Compact non-symmetric and symmetric stencils based on integrated radial basis functions for differential problems

Compact non-symmetric and symmetric stencils based on integrated radial basis functions for differential
problems

This PhD project is concerned with the development of compact local stencils based on integrated radial basis functions (IRBFs) for both spatial and temporal discretisations of partial differential equations (PDEs), and their applications in heat transfer and fluid flows. The proposed approximation stencils are effective and efficient since (i) Cartesian grids are employed to represent both rectangular and non-rectangular domains; (ii) high levels of accuracy of the solution and sparseness of the resultant algebraic system are achieved together; and (iii) time derivatives are discretised with high order approximation.

For spatial discretisation, a compact non-symmetric flat-IRBF stencil is developed. Significant improvements in the matrix condition number, solution accuracy and convergence rate with grid refinement over the usual approaches are obtained. Furthermore, IRBFs are used for Hermite interpolation in the solution of PDEs, resulting in symmetric stencils defined on structured/random nodes. For temporal discretisation, a compact IRBF stencil is proposed, where the time derivative is approximated in terms of, not only nodal function values at the current and previous time levels, but also nodal derivative values at the previous time level. When dealing with moving boundary problems (e.g. particulate suspensions and fluid structure interacting problems), to avoid the grid regeneration issue, an IRBF-based domain embedding method is also developed, where a geometrically-complex domain is extended to a larger, but simpler shaped domain, and a body force is introduced into the momentum equations to represent the moving boundaries.

The proposed methods are verified in the solution of differential problems defined on simply- and multiply-connected domains. Accurate results are achieved using relatively coarse Cartesian grids and relatively large time steps. The rate of convergence with grid refinement can be up to the order of about 5. Converged solutions are obtained in the simulation of highly nonlinear fluid flows and they are in good agreement with benchmark/well-known existing solutions.