A Three-point Coupled Compact Integrated RBF Scheme for Second-order Differential Problems

In this paper, we propose a three-point coupled compact integrated radial basis function (CCIRBF) approximation scheme for the discretisation of second-order differential problems in one and two dimensions. The CCIRBF employs integrated radial basis functions (IRBFs) to construct the approximations for its first and second derivatives over a three-point stencil in each direction. Nodal values of the first and second derivatives (i.e. extra information), incorporated into approximations by means of the constants of integration, are simultaneously employed to compute the first and second derivatives. The essence of the CCIRBF scheme is to couple the extra information of the nodal first and second derivative values via their identity equations. Owing to its coupling of the information of the nodal first and second derivatives, the CCIRBF scheme becomes more accurate, stable and efficient than the normal compact integrated radial basis function (CIRBF) schemes proposed by [Thai-Quang, Mai-Duy, Tran, and Tran-Cong (2012)]. The main features of the CCIRBF scheme include: three-point, high-order accuracy, stability, efficiency and inclusion of boundary values. Several analytic problems are considered to verify the present scheme and to compare its accuracy, stability and efficiency with those of the CIRBF, higher-order compact finite difference (HOC) and some other high-order schemes. Numerical results show that highly accurate and stable results are obtained with the proposed scheme. Additionally, the present scheme also takes less time to achieve target accuracy in comparison with the ORBF and HOC schemes.