Tridiagonal Systems Research Articles

High Performance Solution of Tridiagonal Systems on the GPU

In this paper, we present PfSolve — a new, performant, cross-platform, and open-source implementation of tridiagonal and bidiagonal matrix solvers for the GPU architecture. Released as a stand-alone library, PfSolve can solve systems of arbitrary size that fit into the memory of a single GPU with a potential extension to multi-GPU support in the future. The code works in single, double, and double-double emulation of quad precision using only \(0.1\% \) of the original system size as additional memory. PfSolve is based on the in-house implementation of the Parallel Thomas algorithm optimized for GPU execution by using warp-level instructions and occupancy optimizations, which are discussed in detail in the paper. This work also presents an accuracy analysis of the Parallel Thomas algorithm for tridiagonal matrices with various dominance factors (approximately, the ratio of the off-diagonal to diagonal terms) and demonstrates that PfSolve achieves a considerable speedup over vendor solutions on modern HPC GPUs like Nvidia H100 and AMD MI210. The source code for PfSolve is available on GitHub.

A NUMERICAL STUDY OF CASSON NANOFLUID FLOW EMBEDDED IN A RIGA PLATE WITH IRREGULAR BOUNDARIES

This study analyzes the effects of Darcy-Forchheimer on a convective engine oil-based hybrid nanofluid (Ag-TiO<sub>2</sub>) flow across an upright porous Riga plate. The investigation is carried out under the consideration of hybrid nanofluid's temperature-sensitive thermo-physical characteristics, as the practical scenario demands. This article also considered the magnetohydrodynamics (MHD), radiation, buoyancy, and Soret effects to encompass a broad area of practical applications. The governing mathematical equations that occur as immensely nonlinear PDEs are converted into linear form using the quasi-linearization method. Further, those equations are transformed into a tri-diagonal matrix system implementing implicit finite difference methods. Finally, the Vargas algorithm is adopted to solve the resulting system. The numerical findings are illustrated in graphical representations and briefly summarized, facilitating clear comprehension for ease of understanding. An enhancement in the Dufour number (Df) led to an increase in the velocity and heat rate in the flow. Furthermore, the buoyancy ratio (Nr) has been found to enhance the temperature but decay the velocity profile. The concentration profile decays and elevates subject to the augmentation of Lewis (Le) and Stuart numbers (St), respectively.

Paired piecewise linear regression model with fixed nodes

In this paper we develop a method for constructing a paired regression model in the class of piecewise linear continuous functions with fixed nodes. The concept of linear division of the correlation field, its partial sections and its nodes is introduced into consideration. Using the linear division of the correlation field, a class of piecewise linear functions with fixed nodes is determined. The linear division is revealed during the visual analysis of the correlation field. Using the least squares method, a system of linear equations is compiled to find point estimates of the parameters of the approximating function. With the exception of two unknown angular coefficients (unknown) this system is reduced to a tridiagonal system of equations, the unknowns of which are the nodal values of the approximating function. The tridiagonal system is solved by the run-through method. An algorithm was constructed to demonstrate the operation of the developed method. Its initial data is arrays of the corresponding values of the explanatory and dependent variables, as well as an array of numbers of the right ends of the intervals defining the nodes. An array of fixed nodes is constructed from an array of values of the explanatory variable and an array of numbers of the right ends of the intervals. Next, an array of point estimates of the parameters of the approximating function is constructed. This algorithm is implemented in Python in the form of user-defined functions.

Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix

Abstract The aim of this work is to study the global existence in time of solutions for the tridiagonal system of reaction-diffusion by order m m . Our techniques of proof are based on compact semigroup methods and some L 1 {L}^{1} -estimates. We show that global solutions exist. Our investigation can be applied for a wide class of nonlinear terms of reaction.

Exploring the Efficacy of the Weighted Average Method for Solving Nonlinear Partial Differential Equations: A Study on the Burger-Fisher Equation

This study explores the application of the weighted average method for solving the Burger-Fisher equation, a nonlinear partial differential equation (PDE) of significant interest in various scientific disciplines. Nonlinear PDEs, such as the Burger-Fisher equation, are fundamental in describing complex physical, biological, and engineering phenomena but pose challenges for both analytical and numerical solutions. The weighted average method, known for its ability to converge rapidly to exact solutions, offers a promising approach for tackling such equations. By discretizing both spatial and temporal derivatives using a combination of forward, backward, and central differences, the method approximates solutions with high accuracy and stability. Conducting convergence and stability analyses, this study elucidates the computational requirements and performance characteristics of the weighted average method. Utilizing mathematical software like MATLAB and MAPLE, the method's implementation involves solving a tridiagonal matrix system at each time step. Comparison between numerical solutions obtained using the method and exact solutions demonstrates the method's accuracy, with negligible errors observed. Visual representations further illustrate the close agreement between the numerical and exact solutions, validating the method's reliability for practical applications. The study's findings underscore the practical utility of the weighted average method in solving the Burger-Fisher equation and similar nonlinear PDEs. Its ability to accurately approximate solutions while maintaining stability highlights its efficacy as a computational tool for addressing complex mathematical problems across diverse scientific and engineering fields. The study contributes to advancing the understanding and application of numerical methods for nonlinear PDEs, offering valuable insights for researchers and practitioners seeking precise and reliable solutions to complex mathematical models. Overall, the study emphasizes the importance of fine-tuning numerical parameters and leveraging computational resources to achieve optimal accuracy when utilizing the weighted average method for solving nonlinear PDEs.

