On the accuracy of integral representation of differential operators in Lagrangian blob mesh-less methods

We explore novel ideas to improve the accuracy of the integral approximation of differential operators (Gradient and Laplacian) in the simulation of thermal viscous problems with Lagrangian Blob mesh-less methods. Basically we investigate and develop a novel convolution integral discretization of the differential operators by using 2D-Taylor series expansions and a Gaussian like kernel function defined on a compact support around the blob centre of a given particle. This allows us to overtake: deficiency of cells in the compact domain due to irregular distribution of the particles around the given blob, deficiency of cells in the compact domain caused by the presence of a boundary cutting the support of a nearby blob. The accuracy and order of approximation of such a discretization are determined in regular and randomly distorted grids of various sizes, and compared with the widely used PSE (Particle Strength Exchange) formulation. Results obtained in the solution of thermal buoyant problems at realistic values of the Grashoff number demonstrate validity and benefits of the novel findings.