A Generalised RBF Finite Difference Approach to Solve Nonlinear Heat Conduction Problems on Unstructured Datasets

Radial Basis Functions have traditionally been used to provide a continuous interpolation of scattered data sets. However, this interpolation also allows for the reconstruction of partial derivatives throughout the solution field, which can then be used to drive the solution of a partial differential equation. Since the interpolation takes place on a scattered dataset with no local connectivity, the solution is essentially meshless. RBF-based methods have been successfully used to solve a wide variety of PDEs in this fashion. Such full-domain RBF methods are highly flexible and can exhibit spectral convergence rates Madych & Nelson (1990). However, in their traditional implementation the fully-populated matrix systems which are produced lead to computational complexities of at least order-N2 with datasets of size N. In addition, they suffer fromincreasingly poor numerical conditioning as the size of the dataset grows, and also with increasingly flat interpolating functions. This is a consequence of ill-conditioning in the determination of RBF weighting coefficients (as demonstrated in Driscoll & Fornberg (2002)), and is described by Robert Schaback Schaback (1995) as the uncertainty relation; better conditioning is associated with worse accuracy, and worse conditioning is associated with improved accuracy. Many techniques have been developed to reduce the effect of the uncertainty relation in the traditional RBF formulation, such as RBF-specific preconditioners Baxter (2002); Beatson et al. (1999); Brown (2005); Ling & Kansa (2005), or adaptive selection of data centres Ling et al. (2006); Ling & Schaback (2004). However, at present the only reliable methods of controlling numerical ill-conditioning and computational cost as problem size increases are domain decomposition Hernandez Rosales & Power (2007); Wong et al. (1999); Zhang (2007); Zhou et al. (2003), or the use of locally supported basis functions Fasshauer (1999); Schaback (1997); Wendland (1995); Wu (1995). In this work the domain decomposition principle is applied, forming a large number of heavily overlapping systems that cover the solution domain. A small RBF collocation system is formed around each global data centre, with each collocation system used to approximate the governing PDE at its centrepoint, in terms of the solution value at surrounding collocation points. This leads to a sparse global linear system which may be solved using a variety of standard solvers. In this way, the proposed formulation emulates a finite difference method, with the RBF collocation systems replacing the polynomial interpolation functions used in traditional finite difference methods. However, unlike such polynomial functions RBF collocation is well suited to scattered data, and the method may be applied to both structured and unstructured datasets without modification. The method is applied here to solve the nonlinear heat conduction equation. In order to reduce the nonlinearity in the governing equation the Kirchhoff integral transformation is applied, and the transformed equation is solved using a Picard iterative process. The application of the Kirchhoff transform necessitates that the thermal property functions be transformed to Kirchhoff space also. If the thermal properties are a known and integrable function of temperature then the transformation may be performed analytically. Otherwise, an integration-interpolation procedure can be performed using 1D radial basis functions, as described in Stevens & Power (2010). In recent years a number of local RBF collocation techniques have been proposed, and applied a wide variety of problems (for example; Divo & Kassab (2007); Lee et al. (2003); Sarler & Vertnik (2006); Wright & Fornberg (2006)). A more comprehensive review of such methods is given in Stevens et al. (2009). Unlike most local RBF collocation methods that are used in the literature, the technique described here utilises the Hermitian RBF collocation formulation (see section 2 for more details), and allows both the PDE-boundary and PDE-governing operators to be included within in the local collocation systems. This inclusion of the governing PDE within the basis functions is shown in Stevens et al. (2009) to significantly improve the accuracy and stability of solutions obtained for linear transport problems. Additionally, the incorporation of information about the convective velocity field into the basis functionswas shown to have a stabilising effect, similar to traditional upwinding methods but without the requirement to alter the stencil configuration based on the local convective field. The standard approach to the solution of linear and nonlinear heat conduction problems is the use of finite difference and finite volume methods with simple polynomial interpolants Bejan (1993); Holman (2002); Kreith & Bohn (2000). Due to the dominance of diffusion in most cases, central differencing techniques are commonly used to compute the heat fluxes. However, limiter methods (such as the unconditionally stable TVD schemes) may be used for nonlinear heat conduction problems where the effective convection term, which results from the non-zero variation of thermal conductivity with temperature, can be expected to approach the magnitude of the diffusive term (see, for example, Shen & Han (2002)). Full-domain RBF methods have also been examined for use with nonlinear heat conduction problems (see Chantasiriwan (2007)), however such methods are restricted to small dataset sizes, due to the computational cost and numerical conditioning experienced by full-domain RBF techniques on large datasets. The present work demonstrates how local RBF collocation may be used as an alternative to traditional finite difference and finite volume methods, for nonlinear heat conduction problems. The described method retains freedom from a volumetric mesh, while allowing solution over unstructured datasets. A central stencil configuration is used in each case, and the solution is stabilised via the inclusion of the governing and boundary PDEs within the local collocation systems (“implicit upwinding”), rather than by adjusting the stencil configuration based on the local solution field (“traditional upwinding”). The method is validated using a transient numerical example with a known analytical solution (see section 4), and the ability of the formulation to handle strongly nonlinear problems is demonstrated in the solution of a food freezing problem (see section 5).