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Explicit and implicit meshless methods for linear advection-diffusion-type partial differential equations. (English) Zbl 0968.65053
The authors consider the approximate solution of the three-dimensional convection-diffusion equation by applying the theta-scheme (in fact, the explicit scheme and the Crank-Nicolson scheme are used) in time and an expansion into radial basis functions in space, i.e. functions which depend only on the distance from given points. The coefficients in this expansion are found by collocation. The authors show how to apply upwinding, and, as for the specific choice of the radial basis functions, they vote for thin plate splines. Numerical results are provided mainly for the one-dimensional convection-diffusion equation and show that there is no much difference in accuracy for these versions of the radial basis function approach. However, when doubling the number of points, the accuracy improves considerably.
For questions like non-singularity of the decisive matrices, or like stability and convergence the authors refer to C. A. Micchelli [Constructive Approximation 2, 11-22 (1986; Zbl 0625.41005)], G. E. Fasshauer [Le Mehaute, A. (ed.), Surface fixing and multiresolution methods, Vol. 2, 131-138 (1997; Zbl 0938.65140)], C. Franke and R. Schaback [Adv. Comput. Math. 8, No. 4, 381-399 (1998; Zbl 0909.65088); Appl. Math. Comput. 93, No. 1, 73-82 (1998; Zbl 0943.65133)].

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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