A finite-volume method based on compact local integrated radial basis function approximations for second-order differential problems

In this paper, compact local integrated radial basis function (IRBF) stencils reported in [Mai-Duy and Tran-Cong (2011) Journal of Computational Physics 230(12), 4772-4794] are introduced into the finite-volume / subregion - collocation formulation for the discretisation of second-order differential problems defined on rectangular and non-rectangular domains. The problem domain is simply represented by a Cartesian grid, over which overlapping compact local IRBF stencils are utilised to approximate the field variable and its derivatives. The governing differential equation is integrated over non-overlapping control volumes associated with grid nodes, and the divergence theorem is then applied to convert volume integrals into surface/line integrals. Line integrals are evaluated by means of the middle point rule (i.e. second-order integration scheme) and three-point Gaussian quadrature rule (i.e. high-order integration scheme): The accuracy of the proposed method is numerically investigated through the solution of several test problems including natural convection in an annulus. Numerical results indicate that (i) the proposed method produces accurate results using relatively coarse grids and (ii) the three-point integration scheme is generally more accurate than the middle point scheme.