The method of fundamental solutions for non-linear thermal explosions.

*(English)*Zbl 0839.65143On the background of a spontaneous ignition problem, the author considers a technique for solving a nonlinear Poisson equation in general two-dimensional domains. This consists of a Picard iteration to handle the nonlinearity, of a splitting of the remaining linear Poisson equation into an inhomogeneous part (which is solved disregarding boundary conditions by interpolation with thin plate splines) and a Laplace equation for the correction of the boundary values (which is approximated by collocating a sum of fundamental solutions).

This approach requires the solution of two medium-size dense linear systems (for fixed interpolation points in the interior and fixed collocation points on the boundary), is said to be faster and more accurate than the boundary element method and the finite element method (FEM). Results of 3 tests are displaced confirming that accuracy is comparable with that of FEM (however, missing are CPU times; several hundred iterations are reported to have been performed).

This approach requires the solution of two medium-size dense linear systems (for fixed interpolation points in the interior and fixed collocation points on the boundary), is said to be faster and more accurate than the boundary element method and the finite element method (FEM). Results of 3 tests are displaced confirming that accuracy is comparable with that of FEM (however, missing are CPU times; several hundred iterations are reported to have been performed).

Reviewer: G.Stoyan (Budapest)

##### MSC:

65Z05 | Applications to the sciences |

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

35Q72 | Other PDE from mechanics (MSC2000) |

80A25 | Combustion |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

##### Keywords:

method of fundamental solutions; nonlinear thermal explosions; nonlinear Poisson equation; Picard iteration; Laplace equation; collocation; boundary element method; finite element method
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\textit{C. S. Chen}, Commun. Numer. Methods Eng. 11, No. 8, 675--681 (1995; Zbl 0839.65143)

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##### References:

[1] | Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach space, SIAM Rev. 18 pp 620– (1976) · Zbl 0345.47044 |

[2] | Lacey, Mathematical analysis of thermal runaway for spatially inhomogenous reactions, SIAM J. Appl. Math. 43 (6) pp 1350– (1983) · Zbl 0543.35047 |

[3] | Bogomolny, Fundamental solutions method for elliptic boundary value problems, SIAM J. Numer. Anal. 22 pp 644– (1985) · Zbl 0579.65121 |

[4] | Anderson, Spontaneous ignition: finite element solutions for steady and transient conditions, Trans. ASME J. Heat Transf. 96 pp 398– (1974) |

[5] | Partridge, The Dual Reciprocity Boundary Element (1992) · Zbl 0712.73094 |

[6] | Golberg, On a method of Atkinson for evaluating domain integrals in the boundary element method, Appl. Math. Compu. 60 pp 125– (1994) · Zbl 0797.65090 |

[7] | Nowak, The multiple reciprocity method-A new approach for transforming BEM domain integrals to the boundary, Eng. Anal. Bound. Elem. 6 (3) pp 164– (1989) |

[8] | Golberg, The theory of radial basis functions applied to the BEM for inhomogeneous partial differential equations, Commun. Bound. Elem. 5 pp 57– (1994) |

[9] | Powell, Advances in Numerical Analysis 2 pp 105– (1992) |

[10] | Golberg, Boundary Element Technology IX pp 299– (1994) |

[11] | Chan, A numerical method for semilinear singular parabolic quenching problems, Q. Appl. Math. 47 pp 45– (1989) · Zbl 0675.65124 |

[12] | Duchon, Splines minimizing rotation invariant seminorms in Sobolev spaces in constructive theory of functions of several variables, Lect. Notes Math. 571 (1976) |

[13] | Boddington, Thermal theory and spontaneous ignition: criticality in bodies of arbitrary shape, Philos. Trans. R. Soc. Lond. 270 pp 467– (1971) |

[14] | Zinn, Thermal initiation of explosives, J. Appl. Phys. 31 pp 323– (1960) |

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