An exponential high‐order compact ADI method for 3D unsteady convection–diffusion problems

Abstract

In this article, we develop an exponential high order compact alternating direction implicit (EHOC ADI) method for solving three dimensional (3D) unsteady convection-diffusion equations. The method, which requires only a regular seven-point 3D stencil similar to that in the standard second-order methods, is second order accurate in time and fourth-order accurate in space and unconditionally stable. The resulting EHOC ADI scheme in each alternating direction implicit (ADI) solution step corresponding to a strictly diagonally dominant matrix equation can be solved by the application of the one-dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. Numerical experiments for three test problems are carried out to demonstrate the performance of the present method and to compare it with the classical Douglas-Gunn ADI method and the Karaa's high-order compact ADI method.

Finite Difference Implicit Schemes to Coupled Two-Dimension Reaction Diffusion System

“…Since last two decades, there is a challenging hasting among vendors and software development communities to bring improvement in performance for Journal of Applied Mathematics and Physics High performance computing system, by halting the traditional development to increase the clock rate, number of cores that are being increased in the system, however, many cores architecture based different powerful devices such GPU (Graphics Processing Unit) and GPGPU (General Purpose Graphics Processing Unit) by NIVIDA, MIC (Many Integrated Core) by Intel and FPGA (Field programmable Gate Array) have been introduced recently that outperform the conventional CPU processing by thousand folds [29] [38] [39]. In order to utilize these powerful devices, new software stacks, algorithms and frameworks have been introduced, ehereas the modern computing system provides massive parallelism through inter node and intra node computation where inter node processing is performed by MPI (Message Passing Interface), most popular programming model and intra node by OpenMP directive programing model [29] [38] [39].…”

“…Keeping in view, the advantages of this emerging technology, we have introduced Crank Nicolson and ADI schemes with the help of this application by using FUJITSU Primergy RX 350 S7 HPC computer having Intel Xeon E5-2667 processor of 2.80 GHz processing power which contained 16 physical cores and 32 logical cores, main memory size of 32 GB and HDD 4 TB inside it [29] [38] [39], see from Figures 7-9. Moreover, we used 2 Tesla k-80 accelerated NVIDA devices that are capable to deliver not only graphical processing purpose but also for general purpose processing [29] [38] [39]. However, our application structure is designed for MATLAB based on hybrid of tri-hierarchy level tightly coupled programming model containing CUDA (Compute Uniform Device Architecture) for accelerated computation [29] [38] [39].…”

“…Moreover, we used 2 Tesla k-80 accelerated NVIDA devices that are capable to deliver not only graphical processing purpose but also for general purpose processing [29] [38] [39]. However, our application structure is designed for MATLAB based on hybrid of tri-hierarchy level tightly coupled programming model containing CUDA (Compute Uniform Device Architecture) for accelerated computation [29] [38] [39].…”

High-Order Finite Difference Method Applied to the Solution of the Three-Dimensional Heat Transfer Equation and to the Study of Heat Exchangers

“…The Dirichlet boundary and initial conditions are directly taken from this analytical solution. The computational domain is 0  x,y,z  2 and t = 1.25 and the regular mesh with h = x = y = z was used.We adopt h = 0.025 so as to compare the results obtained in this work the ones presented by [Ge, Tian and Zhang (2013)] in which three methodologies were implemented for the solution of convectiondiffusion equation, all based on the alternating direction implicit method. We may note from Table 5, the results of this study in comparison with the literature, have an equal or higher accuracy for the different values of .…”

“…The example presented numerical example shows that a decrease in the value of the time step, always below the stability limits, leads to a decrease in the global error. (Ge et al, 2013) developed an exponential high order compact alternating direction implicit method with fourth-order in space, second order in time and unconditionally stable and solved three numerical problems to demonstrate the high accuracy and efficiency and to show its superiority over the classical Douglas-Gunn ADI scheme and the Karaa's high order ADI scheme.…”