Tridiagonal Systems Research Articles

A numerical scheme to solve mixed parabolic-elliptic problem involving singular perturbation

In this article, a singularly perturbed mixed-type problem of parabolic-elliptic is considered on a rectangular domain. The solution of the problem possesses both boundary and interior layers which appears in the spatial variable. To approximate the time derivative, first the implicit Euler scheme and then the Crank–Nicolson scheme on uniform mesh in time direction is used. For approximating the spatial derivative, we use the central difference scheme in the first sub-domain and a hybrid scheme in the second sub-domain on Shishkin type meshes (standard Shishkin mesh and Bakhvalov–Shishkin mesh). We prove that both the numerical schemes converge uniformly with respect to the perturbation parameter and are of second-order accurate. Thomas algorithm is used to solve the tri-diagonal system. The numerical results and the error bounds are illustrated through few tables and graphs.

Newton-Based Methods for Finding the Positive Ground State of Gross-Pitaevskii Equations

The discretization of Gross-Pitaevskii equations (GPE) leads to a nonlinear eigenvalue problem with eigenvector nonlinearity (NEPv). We use two Newton-based methods to compute the positive ground state of GPE. The first method comes from the Newton-Noda iteration for saturable nonlinear Schrödinger equations proposed by Ching-Sung Liu, which can be transferred to GPE naturally. The second method combines the idea of the root-finding methods and the idea of Newton method, in which, each subproblem involving block tridiagonal linear systems can be solved easily. We give an explicit convergence and computational complexity analysis for it. Numerical experiments are provided to support the theoretical results.

A numerical study on MHD double diffusive nonlinear mixed convective nanofluid flow around a vertical wedge with diffusion of liquid hydrogen

The present study focuses on double diffusive nonlinear (quadratic) mixed convective flow of nanoliquid about vertical wedge with nonlinear temperature-density-concentration variations. This study is found to be innovative and comprises the impacts of quadratic mixed convection, magnetohydrodynamics, diffusion of nanoparticles and liquid hydrogen flow around a wedge. Highly coupled nonlinear partial differential equations (NPDEs) and boundary constraints have been used to model the flow problem, which are then transformed into a dimensionless set of equations utilizing non-similar transformations. Further, a set of NPDEs would be linearized with the help of Quasilinearization technique, and then, the linear partial differential equations are transformed into a block tri-diagonal system through using implicit finite difference scheme, which is solved using Verga’s algorithm. The study findings were explored through graphs for the fluid velocity, temperature, concentration, nanoparticle volume fraction distributions and its corresponding gradients. One of the important results of this study is that the higher wedge angle values upsurge the friction between the particles of the fluid and the wedge surface. Rising Schmidt number declines the concentration distribution and enhances the magnitude of Sherwood number. Nanofluid’s temperature increases with varying applied magnetic field. The present study has notable applications in the designing and manufacturing of wedge-shaped materials in space aircrafts, construction of dams, thermal systems, oil and gas industries, etc.

A hybrid triangulation method for banded linear systems

We propose a fast solution method for banded linear systems that transforms the original system into an equivalent one with an almost block triangular coefficient matrix, and then constructs a preconditioner based on this formulation. We analyze the algorithmic complexity of the new method and the eigenvalue distribution of the resulting preconditioned matrix. Numerical examples involving block tridiagonal, block Hessenberg and block pentadiagonal systems are illustrated to demonstrate the computational performance and the efficiency of the new matrix solver.

Numerical Solution of Two-Parameter Singularly Perturbed Convection-Diffusion Boundary Value Problems via Fourth Order Compact Finite Difference Method

In this study, we have developed a fourth order compact finite difference method for finding the numerical solution of two-parameter singularly perturbed convection-diffusion boundary value problems. We have used fourth order compact finite difference method on uniform mesh which provides a tridiagonal linear system of equations. The convergence analysis of the proposed method is established through a matrix analysis approach and it is proved that present method gives fourth order convergence results. Present method is implemented on two numerical examples for checking the efficiency and precision of the method. Numerical outcomes are exhibited which supports the theoretical outcomes. Numerical outcomes are compared with other existing methods and found that present method gives more accurate approximate solution as compare to the other existing methods.

