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On Meshless Collocation Approximations of Conservation Laws: Preliminary Investigations on Positive Schemes and Dissipation Models
Author(s) -
Fürst J.,
Sonar Th.
Publication year - 2001
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/1521-4001(200106)81:6<403::aid-zamm403>3.0.co;2-t
Subject(s) - conservation law , discretization , regularized meshless method , dissipation , mathematics , collocation (remote sensing) , scalar (mathematics) , grid , moving least squares , collocation method , mathematical optimization , computer science , singular boundary method , mathematical analysis , finite element method , physics , geometry , machine learning , boundary element method , thermodynamics , differential equation , ordinary differential equation
We consider meshless collocation methods for the numerical solution of transport processes described by hyperbolic conservation laws. The future goal is the construction of a robust and reliable meshfree discretization method for the equations of gas dynamics in complex geometries. In this paper we start with the simplest scalar model problems and analyze basic problems occurring in the grid‐free approach. A topological condition on clouds of points is derived and several possible versions of a generalized Lax‐Friedrichs scheme are discussed with respect to their numerical dissipation. A moving least‐squares approach is followed to construct a positive discretization for solutions with shocks which is thoroughly analyzed and applied to test problems.