Improving the Convergence Order of the Meshless Approach for the Cell Method for Numerical Integration of Discrete Conservation Laws

In thiswork, the problem of increasing the convergence order of the integral meshless method already proposed by the same authors is addressed. Solutions are determined through equations directly written in discrete form over a tributary region represented by the circle with center in the generic node and radius given by the average of the distances between the node itself and its neighbors, thus allowing a considerable ease in writing the discrete form of the governing equations. The proposed approach, besides avoiding global mesh generation, adopts interpolating polynomials, which exactly reproduce nodal values of field variables, and eliminates some problems typically encountered when posing Dirichlet and Neumann boundary conditions with the Finite Element Method. Several numerical schemes adopting extended or compact computational cells are proposed and tested for the Laplace equation, in line with the previous papers. Results show that, when using interpolating polynomials that satisfy also the differential operator in some nodes, compact computational cells characterized by the fifth-order of convergence may be constructed.