Three-points interfacial quadrature for geometrical source terms on nonuniform grids Application to finite volume schemes for parameter-dependent differential equations

This paper deals with numerical (finite volume) approximations, on nonuniform meshes, for ordinary differential equations with parameter-dependent fields. Appropriate discretizations are constructed over the space of parameters, in order to guarantee the consistency in presence of variable cells’ si...

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Bibliographic Details
Published in:Calcolo Vol. 49; no. 3; pp. 149 - 176
Main Authors: Katsaounis, Theodoros, Simeoni, Chiara
Format: Journal Article
Language:English
Published: Milan Springer Milan 01.09.2012
Springer Verlag
Subjects:
Differential equations
Dynamical systems
Dynamics
Finite element method
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Nonuniform
Numerical Analysis
Regularity
Theory of Computation
65L05
65M08
34A34
34A45
Consistency
Nonuniform grids
Optimal convergence rates
65L70
Geometrical source terms
65L20
34C60
45J99
Finite volume schemes
geometrical source terms
finite volume schemes
nonuniform grids
consistency
optimal convergence rates
ISSN:0008-0624, 1126-5434
Online Access:Get full text
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Summary:This paper deals with numerical (finite volume) approximations, on nonuniform meshes, for ordinary differential equations with parameter-dependent fields. Appropriate discretizations are constructed over the space of parameters, in order to guarantee the consistency in presence of variable cells’ size, for which L p -error estimates, 1≤ p <+∞, are proven. Besides, a suitable notion of (weak) regularity for nonuniform meshes is introduced in the most general case, to compensate possibly reduced consistency conditions, and the optimality of the convergence rates with respect to the regularity assumptions on the problem’s data is precisely discussed. This analysis attempts to provide a basic theoretical framework for the numerical simulation on unstructured grids (also generated by adaptive algorithms) of a wide class of mathematical models for real systems (geophysical flows, biological and chemical processes, population dynamics).
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ISSN:0008-0624
1126-5434
DOI:10.1007/s10092-011-0049-6