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Upwind strategies for local RBF scheme to solve convection dominated problems. (English) Zbl 1403.65175
Summary: The most common strategy existing in the literature for solving convection dominated Convection-Diffusion Equations (CDE) is using central approximation to the diffusive terms and upwind approximation to the convective terms. In the present work, we propose a multiquadric local RBF based grid-free upwind \((\mathrm{LRBF}_{-}\mathrm U)\) scheme for solving convection dominated CDE. In this method, the entire CDE operator is discretized over the nodes in the upwind local support domain for strongly convection dominant problems. The variable (optimal) shape parameter for \(\mathrm{LRBF}_{-}\mathrm U\) scheme has been obtained by using a local optimization algorithm developed by the authors. It has been observed that for highly convection dominated problems, the \(\mathrm{LRBF}_{-}\mathrm U\) scheme produces stable and accurate results. The proposed scheme is also been compared with the conventional Central-Upwind combined scheme, to demonstrate its superiority in generating high accurate solutions than the latter.

MSC:
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
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[1] Shu, C.; Ding, H.; Yeo, K. S., Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations, Comput Methods Appl Mech Eng, 192, 3, 941-954, (2003) · Zbl 1025.76036
[2] Wright, G. B.; Fornberg, B., Scattered node compact finite difference-type formulas generated from radial basis functions, J Comput Phys, 212, 99-123, (2006) · Zbl 1089.65020
[3] Chandini, G.; Sanyasiraju, Y. V.S. S., Local RBF-FD solutions for steady convection-diffusion problems, Int J Numer Meth Eng, 72, 352-378, (2007) · Zbl 1194.76174
[4] Sanyasiraju, Y. V.S. S.; Chandini, G., Local radial basis function based gridfree scheme for unsteady incompressible viscous flows, J Comput Phys, 227, 8922-8948, (2008) · Zbl 1146.76045
[5] Shen, Q., Local RBF-based differential quadrature collocation method for boundary layer problems, Eng Anal Bound Elem, 34, 213-228, (2010) · Zbl 1244.65118
[6] Stevens, D.; Power, H.; Lees, M.; Morton, H., The use of PDE centers in the local RBF Hermitian method for 3D convection-diffusion problems, J Comput Phys, 228, 4606-4624, (2009) · Zbl 1167.65447
[7] Franke, R., Scattered data interpolationtest of some methods, Math Comput, 38, 181-200, (1982) · Zbl 0476.65005
[8] Kansa, E. J., Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics-iisolutions to parabolic, hyperbolic and elliptic partial differential equations, Comput Math Appl, 19, 147-161, (1990) · Zbl 0850.76048
[9] Hardy, R. L., Multiquadric equations of topography and another irregular surfaces, J Geophys Res, 76, 1905-1915, (1971)
[10] Rippa, S., An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Adv Comput Math, 11, 193-210, (1999) · Zbl 0943.65017
[11] Kansa, E. J.; Carlson, R. E., Improved accuracy of multiquadric interpolation using variable shape parameters, Comput Math Appl, 24, 12, 99-120, (1992) · Zbl 0765.65008
[12] Fornberg, B.; Zuev, Julia, The Runge phenomenon and spatially variable shape parameters in RBF interpolation, Comput Math Appl, 54, 379-398, (2007) · Zbl 1128.41001
[13] Fasshauer, G. E.; Zhang, J. G., On choosing “optimal” shape parameters for RBF approximation, Numer Algorithms, 45, 345-368, (2007) · Zbl 1127.65009
[14] Roque, R. M.C.; Ferreira, A. J.M., Numerical experiments on optimal shape parameters for radial basis functions, Numer Methods Partial Differ Equ, 26, 3, 675-689, (2010) · Zbl 1190.65178
[15] Sanyasiraju, Y. V.S. S.; Chandini, G., A note on two upwind strategies for RBF-based grid-free schemes to solve steady convection-diffusion equations, Int J Numer Meth Fluids, 61, 1053-1062, (2009) · Zbl 1252.65195
[16] Sanyasiraju, Y. V.S. S.; Satyanarayana, Ch., On optimization of the RBF shape parameter in a grid-free local scheme for convection dominated problems over non-uniform centers, Appl Math Modell, 37, 7245-7272, (2013) · Zbl 1427.65300
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