Local meshless method for convection dominated steady and unsteady partial differential equations

In this paper, we propose a mildly shock-capturing stabilized local meshless method (SLMM) for convection-dominated steady and unsteady PDEs. This work is extension of the numerical procedure, which was designed only for steady state convection-dominated PDEs (Siraj-ul-Islam and Singh in Int J Comput Methods 14(6):1750067, 2017). The proposed meshless methodology is based on employing different type of stencils embodying the already known flow direction. Numerical experiments are performed to compare the proposed method with the finite-difference method on special grid (FDSG) and other numerical methods. Numerical results confirm that the new approach is accurate and efficient for solving a wide class of one- and two- dimensional convection-dominated PDEs having sharp corners and jump discontinuities. Performance of the SLMM is found better in some cases and comparable in other cases, than the mesh-based numerical methods.