Two dimensional solution of the advection-diffusion equation using two collocation methods with local upwinding RBF
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The two-dimensional advection-diffusion equation is solved using two local collocation methods with Multiquadric (MQ)Radial Basis Functions (RBFs). Although both methods use upwinding, the first one, similar to the method of Kansa, approximates the dependent variable with a linear combination of MQs. The nodes are grouped into two types of stencil: cross-shaped stencil to approximate the Laplacian of the variable and circular sector shape stencil to approximate the gradient components. The circular sector opens in opposite to the flow direction and therefore the maximum number of nodes and the shape parameter value are selected conveniently. The second method is based on the Hermitian interpolation where the approximation function is a linear combination of MQs and the resulting functions of applying partial differential equation (PDE) and boundary operators to MQs, all of them centred at different points. The performance of these methods is analysed by solving several test problems whose analytical solutions are known. Solutions are obtained for different Peclet numbers, Pe, and several values of the shape parameter. For high Peclet numbers the accuracy of the second method is affected by the ill-conditioning of the interpolation matrix while the first interpolation method requires the introduction of additional nodes in the cross stencil. For low Pe both methods yield accurate results. Moreover, the first method is employed to solve the twodimensional Navier-Stokes equations in velocity-vorticity formulation for the lid-driven cavity problem moderate Pe.
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