Local variably scaled Newton basis functions collocation method for solving Burgers’ equation

In the present paper, a meshless method which constructs derivatives discretizations based on variably scaled Newton basis functions (VSNBFs) interpolants for localized sets of nodes is provided to solve Burgers’ equation in the “native” Hilbert space of the reproducing kernel. Using such formulas for discretizations of PDEs when each internal node considered as a center, leads to a sparse global of system of ordinary differential equations (ODEs) to be formed by a simple assembly of function values at the local system centrepoints. For efficiency and stability reasons, we use the interpolation with variably scaled kernels, which are introduced recently by Bozzini et al. The VSNBFs can be constructed from a symmetric positive definite variably scaled kernel. The main advantage of using the local method is that the overlapping domains of influence result 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, sparse global derivative matrices from local contributions. Therefore, the method needs less computer storage and flops. The method is tested on six benchmark problems on Burgers’ equation with mixed boundary conditions. Five number of nodes are used in the domain of influence of the local support. The numerical results for different values of Reynolds numbers (Re) are compared with analytical solutions 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. Numerical results also show that using variably scaled kernels leads to results that are better than using constant scaled kernels, with respect to both stability and error.