Meshless techniques for convection dominated problems.

*(English)*Zbl 1138.76402Summary: In this paper, the stability problem in the analysis of the convection dominated problems using meshfree methods is first discussed through an example problem of steady state convection-diffusion. Several techniques are then developed to overcome the instability issues in convection dominated phenomenon simulated using meshfree collocation methods. These techniques include: the enlargement of the local support domain, the upwind support domain, the adaptive upwind support domain, the biased support domain, the nodal refinement, and the adaptive analysis. These techniques are then demonstrated in one- and two-dimensional problems. Numerical results for example problems demonstrate the techniques developed in this paper are effective to solve convection dominated problems, and in these techniques, using adaptive local support domain is the most effective method. Comparing with the conventional finite difference method (FDM) and the finite element method (FEM), the meshfree method has found some attractive advantages in solving the convection dominated problems, because it easily overcomes the instability issues.

##### MSC:

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

76R99 | Diffusion and convection |

##### Keywords:

Meshless method; meshfree method; convection-diffusion; convection dominated; numerical analysis##### Software:

Mfree2D
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\textit{Y. T. Gu} and \textit{G. R. Liu}, Comput. Mech. 38, No. 2, 171--182 (2006; Zbl 1138.76402)

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