics with Applications*,

*38*(9), 79-85. https://doi.org/10.1016/S0898-1221(99)00263-1

**Optimal mesh choice in the numerical solution of the heat equation.** / Faragó, I.; Horváth, R.

Research output: Contribution to journal › Article

*Computers and Mathematics with Applications*, vol. 38, no. 9, pp. 79-85. https://doi.org/10.1016/S0898-1221(99)00263-1

}

TY - JOUR

T1 - Optimal mesh choice in the numerical solution of the heat equation

AU - Faragó, I.

AU - Horváth, R.

PY - 1999/11

Y1 - 1999/11

N2 - We consider the one-dimensional heat conduction equation. The so-called θ-method will be applied to the numerical solution of the problem. Here, the question is the suitable choice of the mesh on which the continuous problem is discretized, that is, the choice of the stepsize of the discretization in both variables. The basic condition comes from the condition of the convergence. Moreover, it is reasonable to choose such a convergent numerical method which is optimal in a certain sense. For any given implicit finite difference method with fixed number of arithmetic operations, we introduce an optimal parameter choice and define the optimal mesh in this sense of values of the stepsizes. We compare our results with the bounds obtained for the preservation of basic qualitative properties. As a result, we obtain the parameter choice for any convergent method being both optimal and preserving the main qualitative properties. Finally, a numerical example is given.

AB - We consider the one-dimensional heat conduction equation. The so-called θ-method will be applied to the numerical solution of the problem. Here, the question is the suitable choice of the mesh on which the continuous problem is discretized, that is, the choice of the stepsize of the discretization in both variables. The basic condition comes from the condition of the convergence. Moreover, it is reasonable to choose such a convergent numerical method which is optimal in a certain sense. For any given implicit finite difference method with fixed number of arithmetic operations, we introduce an optimal parameter choice and define the optimal mesh in this sense of values of the stepsizes. We compare our results with the bounds obtained for the preservation of basic qualitative properties. As a result, we obtain the parameter choice for any convergent method being both optimal and preserving the main qualitative properties. Finally, a numerical example is given.

UR - http://www.scopus.com/inward/record.url?scp=0033224964&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033224964&partnerID=8YFLogxK

U2 - 10.1016/S0898-1221(99)00263-1

DO - 10.1016/S0898-1221(99)00263-1

M3 - Article

VL - 38

SP - 79

EP - 85

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 9

ER -