A novel hybrid difference method for an elliptic equation

Hybrid difference methods are a kind of finite difference methods which is similar to hybrid discontinuous Galerkin methods introduced by Jeon and Park (SIAM J. Numer. Anal., 2010). In the previous hybrid difference method, the approximate solution is only defined on the lines parallel to the coordinate axes, but the approximation is not defined at the corner/edge nodes. Thus, it is hard to calculate the approximation value without loosing accuracy at any point in the domain due to the aforementioned missing information. To overcome this issue, we first propose a novel hybrid difference method by imposing some conditions to determine the missing information. With this, we are able to provide not only continuous approximations in the whole domain, but also the gradient of the numerical solution becomes continuous under certain conditions. Next, we prove that the proposed method is stable and locally conservative. The proposed method can be also viewed as the existing hybrid difference method with a simple postprocessing. The postprocessing not only induces a simple tridiagonal system on each mesh line, but also it can be done efficiently line by line or in parallel. Lastly, a reliable and efficient a posteriori error estimate is established for computational efficiency. Several numerical results are presented to confirm our findings.

Three-dimensional parabolic equation for infrasound propagation above topography

The parabolic equation (PE) is one of the most popular methods for the modeling of low-frequency sound in the atmosphere. Over the past decades, considerable efforts have been dedicated to the inclusion of medium inhomogeneities, such as wind and turbulence, extending the PE to more realistic atmospheric conditions. On the other hand, few models take topography into account, while its effects on infrasound propagation are important in some cases. A new development has enabled the inclusion of irregular terrain in the narrow-angle 3DPE for a moving inhomogeneous medium [Khodr et al., JASA (2020)]. A coordinate transform similar to the Beilis-Tappert mapping used in 2D [Parakkal et al., JASA (2012)] is introduced to account for the variation in topography. The numerical solution relies on a Crank–Nicolson implicit scheme along the propagating direction. The pressure field at every step is computed with an iterative fixed-point algorithm which consists of a succession of tridiagonal systems that can be efficiently solved. A validation is performed against 3D Boundary Element simulations for propagation above a Gaussian hill in a homogeneous atmosphere. The existence of transversal scattering in the shadow zone, not represented by 2D models, is highlighted. Further developments include the extension of the method to a split-step wide-angle formulation. [This work was conducted at the University of Bristol in partnership with AWE Blacknest.]

A Low-Communication-Overhead Parallel DNS Method for the 3D Incompressible Wall Turbulence

This paper presents a low-communication-overhead parallel method for direct numerical simulations of the 3D incompressible wall turbulence. A fully explicit projection method with second-order space-time accuracy is adopted. Combined with fast Fourier transforms, the parallel diagonal dominant (PDD) algorithm for the tridiagonal system is employed to solve the pressure Poisson equation. The number of all-to-all communications is decreased to only two, in a 2D pencil-like domain decomposition. The resulting MPI/OpenMP hybrid parallel code shows excellent strong scalability up to 104 cores and small wall-clock time per timestep. Numerical simulations of turbulent channel flow at different friction Reynolds numbers () have been conducted and the statistics are in good agreement with the reference data and the recent scaling theories [Chen, X. and K. R. Sreenivasan. 2021. “Reynolds Number Scaling of the Peak Turbulence Intensity in Wall Flows.” Journal of Fluid Mechanics 908: R3. doi:10.1017/jfm.2020.991].

Highly efficient parallel algorithms for solving the Bates PIDE for pricing options on a GPU

In this paper we investigate faster and memory efficient parallel techniques to numerically solve the Bates model for European options. We have followed method-of-linesapproach and implemented the numerical algorithms on a graphics processing unit (GPU). Two second order finite difference (FD) schemes are taken into account that yield discretization matrices with tridiagonal and pentadiagonal block structures. Three recent adaptations of an alternating direction implicit scheme are employed for time-stepping. Spatial and temporal errors corresponding to our chosen FD and time-stepping schemes are numerically studied. For parallel computation of solutions we have applied the well-know parallel cyclic reduction (PCR) algorithm for tridiagonal systems and our novel PCR algorithm for pentadiagonal systems. Ample numerical experiments are performed to study speed and accuracy on three platforms: single GPU using CUDA, multi-core CPU using OpenMP and an efficient sequential algorithm on a single core using MATLAB, where substantial speed-up is observed on the GPU. Sensitivities of computational times of the sequential algorithm in MATLAB with respect to certain parameters in the Bates model are also analysed.