Numerical Solution of Two-Dimensional Nonlinear Unsteady Advection-Diffusion-Reaction Equations with Variable Coefficients

The advection-diffusion-reaction (ADR) equation is a fundamental mathematical model used to describe various processes in many different areas of science and engineering. Due to wide applicability of the ADR equation, finding accurate solution is very important to better understand a physical phenomenon represented by the equation. In this study, a numerical scheme for solving two-dimensional unsteady ADR equations with spatially varying velocity and diffusion coefficients is presented. The equations include nonlinear reaction terms. To discretize the ADR equations, the Crank–Nicolson finite difference method is employed with a uniform grid. The resulting nonlinear system of equations is solved using Newton’s method. At each iteration of Newton’s method, the Gauss–Seidel iterative method with sparse matrix computation is utilized to solve the block tridiagonal system and obtain the error correction vector. The consistency and stability of the numerical scheme are investigated. MATLAB codes are developed to implement this combined numerical approach. The validation of the scheme is verified by solving a two-dimensional advection-diffusion equation without reaction term. Numerical tests are provided to show the good performances of the proposed numerical scheme in simulation of ADR problems. The numerical scheme gives accurate results. The obtained numerical solutions are presented graphically. The result of this study may provide insights to apply numerical methods in solving comprehensive models of physical phenomena that capture the underlying situations.

Numerical algorithms for the fast and reliable solution of periodic tridiagonal Toeplitz linear systems

Numerical algorithms for the fast and reliable solution of periodic tridiagonal Toeplitz linear systems

A Five-Step Block Method Coupled with Symmetric Compact Finite Difference Scheme for Solving Time-Dependent Partial Differential Equations

In this article, we present a five-step block method coupled with an existing fourth-order symmetric compact finite difference scheme for solving time-dependent initial-boundary value partial differential equations (PDEs) numerically. Firstly, a five-step block method has been designed to solve a first-order system of ordinary differential equations that arise in the semi-discretisation of a given initial boundary value PDE. The five-step block method is derived by utilising the theory of interpolation and collocation approaches, resulting in a method with eighth-order accuracy. Further, characteristics of the method have been analysed, and it is found that the block method possesses A-stability properties. The block method is coupled with an existing fourth-order symmetric compact finite difference scheme to solve a given PDE, resulting in an efficient combined numerical scheme. The discretisation of spatial derivatives appearing in the given equation using symmetric compact finite difference scheme results in a tridiagonal system of equations that can be solved by using any computer algebra system to get the approximate values of the spatial derivatives at different grid points. Two well-known test problems, namely the nonlinear Burgers equation and the FitzHugh-Nagumo equation, have been considered to analyse the proposed scheme. Numerical experiments reveal the good performance of the scheme considered in the article.

Efficient numerical methods for semilinear one dimensional parabolic singularly perturbed convection-diffusion systems

In this work we deal with the numerical solution of one dimensional semilinear parabolic singularly perturbed systems of convection-diffusion type. We assume that the coupling in the convection terms is weak and also that the coupling reaction terms are nonlinear. In the case of considering different small diffusion parameters at each equation with different orders of magnitude, the exact solution usually shows overlapping boundary layers on the outflow of the spatial domain. To approximate it, we construct a numerical scheme which combines the upwind finite difference scheme, defined on a piecewise uniform Shishkin mesh, to discretize in space and a linearized version of the fractional implicit Euler method together with an appropriate splitting by components to discretize in time. Then, the fully discrete method is uniformly convergent with respect to both diffusion parameters, having first order in time and almost first order in space. The choice of this time integrator provokes that only tridiagonal linear systems must be solved at each time step; in this way, the computational cost of the algorithm is considerably lower than this one associated to classical implicit methods. The numerical results obtained for different test problems corroborate in practice the good performance of the numerical algorithm.

