Finite difference method

Abstract: Current study contains adaption of Haar wavelet discretization method (HWDM) for FG beams and its accuracy estimates. The convergence analysis is performed for differential equations covering a wide class of composite and nanostructures. Corresponding error bound has been derived. It has been shown that the order of convergence of the HWDM can be increased from two to four by applying Richardson extrapolation method. The theoretical estimates are validated by numerical samples considering FGM beam as a model problem. The results obtained by applying HWDM are compared with the results of finite difference method (FDM).

Abstract: A technique derived from two related methods suggested earlier by some of the authors for optimization of finite-difference grids and absorbing boundary conditions is applied to discretization of perfectly matched layer (PML) absorbing boundary conditions for wave equations in Cartesian coordinates. We formulate simple sufficient conditions for optimality and implement them. It is found that the minimal error can be achieved using pure imaginary coordinate stretching. As such, the PML discretization is algebraically equivalent to the rational approximation of the square root on [0,1] conventionally used for approximate absorbing boundary conditions. We present optimal solutions for two cost functions, with exponential (and exponential of the square root) rates of convergence with respect to the number of the discrete PML layers using a second order finite-difference scheme with optimal grids. Results of numerical calculations are presented.

Abstract: To retain intact state information of the district heating network (DHN) as possible in the optimization of heat and electricity integrated energy system (HE-IES), this article combines the transient heat flow and steady-state electric power flow to formulate the dynamic optimal energy flow (OEF) model of HE-IES. For efficient and standardized solution, the finite difference method is applied to convert the partial differential equation constraint (introduced by the temperature dynamics in DHN) into a set of linear equality constraints. The structure of applicable difference schemes for system optimization is analyzed, based on which, a scheme with balanced performances in stability, convergence, simulation accuracy, and computation burden is developed. Moreover, with the proposed multi-objective optimization based method to select proper spatial and temporal calculation step sizes, a compromise between model precision and solution complexity can be reached. Case studies validate the feasibility and benefits of our proposed dynamic OEF computing method, which can further provide support for making optimal planning and operating strategies.