Class Of Compact Schemes Research Articles

Parallel FFT algorithms for high-order approximations on three-dimensional compact stencils

The recent development of multicore technologies on modern desktop computers makes parallelization of the proposed numerical approaches a priority in algorithmic research. The main performance improvement of personal computers in the upcoming years will be made based on the increasing number of cores on modern CPUs. This shifts the focus of algorithmic research from the development of sequential numerical methods to parallel methodology. This paper presents an efficient parallel direct algorithm with near-optimal complexity for the compact fourth and sixth-order approximation of the three-dimensional Helmholtz equations (Turkel et al., 2013) with the problem coefficient depending on only one of the coordinate directions. The developed method is based on a combination of the separation of variables technique and a Fast Fourier Transform (FFT) type method. Similar direct solvers for the lower-order approximations of the two and three-dimensional Helmholtz equation were considered in several previous publications by the authors and other researchers (see, e.g. Gryazin et al. (2000); Gryazin (2014); Elman and O’Leary (1998); Elman and O’Leary (1999); Toivanen and Wolfmayr (2020)). The authors also consider a generalization of the presented algorithm to the solution of a wide class of linear systems obtained from approximation on the compact 27-point three-dimensional stencils on the rectangular grids with similar requirements on the stencil coefficients. The general restrictions on the coefficients in the considered class of compact schemes are developed and presented. This class includes the second, fourth and sixth-order compact approximation schemes for the three-dimensional Helmholtz equation considered in this paper and our previous publications (Gryazin et al., 2000; Gryazin, 2014; Gryazin, 2014). As an example of the diversity of applications of the developed general method, the direct parallel implementation of a compact fourth-order approximation scheme for a convection–diffusion equation is considered. Another goal of this paper is to investigate the scalability of the proposed technique in the case of a large linear system using different parallel programming extensions. The results of the implementation of this method in OpenMP, MPI and hybrid programming environments on the multicore computers and multiple node clusters are presented and discussed. The results demonstrate the high efficiency of the proposed direct solvers for many important applications on the structured grid with the corresponding 27-diagonal matrices of sizes up to 1011 by 1011.

A Maximum-Principle-Satisfying High-Order Finite Volume Compact WENO Scheme for Scalar Conservation Laws with Applications in Incompressible Flows

In this paper, a maximum-principle-satisfying finite volume compact scheme is proposed for solving scalar hyperbolic conservation laws. The scheme combines weighted essentially non-oscillatory schemes (WENO) with a class of compact schemes under a finite volume framework, in which the nonlinear WENO weights are coupled with lower order compact stencils. The maximum-principle-satisfying polynomial rescaling limiter in Zhang and Shu (J Comput Phys 229:3091---3120, 2010, Proc R Soc A Math Phys Eng Sci 467:2752---2776, 2011) is adopted to construct the present schemes at each stage of an explicit Runge---Kutta method, without destroying high order accuracy and conservativity. Numerical examples for one and two dimensional problems including incompressible flows are presented to assess the good performance, maximum principle preserving, essentially non-oscillatory and high resolution of the proposed method.

A new compact scheme for parallel computing using domain decomposition

Direct numerical simulation (DNS) of complex flows require solving the problem on parallel machines using high accuracy schemes. Compact schemes provide very high spectral resolution, while satisfying the physical dispersion relation numerically. However, as shown here, compact schemes also display bias in the direction of convection – often producing numerical instability near the inflow and severely damping the solution, always near the outflow. This does not allow its use for parallel computing using domain decomposition and solving the problem in parallel in different sub-domains. To avoid this, in all reported parallel computations with compact schemes the full domain is treated integrally, while using parallel Thomas algorithm (PTA) or parallel diagonal dominant (PDD) algorithm in different processors with resultant latencies and inefficiencies. For domain decomposition methods using compact scheme in each sub-domain independently, a new class of compact schemes is proposed and specific strategies are developed to remove remaining problems of parallel computing. This is calibrated here for parallel computing by solving one-dimensional wave equation by domain decomposition method. We also provide the error norm with respect to the wavelength of the propagated wave-packet. Next, the advantage of the new compact scheme, on a parallel framework, has been shown by solving three-dimensional unsteady Navier–Stokes equations for flow past a cone-cylinder configuration at a Mach number of 4. Additionally, a test case is conducted on the advection of a vortex for a subsonic case to provide an estimate for the error and parallel efficiency of the method using the proposed compact scheme in multiple processors.