Compact Integrated Radial Basis Function Research Articles

A control volume scheme using compact integrated radial basis function stencils for solving the Richards equation

A new control volume approach is developed based on compact integrated radial basis function (CIRBF) stencils for solution of the highly nonlinear Richards equation describing transient water flow in variably saturated soils. Unlike the conventional control volume method, which is regarded as second-order accurate, the proposed approach has high-order accuracy owing to the use of a compact integrated radial basis function approximation that enables improved flux predictions. The method is used to solve the Richards equation for transient flow in 1D homogeneous and heterogeneous soil profiles. Numerical results for different boundary conditions, initial conditions and soil types are shown to be in good agreement with Warrick’s semi-analytical solution and simulations using the HYDRUS-1D software package. Results obtained with the proposed method were far less dependent upon the grid spacing than the HYDRUS-1D finite element solutions.

A compact integrated RBF method for time fractional convection–diffusion–reaction equations

In this paper, a local compact integrated radial basis function (CIRBF) method is proposed to solve the time fractional convection–diffusion–reaction equations. The proposed CIRBF scheme derives high accuracy and fast convergence rate with grid refinement. The spatial derivatives are discretized by combining with the integrated RBF approximation and compact approximation in a stencil. The nodal function values and the nodal second derivatives are used to establish the relation between the physical space and the RBF weight space. The time discretization is replaced by the second order shifted Grünwald scheme. Moreover, the unconditional stability and convergence of the proposed method are shown. Finally, some numerical examples on 2D and 3D domains are carried out to verify the accuracy and efficiency of the proposed algorithm for the fractional differential equations.

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, N.; Mai-Duy, N.; Tran, C.-D.; Tran-Cong, T. (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 CIRBF and HOC schemes.

A direct forcing immersed boundary method employed with compact integrated RBF approximations for heat transfer and fluid flow problems

In this paper, we present a numerical scheme, based on the direct forcing immersed boundary (DFIB) approach and compact integrated radial basis function (CIRBF) approximations, for solving the Navier-Stokes equations in two dimensions. The problem domain of complicated shape is embedded in a Cartesian grid containing Eulerian nodes. Non-slip conditions on the inner boundaries, represented by Lagrangian nodes, are imposed by means of the DFIB method, in which a smoothed version of the discrete delta functions is utilised to transfer the physical quantities between two types of nodes. The velocities and pressure variables are approximated locally on Eulerian nodes using 3-node CIRBF stencils, where first- and second-order derivative values of the field variables are also included in the RBF approximations. The present DFIB-CIRBF scheme is verified through the solution of several test problems including Taylor-Green vortices, rotational flow, lid-driven cavity flow with multiple solid bodies, flow between rotating circular and fixed square cylinders, and natural convection in an eccentric annulus between two circular cylinders. Numerical results obtained using relatively coarse grids are in good agreement with available data in the literature.

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

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.

High-order Alternating Direction Implicit Method Based on Compact Integrated-RBF Approximations for Unsteady/Steady Convection-Diffusion Equations

In this paper, the alternating direction implicit (ADI) method reported in [You (2006)] for the convection-diffusion equation is implemented in the context of compact integrated radial basis function (CIRBF) approximations. The CIRBF approximations are constructed over 3-point stencils, where extra information is incorporated via two forms: only nodal second-order derivative values (Scheme 1), and both nodal first- and second-order derivative values (Scheme 2). The resultant algebraic systems are sparse, especially for Scheme 2 (tridiagonal matrices). Several steady and non-steady problems are considered to verify the present schemes and to compare their accuracy with some other ADI schemes. Numerical results show that highly accurate results are obtained with the proposed methods.