An implicit upwinding volume element method based on meshless radial basis function techniques for modelling transport phenomena

Abstract

SUMMARYThis work presents a control volume (CV) method for transient transport problems where the cell surface fluxes are reconstructed using local interpolation functions that besides interpolating the nodal values of the field variable, also satisfies the governing equation at some auxiliary points in the interpolation stencils. The interpolation function relies on a Hermitian radial basis function (HRBF) meshless collocation approach to find the solution of auxiliary local boundary/initial value problems, that are solved using the same time-integration scheme adopted to update the global CV solution. By the use of interpolation functions that approximate the governing equation a form of analytical upwinding scheme is achieved, without the need of using predefined interpolation stencils according to the magnitude and direction of the local advective velocity. In this way, the interpolation formula retains the desired information about the advective velocity field allowing the use of centrally defined stencils even in the cases of advective dominant problems. This new CV approach is referred to as the CV-HRBF method. Two time-stepping formulations are considered: a full implicit approach and the weighted Crank-Nicholson one. The implicit upwinding scheme, intrinsic to the proposed CV-HRBF, is tested by solving a travelling front problem at the Péclet number equal to 500, 1000 and infinity; with the latter corresponding to a shock front. Finally, the accuracy of the numerical method is validated against one-and three-dimensional reactive transport problems characterized by smooth solutions.

Control volume-radial basis function method for two-dimensional non-linear heat conduction and convection problems

“…In this section, the formulation of the CV-RBF method developed by Orsini et al [20] applied to the heat equation is presented.…”

“…In the Finite Volume Method (FVM), the RBF collocation was first implemented as an interpolation scheme by Moroney and Turner [18] to solve the two-dimensional nonlinear anisotropic diffusion equation and later, by Moroney and Turner [19] for the three-dimensional case in the same situation. Recently Orsini et al [20] solved the diffusion convection equation by the FVM in conjunction with the RBF Hermite interpolation scheme. They had the convective and diffusive fluxes in terms of neighboring node values according to the Symmetric method and called this strategy the Control Volume-Radial Basis Function (CV-RBF).…”

Local regular dual reciprocity method for 2D convection-diffusion equation

“…In [7] a modified Control Volume (CV) method which uses a RBF interpolation to improve the prediction of the flux accuracy at the faces of the CV is presented. This method is also more flexible than the classical CV formulations because the boundary conditions are explicitly imposed in the interpolation formula, without the need for artificial schemes (e.g.…”

A control volume technique based on integrated RBFNs for the convection-diffusion equation

“…In recent years, considerable effort has been put into the development of new techniques that enhance the flux prediction (e.g. [3][4][5][6][7][8]). …”

“…There has been an increased level of interest in this type of application. For example, in the context of CVMs, Moroney and Turner [6,7] and Orsini, Power and Morvan [8] replaced linear/quadratic shape functions with RBFs to improve the accuracy of a flux approximation. In addition, to make CV-RBF schemes more accurate, surface integrals were evaluated using a high-order numerical integration scheme (i.e.…”