Solving tridiagonal Toeplitz systems of linear equations on GPU‐accelerated computers

AbstractThe aim of this article is to show that solvers for tridiagonal Toeplitz systems of linear equations can be efficiently implemented for a variety of modern GPU‐accelerated and multicore architectures using OpenACC. We consider two parallel algorithms for solving such systems with special assumptions about coefficient matrices. As the first algorithm, we propose a new, faster implementation of the divide and conquer method. The next algorithm is a new, vectorizable algorithm based on a recently introduced sequential method. We consider the use of both column‐wise and row‐wise storage formats for two‐dimensional arrays and show how to efficiently convert between these two formats using cache memory and improve the overall performance of our implementations. We also show how to tune the performance by predicting the best values of the methods' parameters. Numerical experiments performed on Intel CPUs and Nvidia GPUs show that our new implementations achieve relatively good performance and accuracy.

Effects of the Solder Phase Transformation on the Optimization of Reflow Soldering Parameters and Temperature Profiles

In this study, a heat convection model of the reflow oven and a heat conduction model of the soldering area are proposed based on heat transfer theory, and a dynamic Thomas algorithm is developed for solving linear equations with coefficient matrix evolving over time in the tridiagonal system, which is derived from a heat transfer problem with moving boundaries in the solder phase transition process. We have also carried out numerical simulations for investigating the accuracy of the mathematical model, in which the temperature profiles are calculated and compared for different cases with considering or ignoring phase transformations, respectively. Parameters of reflow soldering, such as the conveyor speed, the set temperature in each zone, and a part of the heating factor, are optimized by the use of the nondominated sorting genetic algorithm II. By comparing the temperature profile and optimal solutions in the two cases, numerical results show that phase transitions of the solder have great impacts on optimal parameters and the slope of temperature profiles. Moreover, the phenomenon that the heating factor varies with the maximum set temperature in a banded distribution is investigated and analyzed, which is an important part of this work.

Crank-Nicolson scheme for two-dimensional in space fractional diffusion equations with functional delay

A two-dimensional in space fractional diffusion equation with functional delay of a general form is considered. For this problem, the Crank-Nicolson method is constructed, based on shifted Grunwald-Letnikov formulas for approximating fractional derivatives with respect to each spatial variable and using piecewise linear interpolation of discrete history with continuation extrapolation to take into account the delay effect. The Douglas scheme is used to reduce the emerging high-dimensional system to tridiagonal systems. The residual of the method is investigated. To obtain the order of the method, we reduce the systems to constructions of the general difference scheme with heredity. A theorem on the second order of convergence of the method in time and space steps is proved. The results of numerical experiments are presented.

A multi-GPU method for ADI-based fractional-step integration of incompressible Navier-Stokes equations

A computational method for GPU-accelerated fractional-step integration of incompressible Navier-Stokes equations based on the Alternating Direction Implicit (ADI) method is presented. Non-iterative, direct solution methods used in the semi-implicit fractional-step method take advantage of tridiagonal systems and Fourier transform whose solution can be computed using fast algorithms on a single GPU. However, when data is distributed to multiple GPUs, all-to-all matrix transposition is required, which increases computational cost significantly. In this work, a new strategy that does not require all-to-all transposition is proposed. The computational domain is divided in the wall-normal direction, and decoupled tridiagonal systems are obtained using Parallel Diagonal Dominant (PDD) and Parallel Partition (PPT) methods. An optimal batch size is determined to maximize the performance of PDD and PPT methods within a given amount of GPU memory. Strengths and weaknesses of this type of domain decomposition are investigated in comparison to conventional ways of dividing the domain along streamwise or spanwise directions. Using 8 NVIDIA Tesla P100 GPUs, the utility of the present method is demonstrated in a direct numerical simulation (DNS) of a canonical zero-pressure-gradient turbulent boundary layer and a DNS of a K-type boundary-layer transition on 1.4 billion grid cells.

