An integral Chebyshev expansion method for boundary-value problems of O.D.E. type

Numerical modelling of axisymmetric electromagnetically driven flows in thin layers

“…An example of the azimuthal velocity profiles obtained by solving equations (1)(2)(3)(4)(5) is shown in Figure 1. We refer to this solution as Type 1 to distinguish it from another solution termed as Type 2 that will be discussed later.…”

“…As shown by Ku and Hatziavramidis [4,5] these values uniquely define the coefficients in the linear combinations of Chebyshev polynomials used in the spectral approximation of the solution. Therefore, the function values at any point within a computational domain can be found since the explicit expressions for Chebyshev polynomials are readily available.…”

“…Thus the spectral accuracy of the method is ensured. At the same time as discussed by Ku and Hatziavramidis [4,5] the collocation formulation enables one to reduce the integration and differentiation procedures to a simple matrixvector multiplication. Namely, introducing the vector…”

“…The governing equations are solved using the multi-dimensional version of the Chebyshev pseudospectral collocation method initially proposed by Ku and Hatziavramidis [4,5] and implemented by Suslov and Paolucci [7,8]. In this method the solution is represented by a linear combination of Chebyshev polynomials of degree ranging from 0 to the pre-set values N r and N z that define the total number of Chebyshev modes used for approximating the solutions in the radial and depth directions, respectively.…”

“…, and the standard N r × N r matricesĜ N r ×N r 1 and W N r ×N r are defined by Ku and Hatziavramidis [4,5]. Differentiation and integration in z is performed using the similar N z ×N z matrices.…”