Introduction to Finite Element, Boundary Element, and Meshless Methods: With Applications to Heat Transfer and Fluid Flow
3.5 The Finite-Differencing Enhanced LCMM
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Even after optimization of the shape parameter, c, and careful reduction of oscillative behavior through smoothing and conditioning, the LCMM still has some difficulties when dealing with steep gradients and highly convective fields. This is in part due to the fact that there is little control over the way the information is passed to the data center from the scattered points in the topology. A practical and general way around this problem is to take full advantage of the highly accurate field variable interpolation capabilities of the RBF but do not use them directly to determine the derivative fields. Instead, the field variable can be RBF-interpolated to locations where it can be used to pass information to the data center in a controlled fashion, like, for instance, to a set of locations where finite-differencing approximation of derivatives can be performed. This approach is general in the sense that it can be implemented in the same localized topologies defined for the LCMM and formulated, as it will be seen shortly, to yield derivative interpolation vectors in the same way the LCMM formulation was rendered. In addition, it will provide the necessary degree of control over the topology to pass the desired information to the data center as it will be shown in the following section when upwinding schemes are presented. This is the approach, in combination with smoothing and refinement schemes, that will be followed from here on to implement the LCMM method in all the applications presented after the framework is fully formulated.