Finite difference method

Abstract: A meshless numerical model is developed for analyzing transient heat conduction in non-homogeneous functionally graded materials (FGM), which has a continuously functionally graded thermal conductivity parameter First, the analog equation method is used to transform the original non-homogeneous problem into an equivalent homogeneous one at any given time so that a simpler fundamental solution can be employed to take the place of the one related to the original problem Next, the approximate particular and homogeneous solutions are constructed using radial basis functions and virtual boundary collocation method, respectively Finally, by enforcing satisfaction of the governing equation and boundary conditions at collocation points of the original problem, in which the time domain is discretized using the finite difference method, a linear algebraic system is obtained from which the unknown fictitious sources and interpolation coefficients can be determined Further, the temperature at any point can be easily computed using the results of fictitious sources and interpolation coefficients The accuracy of the proposed method is assessed through two numerical examples

Abstract: We suggest a pseudospectral method for solving the three-dimensional time-dependent Gross–Pitaevskii (GP) equation, and use it to study the resonance dynamics of a trapped Bose–Einstein condensate induced by a periodic variation in the atomic scattering length. When the frequency of oscillation of the scattering length is an even multiple of one of the trapping frequencies along the x, y or z direction, the corresponding size of the condensate executes resonant oscillation. Using the concept of the differentiation matrix, the partial-differential GP equation is reduced to a set of coupled ordinary differential equations, which is solved by a fourth-order adaptive step-size control Runge–Kutta method. The pseudospectral method is contrasted with the finite-difference method for the same problem, where the time evolution is performed by the Crank–Nicholson algorithm. The latter method is illustrated to be more suitable for a three-dimensional standing-wave optical-lattice trapping potential.

Abstract: This paper analyzes the effects of complex geometries on three-dimensional (3D) slope stability using an elastoplastic finite difference method (FDM) with a strength reduction technique. A series o...