Roots of Characteristic Polynomial Sequences in Iterative Block Cyclic Reductions

The block cyclic reduction method is a finite-step direct method used for solving linear systems with block tridiagonal coefficient matrices. It iteratively uses transformations to reduce the number of non-zero blocks in coefficient matrices. With repeated block cyclic reductions, non-zero off-diagonal blocks in coefficient matrices incrementally leave the diagonal blocks and eventually vanish after a finite number of block cyclic reductions. In this paper, we focus on the roots of characteristic polynomials of coefficient matrices that are repeatedly transformed by block cyclic reductions. We regard each block cyclic reduction as a composition of two types of matrix transformations, and then attempt to examine changes in the existence range of roots. This is a block extension of the idea presented in our previous papers on simple cyclic reductions. The property that the roots are not very scattered is a key to accurately solve linear systems in floating-point arithmetic. We clarify that block cyclic reductions do not disperse roots, but rather narrow their distribution, if the original coefficient matrix is symmetric positive or negative definite.

A COMPARISON OF SERIAL AND PARALLEL SOLUTIONS OF TWO DIMENSIONAL HEAT CONDUCTION EQUATION

We study a comparison of serial and parallel solution of 2D-parabolic heat conduction equation using a Crank-Nicolson method with an Alternating Direction Implicit (ADI) scheme. The two-dimensional Heat equation is applied on a thin rectangular aluminum sheet. The forward difference formula is used for time and an averaged second order central difference formula for the derivatives in space to develop the Crank-Nicolson method. FORTRAN serial codes and parallel algorithms using OpenMP are used. Thomas tridigonal algorithm and parallel cyclic reduction methods are employed to solve the tridigonal matrix generated while solving heat equation. This paper emphasize on the run time of both algorithms and their difference. The results are compared and evaluated by creating GNU-plots (Command-line driven graphing utility).

PARALLEL ALGORITHM ON CUDA FOR SOLVING MULTIPHASE, MULTICOMPONENT FLUID FILTRATION PROBLEMS IN POROUS MEDIA

The paper discusses the implementation of alternating direction implicit parallel algorithm with CUDA technology to solve the problems of multiphase filtration of a multicomponent fluid in porous media. To solve the tridiagonal equations systems of alternating direction implicit method, cyclic and parallel cyclic reduction methods were used. The results of the study showed that the implementation of cyclic and parallel cyclic reduction algorithms on modern GPUs is much more efficient than on CPUs.

Response of redox zonation to recharge in a riverbank filtration system: a case study of the Second Songhua river, NE China

Bank filtration induced by groundwater pumping results in redox zonation along the groundwater flow path. Besides the river water, recharge from other sources can change local redox conditions; therefore, redox zonation is likely to be complex within the riverbank filtration (RBF) system. In this study, hydrodynamics, hydrogeochemistry, and environmental stable isotopes were combined together to identify the redox conditions at an RBF site. The recharge characteristics and redox processes were revealed by monitoring the variations of water level, δ2H and δ18O, and redox indexes along shallow and deep flow paths. The results show that local groundwater is recharged from river, regional groundwater, and precipitation. The responses of redox zonation are sensitive to different sources. In the river water recharge zone near shore, O2, , Mn(IV), Fe(III), and are reduced in sequence, the ranges of each reaction are wider in deep groundwater because of the high-velocity deep flow. In the precipitation vertical recharge zone, precipitation intermittently drives O2, , and organic carbon to migrate through vadose zone, thereby decreasing the groundwater reducibility. In the regional groundwater lateral recharge zone in the depression cone, the reductive regional groundwater is continuously recharging local groundwater, leading to the cyclic reduction of Mn(IV) and Fe(III).

Parallel Thomas approach development for solving tridiagonal systems in GPU programming − steady and unsteady flow simulation

The solution of tridiagonal system of equations using graphic processing units (GPU) is assessed. The parallel-Thomas-algorithm (PTA) is developed and the solution of PTA is compared to two known parallel algorithms, i.e. cyclic-reduction (CR) and parallel-cyclic-reduction (PCR). Lid-driven cavity problem is considered to assess these parallel approaches. This problem is also simulated using the classic Thomas algorithm that runs on a central processing unit (CPU). Runtimes and physical parameters of the mentioned GPU and CPU algorithms are compared. The results show that the speedup of CR, PCR and PTA against the CPU runtime is 4.4x,5.2x and 38.5x, respectively. Furthermore, the effect of coalesced and uncoalesced memory access to GPU global memory is examined for PTA, and a 2x-speedup is achieved for the coalesced memory access. Additionally, the PTA performance in a time dependent problem, the unsteady flow over a square, is assessed and a 9x-speedup is obtained against the CPU.