A numerical scheme based on compact integrated-RBFs and Adams–Bashforth/Crank–Nicolson algorithms for diffusion and unsteady fluid flow problems

Abstract

Abstract This paper presents a high-order approximation scheme based on compact integrated radial basis function (CIRBF) stencils and second-order Adams–Bashforth/Crank–Nicolson algorithms for solving time-dependent problems in one and two space dimensions. We employ CIRBF stencils, where the RBF approximations are locally constructed through integration and expressed in terms of nodal values of the function and its derivatives, to discretise the spatial derivatives in the governing equation. We adopt the Adams–Bashforth and Crank–Nicolson algorithms, which are second-order accurate, to discretise the temporal derivatives. The performance of the proposed scheme is investigated numerically through the solution of several test problems, including heat transfer governed by the diffusion equation, shock wave propagation and shock-like solution governed by the Burgers' equation, and torsionally oscillating lid-driven cavity flow governed by the Navier–Stokes equation in the primitive variables. Numerical experiments show that the proposed scheme is stable and high-order accurate in reference to the exact solution of analytic test problems and achieves a good agreement with published results for other test problems.