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.
KeywordsLocal meshless collocation method Radial basis function Boundary layer Convection–diffusion–PDEs TVD Runge–Kutta method
- 3.Qamar S, Noor S, Rehman M, Seidel-Morgenstern A (2010) Numerical solution of a multi-dimensional batch crystallization model with fines dissolution. Comput Chem Eng 36(3):1148–1160Google Scholar
- 16.Dehghan M, Mohammadi V (2017) A numerical scheme based on radial basis function finite difference (RBF-FD) technique for solving the high-dimensional nonlinear Schrödinger equations using an explicit time discretization: Runge–Kutta method. Comput Phys Commun 217:23–34MathSciNetCrossRefGoogle Scholar
- 24.Dehghan M, Abbaszadeh M (2018) Solution of multi-dimensional Klein–Gordon–Zakharov and Schrödinger/Gross–Pitaevskii equations via local radial basis functions-differential quadrature (RBF-DQ) technique on non-rectangular computational domains. Eng Anal Bound Elem 92:156–170MathSciNetCrossRefGoogle Scholar
- 32.Ali M (2012) Optimal operation of industrial batch crystallizers a nonlinear model-based control approach. Thesis 36(3):1148–1160Google Scholar