An exponentially fitted integration scheme for a class of quasilinear singular perturbation problems

A class of quasilinear singularly perturbed two point boundary value problems in ordinary differential equations exhibiting boundary layer at the left end of the underlying interval is considered and a new exponentially fitted integration scheme on a uniform mesh is devised for its numerical solution. The devised scheme is obtained by introducing a suitable constant fitting factor in a new three term recurrence relationship derived by the application of outer region solution obtained by asymptotic expansion procedure and a combination of the exact and approximate rules of integration with finite difference approximations of the first derivative. Value of the fitting factor is determined using the theory of singular perturbations and used to take care of rapid changes in the solution. Resulting tridiagonal algebraic system of equations is solved by the Thomas algorithm. Convergence of the scheme is analyzed. Three numerical example problems are solved and computational results are tabulated to show the accuracy and efficiency of the method. It is easily observed that the derived scheme is able to produce accurate results with minimal computational effort for all the values of the mesh size h when the perturbation parameter e tends to zero. Both the theoretical and numerical analysis of the method reveals that the method is able to produce uniformly convergent results with quadratic convergence rate.

A fitted numerical method for a class of singularly perturbed differential-difference equations

In this paper , a class of singularly perturbed differential-difference equations having boundary layer at one end is analysed to get its solution numerically by a fitted method. Such types of equations occur very frequently in various fields of applied mathematics and engineering such as fluid dynamics, quantum mechanics, optimal control, chemical reactor theory etc. The basic purpose of this study is to describe a numerical approach for the solution of singularly perturbed differential-difference equation based on deviating argument and interpolation. Thomas algorithm is used to solve the tri-diagonal system. Numerical examples are presented which demonstrate the applicability of this method.

MODELING OF MAXIMALLY PARALLEL STRUCTURES OF ALGORITHMS FOR SOLVING THERMAL PROBLEMS

The paper demonstrates the possibility of creating a maximum parallel form of computational algorithms to solve thermal problems and their mapping to the architecture of multiprocessor systems based on solving thermal problems of mathematical physics. It is shown that an effective tool for studying heat and mass transfer problems in metallurgical production could be parallel computing technologies on distributed cluster systems with a relatively low cost and reasonably easily scalable both in the number of processors and in the amount of RAM. Tridiagonal structure systems' parallelization was implemented by a numerical-analytical approach, which predetermined their maximally parallel algorithmic form. That approach is facilitated by the minimum possible implementation time of the developed algorithm on parallel computing systems. Furthermore, during the arithmetic expressions parallel computations, the developed algorithm separates the error in the output data from rounding operations. Thus, the parallelization of tridiagonal systems based on numerical-analytical discretization methods does not impose any restrictions on the topology of the mesh nodes of the computational domain.Furthermore, as applied to the parallel computation of arithmetic expressions, it separates the initial data error from a real PC's rounding operations. That approach eliminates the recurrent structure of computing the sought-for decision vectors, which, as a rule, leads to the round-off errors accumulation. Such a parallel form of the constructed algorithm is maximal and has the shortest possible implementation time of the algorithm on parallel computing systems. The developed approach to parallelizing the mathematical model is stable for various types of input data. It has the most parallel form and is distinguished by the minimum time for solving the problem as applied to multiprocessor computing systems. That is explained as follows. If it is hypothesized that one processor can be assigned to one processor and one processor can be assigned to one node of the computational mesh domain, the computations can be processed in parallel and simultaneously for all nodes of the computational mesh domain. The simulation process was implemented on a PC cluster. It follows from the simulation results analysis that the developed method for solving the heat conduction problem effectively minimizes residuals.