The solution for singularly perturbed differential-difference equation with boundary layers at both ends by a numerical integration method

This paper presents a numerical integration method for the solution of ‘Singularly Perturbed Differential-Difference Equations' having dual layers. It is well known that when we use already existing numerical methods to solve such problems, we get oscillatory or unsatisfactory results unless we take a very small step size, which is time-consuming and costly. To get a numerical solution for such a problem, first, the delay and advanced parameters present in the SPDDE are approximated by Taylor’s series to get an equivalent ‘Singularly Perturbed Differential Equation' of second order. Second, an asymptotically equivalent first-order differential equation is obtained from SPDE using Taylor’s transformation. Composite Simpson’s 1/3 rule is implemented to get a three-term recurrence relation. The Thomas algorithm is applied to get the solution of the tri-diagonal system of equations. Several model examples are tested and it was found that the numerical solution approximates the available/exact solution very well.

A general solution for a special tri-diagonal linear system

The tri-diagonal linear system with coefficients of (–1, 2, –1) along the bandwidth and the transposed vector of (1k, 2k, …., nk) as the right hand sides is solved in a general form linking with the classical Faulhaber’s Formula. We derive the general solution for any k and conduct computational experiments compared with using the direct Thomas’s algorithm. The result shows that in the case where k = 1 to 16, the general solution clearly performs more efficiently. For cases where k = 1, the results also indicate that as the problem size continuously expands, there will always be a point at which the general solution processes faster.

A NUMERICAL APPROACH OF FORCED CONVECTIVE MHD CASSON HYBRID NANOFLUID FLOWS EXPOSED TO JOULE HEATING AND VISCOUS DISSIPATION OVER DIVERGING CHANNEL

The present study deals with boundary layer flows of buoyancy-driven magnetohydrodynamic, chemical-radiative, and temperature-sensitive Casson hybrid nanofluid over diverging channel. Copper (Cu) and aluminum oxide (Al<sub>2</sub>O<sub>3</sub>) nanoparticles are suspended upon ethylene glycol-based non-Newtonian Casson fluid. The proposed model is applicable in power transmission systems the design of nuclear reactors where a moving plate is used as a control rod, and the design of compression molding processes. The boundary layer governing equations undergo nonsimilar transformations followed by a quasilinearization technique and an implicit finite difference scheme. Varga's algorithm is applied on the obtained block tri-diagonal system of equations. The study pertinent to dimensionless parameters like Reynolds number, Eckert number, Casson parameter, and Richardson's number on velocity, temperature, drag coefficient, and heat transfer profiles. Also surface plots are plotted for varied values of Casson parameter and magnetic parameter on skin friction and heat transfer coefficients. It is to be noted that for enhanced values of Casson parameter β, the velocity profile is augmented, and the temperature profile is declined. It is observed that the temperature profile is enhanced at the center of the channel for enhanced values of viscous dissipation parameter Ec.

Interface Splitting Algorithm: A Parallel Solution to Diagonally Dominant Tridiagonal Systems

We present an interface-splitting algorithm (ITS) for solving diagonally dominant tridiagonal systems in parallel. The construction of the ITS algorithm profits from bidirectional links in modern networks, and it only needs one synchronization step to solve the system. The algorithm trades some necessary accuracy for better parallel performance. The accuracy and the performance of the ITS algorithm are evaluated on four different parallel machines of up to 2048 processors. The proposed algorithm scales very well, and it is significantly faster than the algorithm used in ScaLAPACK. The applicability of the algorithm is demonstrated in the three-dimensional simulations of turbulent channel flow at Reynolds number 41,430.

SwPTS: an efficient parallel Thomas split algorithm for tridiagonal systems on Sunway manycore processors

swPTS: an efficient parallel Thomas split algorithm for tridiagonal systems on Sunway manycore processors

Graded mesh B-spline collocation method for two parameters singularly perturbed boundary value problems

The solutions of two parameters singularly perturbed boundary value problems typically exhibit two boundary layers. Because of the presence of these layers standard numerical methods fail to give accurate approximations. This paper introduces a numerical treatment of a class of two parameters singularly perturbed boundary value problems whose solution exhibits boundary layer phenomena. A graded mesh is considered to resolve the boundary layers and collocation method with cubic B-splines on the graded mesh is proposed and analyzed. The proposed method leads to a tri-diagonal linear system of equations. The stability and parameters uniform convergence of the present method are examined. To verify the theoretical estimates and efficiency of the method several known test problems in the literature are considered. Comparisons to some existing results are made to show the better efficiency of the proposed method. Summing up:•The present method is found to be stable and parameters uniform convergent and the numerical results support the theoretical findings.•Experimental results show that the present method approximates the solution very well and has a rate of convergence of order two in the maximum norm.•Experimental results show that cubic B-spline collocation method on graded mesh is more efficient than cubic B-spline collocation method on Shishkin mesh and some other existing methods in the literature.

