From Time-Collocated to Leapfrog Fundamental Schemes for ADI and CDI FDTD Methods

The leapfrog schemes have been developed for unconditionally stable alternating-direction implicit (ADI) finite-difference time-domain (FDTD) method, and recently the complying-divergence implicit (CDI) FDTD method. In this paper, the formulations from time-collocated to leapfrog fundamental schemes are presented for ADI and CDI FDTD methods. For the ADI FDTD method, the time-collocated fundamental schemes are implemented using implicit E-E and E-H update procedures, which comprise simple and concise right-hand sides (RHS) in their update equations. From the fundamental implicit E-H scheme, the leapfrog ADI FDTD method is formulated in conventional form, whose RHS are simplified into the leapfrog fundamental scheme with reduced operations and improved efficiency. For the CDI FDTD method, the time-collocated fundamental scheme is presented based on locally one-dimensional (LOD) FDTD method with complying divergence. The formulations from time-collocated to leapfrog schemes are provided, which result in the leapfrog fundamental scheme for CDI FDTD method. Based on their fundamental forms, further insights are given into the relations of leapfrog fundamental schemes for ADI and CDI FDTD methods. The time-collocated fundamental schemes require considerably fewer operations than all conventional ADI, LOD and leapfrog ADI FDTD methods, while the leapfrog fundamental schemes for ADI and CDI FDTD methods constitute the most efficient implicit FDTD schemes to date.

Issues related to the numerical simulation of herringbone grooved journal bearing including cavitation condition

This study aims to address the complexities involved in determining the numerical solution of herringbone grooved journal bearing considering cavitation. A modification is made to the Reynolds’ equation to include the effect of cavitation as given by Elrod's cavitation algorithm. Grid transformation is performed to consider the effect of the inclined grooves. The partial differential equation is discretised using finite-difference method. Then, the solution of the resulting set of equations is determined by the alternating-direction implicit method and the pressure, load capacity and attitude angle are obtained. Time step (Δ t) and Bulk modulus have a significant impact on the convergence of the numerical solution incorporating Elrod's cavitation algorithm. Use of alternating-direction implicit method over point by point method like Gauss–Seidel is essential to obtain convergence. Load capacity of the herringbone grooved journal bearing rises with the rise in eccentricity ratio. As compared to the Reynolds boundary conditions, Elrod's model results into lower attitude angle for herringbone grooved journal bearing. Cavitation distribution for herringbone grooved journal bearing is much lower than that of plain journal bearing. The effect of variation of groove angle on the herringbone grooved journal bearing's load capacity, side leakage and friction parameter is also determined. A detailed discussion on the various complexities such as treatment at groove ridge boundaries; numerical oscillations; choice of time step and bulk modulus; and influence of compressibility in the Couette term in full film region in the numerical analysis of herringbone grooved journal bearing specifically considering cavitation is given in this work. Multiple methods to deal with the aforementioned complexities are examined and appropriate solutions are obtained.

Numerical Method for 2D Quasi-linear Hyperbolic Equation on an Irrational Domain: Application to Telegraphic Equation

We develop a novel three-level compact method (implicit) of second order in time and space directions using unequal grid for the numerical solution of 2D quasi-linear hyperbolic partial differential equations on an irrational domain. The stability analysis of the model problem for unequal mesh is discussed and it is revealed that the developed scheme is unconditionally stable for the Telegraphic equation. For linear difference equations on an irrational domain, the alternating direction implicit method is discussed. The projected technique is scrutinized on several physical problems on an irrational domain to exhibitthe accuracy and effectiveness of the suggested method. Dhaka Univ. J. Sci. 69(2): 116-123, 2021 (July)

Alternating Direction Implicit (ADI) Methods for Solving Two-Dimensional Parabolic Interface Problems with Variable Coefficients

The matched interface and boundary method (MIB) and ghost fluid method (GFM) are two well-known methods for solving elliptic interface problems. Moreover, they can be coupled with efficient time advancing methods, such as the alternating direction implicit (ADI) methods, for solving time-dependent partial differential equations (PDEs) with interfaces. However, to our best knowledge, all existing interface ADI methods for solving parabolic interface problems concern only constant coefficient PDEs, and no efficient and accurate ADI method has been developed for variable coefficient PDEs. In this work, we propose to incorporate the MIB and GFM in the framework of the ADI methods for generalized methods to solve two-dimensional parabolic interface problems with variable coefficients. Various numerical tests are conducted to investigate the accuracy, efficiency, and stability of the proposed methods. Both the semi-implicit MIB-ADI and fully-implicit GFM-ADI methods can recover the accuracy reduction near interfaces while maintaining the ADI efficiency. In summary, the GFM-ADI is found to be more stable as a fully-implicit time integration method, while the MIB-ADI is found to be more accurate with higher spatial and temporal convergence rates.

Inverse Problem for a Two-Dimensional Anomalous Diffusion Equation with a Fractional Derivative of the Riemann–Liouville Type

The article presents a method for solving the inverse problem of a two-dimensional anomalous diffusion equation with a Riemann–Liouville fractional-order derivative. In the first part of the present study, the authors present a numerical solution of the direct problem. For this purpose, a differential scheme was developed based on the alternating direction implicit method. The presented method was accompanied by examples illustrating its accuracy. The second part of the study concerned the inverse problem of recreating the model parameters, including the orders of the fractional derivative, in the anomalous diffusion equation. Equations of this type can be used to describe, inter alia, the heat conductivity in porous materials. The ant colony optimization algorithm was used to solve this problem. The authors investigated the impact of the distribution of measurement points, the use of different mesh sizes, and the input data errors on the obtained results.