A fast forward algorithm for three-dimensional gravity gradient tensor using the compact difference schemes

We propose a fast and accurate 3D large-scale forward modeling method to compute the gravity gradient tensor, in terms of a fourth-order compact finite difference scheme and the extrapolation cascadic multigrid (EXCMG) method. Firstly, the 19-point fourth-order compact finite difference scheme with unequal mesh sizes in different coordinate directions is used to discretize the governing 3D Poisson equation. The resulted symmetric positive definite linear systems are solved by the EXCMG method. Then, the second-order derivatives (gravity gradient tensor) are calculated by solving a series of tridiagonal linear systems resulting from the fourth-order compact finite difference discretization. Finally, numerical result for a cubic model with positive density is used to verify the accuracy of our presented method. It shows that our method can give nearly fourth-order accurate approximations to gravitational potential and gravity gradient tensor. The accuracy is much higher and more efficient than traditional finite difference methods.

Comparative analysis of parallel algorithms for solving oil recovery problem using CUDA and OpenCL

‎In this paper the implementation of parallel algorithm of alternating direction implicit (ADI) method has been considered‎. ‎ADI parallel algorithm is used to solve a multiphase multicomponent fluid flow problem in porous media‎. ‎There are various technologies for implementing parallel algorithms on the CPU and GPU for solving hydrodynamic problems‎. ‎In this paper GPU-based (graphic processor unit) algorithm was used‎. ‎To implement the GPU-based parallel ADI method‎, ‎CUDA and OpenCL were used‎. ‎ADI is an iterative method used to solve matrix equations‎. ‎To solve the tridiagonal system of equations in ADI method‎, ‎the parallel version of cyclic reduction (CR) method was implemented‎. ‎The cyclic reduction is a method for solving linear equations by repeatedly splitting a problem as a Thomas method‎. ‎To implement of a sequential algorithm for solving the oil recovery problem‎, ‎the implicit Thomas method was used‎. ‎Thomas method or tridiagonal matrix algorithm is used to solve tridiagonal systems of equations‎. ‎To test parallel algorithms personal computer installed Nvidia RTX 2080 graphic card with 8 GB of video memory was used‎. ‎The computing results of parallel algorithms using CUDA and OpenCL were compared and analyzed‎. ‎The main purpose of this research work is a comparative analysis of the parallel algorithm computing results on different technologies‎, ‎in order to show the advantages and disadvantages each of CUDA and OpenCL for solving oil recovery problems‎.

A class of compact finite difference schemes for solving the 2D and 3D Burgers’ equations

In this paper, a class of two-level high-order compact finite difference implicit schemes are proposed for solving the Burgers’ equations. Firstly, based on the fourth-order compact finite difference schemes for spatial derivatives and the truncation error remainder correction method for temporal derivative, the high-order compact (HOC) difference method is introduced for solving the one-dimensional (1D) Burgers’ equation. At the same time, the stability of the scheme is analyzed by using the Fourier analysis method. Because only three grid points are involved in each time level. The Thomas algorithm can be directly used to solve the tridiagonal linear system. Then, this method is extended to solve the two-dimensional (2D) and three-dimensional (3D) coupled Burgers’ equations. Finally, numerical experiments are conducted to verify the accuracy and the reliability of the present schemes.

An interpolatory directional splitting-local discontinuous Galerkin method with application to pattern formation in 2D/3D

An efficient computational method to approximate the solution of a general class of nonlinear reaction-diffusion systems in Cartesian grids is presented. The proposed scheme uses the Local Discontinuous Galerkin (LDG) method as spatial discretization and the symmetric Strang operator splitting as time marching scheme. As a result, not only diffusion and reaction are decoupled; but the reaction sub-step reduces to a set of completely independent small local nonlinear systems of ordinary differential equations. To reduce their computational cost a Lagrange interpolatory technique is used. Diffusion sub-steps are approximated by a directional splitting method derived from an algebraic decomposition of the discrete Laplacian operator, which amounts to solve sets of completely independent small block tridiagonal linear systems; one set per direction. Stability of the diffusive step is analyzed. Finally the efficiency and accuracy of the proposed scheme is numerically investigated on well known Turing pattern formation models in 2D/3D.