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**The weighted particle method for convection-diffusion equations. I: The case of an isotropic viscosity.**
*(English)*
Zbl 0676.65121

In the particle method, first introduced to compute the flow of homogeneous incompressible and inviscid fluids [cf. A. Leonard, J. Comput. Phys. 37, 289-339 (1980; Zbl 0438.76009)] the fluid is represented by pointwise vortices which travel with the fluid velocity. The method can also be presented as a generalized finite difference method using Lagrangian coordinates. In the case of a viscous fluid, the particle method must take into account the diffusion effects. To give an answer to this problem, a third degree of freedom, called the strength, is associated with each particle in addition to its position and its volume. The time evolution of both the position and the volume is governed by the convective part of the equation, while the time evolution of the strength is governed by the diffusion part of the equation.

The method consists in approximate replacing the diffusion operator \(\text{div}(b \cdot \text{grad} f)\) by an integral operator and then solving the integro-differential equation by a now classical particle method. The particle method is well known to be better suited to slightly viscous media. Reflecting this, the kernel of the integral operator has to satisfy some scaling conditions, to be compared with the classical approximation of the Boltzmann equation by the Fokker-Planck equation (small angles collision approximation). On the other hand, if the kernel is positive (e.g. if the viscosity is constant), uniformly stable approximation is found without any assumption on the discretization parameter.

The method consists in approximate replacing the diffusion operator \(\text{div}(b \cdot \text{grad} f)\) by an integral operator and then solving the integro-differential equation by a now classical particle method. The particle method is well known to be better suited to slightly viscous media. Reflecting this, the kernel of the integral operator has to satisfy some scaling conditions, to be compared with the classical approximation of the Boltzmann equation by the Fokker-Planck equation (small angles collision approximation). On the other hand, if the kernel is positive (e.g. if the viscosity is constant), uniformly stable approximation is found without any assumption on the discretization parameter.

Reviewer: E.Lanckau

### MSC:

65Z05 | Applications to the sciences |

65C05 | Monte Carlo methods |

76R05 | Forced convection |

35Q99 | Partial differential equations of mathematical physics and other areas of application |