The meshless approach for the cell method: a new way for the numerical solution of discrete conservation laws

A new methodology for the solution of discrete conservation laws, based on a point by point approach, is presented. For each node, a local mesh is firstly generated, made up of all triangles whose vertices coincide with the node itself and its neighbours. The solution is then determined through mass, energy and momentum balances directly written in a discrete form over a tributary region, represented by the polygon whose vertices are the barycenters and/or the circumcenters of the triangles belonging to the local mesh. This approach avoids global mesh generation (computationally much more expensive), and is particularly efficient for non-linear problems, such as fracture mechanics. In the paper, the numerical method is described in detail for Laplace equation, together with the convergence order as a function of the number of nodes and the type of boundary conditions. Finally, in order to further simplify the procedure, it is proposed to consider the tributary area formed by the circle with center in the generic node and radius equal to the average of the distances between the node and its neighbours. This results in a considerable ease in writing the discrete form of the governing equations, while maintaining the same accuracy and order of convergence than the approach based on local triangles.