Compact approximation stencils based on integrated radial basis functions for fluid flows

This research project is concerned with the development of compact local integrated radial basis function (CLIRBF) stencils and its verification in the simulation of Newtonian and non-Newtonian fluid flows governed by the streamfunction-vorticity formulation. In the CLIRBF stencils, nodal values of the governing equations at selected nodes on a stencil are also incorporated into the IRBF approximations with the help of integration constants to enhance their accuracy.The proposed CLIRBF stencils overcome some of the weaknesses of the local RBF stencils (low order accuracy) and the global RBF approximations (full system matrices). Four main research tasks are carried out • Development of CLIRBF stencils for approximating the field variables and their derivatives. • Incorporation of CLIRBF stencils into two formulations for discretisation of ordinary/partial differential equation (ODEs/PDEs), namely point collocation and subregion collocation. • Discretisation of fourth-order elliptic ODEs/PDEs, sets of two second-order PDEs representing fourth-order PDEs, and second-order ODEs/PDEs. • Discretisation of transient problems, where time derivatives are approximated using the Crank-Nicolson/Adams Bashforth scheme. The proposed CLIRBF stencils are verified in a wide range of problems: (i)ODEs/PDEs with analytic solutions; (ii) heat transfer problems; (iii) Newtonian fluid flows (Burger’s equations, lid-driven cavity flows, natural convection in rectangular and nonrectangular domains); and viscoelastic fluid flows (Poiseuille flows and corrugated tube flows of Oldroyd-B fluid). Numerical verifications show that (i) compact local forms are generally much more accurate than local forms and much more efficient than global forms; and (ii) the CLIRBF-based point/subregion collocation methods yield highly accurate results using relatively coarse grids.