A local reproducing kernel method accompanied by some different edge improvement techniques: application to the Burgers’ equation

Abstract

The paper introduces a local reproducing kernel method based on spatial trial space spanned by the Newton basis functions in the “native” Hilbert space of the reproducing kernel. It is a truly meshless approach which uses the local sub clusters of domain nodes, named local domains of influence, for the calculation of the spatial partial derivatives. With the selected domain of influence, an approximation function is introduced as a sum of weighted local Newton basis functions. Then the collocation approach is used to determine weights. After the successful approximation function creation, all the needed differential operators can be constructed by applying an arbitrary operator on the approximation function. The main advantage of using the local method is that the overlapping domains of influence results in many small matrices with the dimension of the number of nodes included in the domain of influence for each center node, instead of a large collocation matrix, and hence the sparse global derivative matrices from local contributions. So the method needs less computer storage and flops. The method leads to a system of ordinary differential equations (ODEs) for the time-dependent partial differential equations (PDEs). This method can be implemented on a set of uniform or random nodes, without any a priori knowledge of node to node connectivity. We have chosen uniform nodal arrangement due to their suitability and better accuracy. To decrease accuracy degradations near boundaries, some different edge improvement techniques are also used. The method is tested on four benchmark problems on Burgers’ equation with mixed boundary conditions. Five nodded domains of influence are used in the local support. The numerical results for different value of Reynolds numbers (Re) is compared with analytical solution as well as some other numerical methods. It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high values of Re.