An improved time accurate numerical estimation for singularly perturbed semilinear parabolic differential equations with small space shifts and a large time lag

The objective of this work is to provide a robust numerical scheme for solving a singularly perturbed semilinear parabolic differential–difference equation having both time delay along with spatial delay, and shift parameters. The small delay and shift arguments in spatial direction are treated with Taylor’s approximation. The technique of quasilinearization is used to treat the semilinearity and the temporal direction is handled with the θ-method. To handle the layer behavior caused by the presence of perturbation parameter, the problem is solved using the upwind scheme on two-layer resolving meshes in the spatial direction providing a first-order accurate result for 0.5≤θ≤1. For θ=0.5, we have the second-order accurate Crank–Nicolson scheme in time. In order to have a higher-order accuracy, the Richardson extrapolation technique is applied in the spatial direction, which in turn provides a global second-order accurate result. Thomas algorithm is used to solve the tridiagonal system of equations formed in the fully discrete scheme. The numerical approach presented is shown to be parameter uniform and convergent for 0.5≤θ≤1. Some numerical tests are presented to show the efficacy of the proposed scheme.

Numerical solution of differential – difference equations having an interior layer using nonstandard finite differences

This paper addresses the solution of a differential-difference type equation having an interior layer behaviour. A difference scheme is suggested to solve this equation using a non-standard finite difference method. Finite differences are derived from the first and second order derivatives. Using these approximations, the given equation is discretized. The discretized equation is solved using the algorithm for the tridiagonal system. The method is examined for convergence. Numerical examples are illustrated to validate the method. Maximum errors in the solution, in contrast to the other methods are organized to justify the method. The layer behaviour in the solution of the examples is depicted in graphs.

A model instability issue in the National Centers for Environmental Prediction Global Forecast System version 16 and potential solutions

Abstract. The National Centers for Environmental Prediction (NCEP) Global Forecast System version 16 (GFSv16) encountered a few model instability failures during the pre-operational real-time parallel runs. The model forecasts failed when an extremely small thickness depth appeared at the model's lowest layer during the landfall of strong tropical cyclones. A quick solution was to increase the value of minimum thickness depth, an arbitrary parameter introduced to prevent numerical instability. This modification solved the model's numerical instability with a small impact on forecast skills. It was adopted in GFSv16 to implement this version of the operational system as planned. Upon further investigation, it was determined that the extremely thin depth was a result of the advection of geopotential heights at the interfaces of model layers. In the Finite-Volume Cubed-Sphere (FV3) dynamical core, the horizontal winds at interfaces for advection are calculated from the layer-mean values by solving a tridiagonal system of equations in the entire vertical column based on the parabolic spline method (PSM) with high-order boundary conditions (BCs). We replaced the high-order BCs with zero-gradient BCs for the interface-wind reconstruction. The impact of the zero-gradient BCs was investigated by performing sensitivity experiments with GFSv16, idealized mountain ridge tests and the Rapid Refresh Forecast System (RRFS). The results showed that zero-gradient BCs can fundamentally solve the instability and have little impact on the forecast performances and the numerical solutions of idealized mountain tests. This option has been added to FV3 and will be utilized in the GFS (GFSv17/Global Ensemble Forecast System version 13 – GEFSv13) and RRFS for operations.

Гидродинамика сетных жестких безузловых конструкций

Rigid netting structures are elements, parts of commercial fishing gear and aquaculture cages. They serve to enclose or filter hydrobionts, they are engineering structures. Rigid netting structures can be used as inserts in commercial fishing gear, as well as selective gratings and elements that prevent hydraulic backwater in trawls. They are also elements of stationary fishing gear, such as winders or other fishing gear or barriers. In aquaculture cages, rigid netting structures, the elements of which have a sufficiently large value of the longitudinal and transverse modulus of elasticity E, are parts that ensure the strength of the netting structures. Mesh rigid structures are made of plastic or aluminum rods in the form of cylinders, which can be either smooth or twisted. The article considers the application of a numerical method for determining the hydrodynamic characteristics of mesh rigid structures using the software developed by the authors. A schematization of a rigid netting structure, a mathematical model based on Navier-Stokes partial differential equations, a computational domain, initial and boundary conditions are proposed. The calculation was carried out on a regular computational grid using an implicit finite-difference scheme using the methods of coordinate-wise splitting, linearization of nonlinear equations with subsequent correction of nonlinear coefficients, and solution of the obtained tridiagonal systems by the sweep method. The results of numerical experiments are presented in the form of visualization of pressure on the surface of various mesh structures at various angles of attack.

PaScaL_TDMA 2.0: A multi-GPU-based library for solving massive tridiagonal systems

PaScaL_TDMA 2.0: A multi-GPU-based library for solving massive tridiagonal systems