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Cubic B-splines collocation method for solving nonlinear parabolic partial differential equations with Neumann boundary conditions

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2012年第12期17卷 4616-4625页

作者： Mittal, R. C. Jain, K. IIT Dept Math Roorkee 247667 Uttar Pradesh India

In this paper, a numerical method is proposed to approximate the solution of the nonlinear parabolic partial differential equation with Neumann's boundary conditions. The method is based on collocation of cubic B-splines over finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range. We apply cubic B-splines for spatial variable and its derivatives, which produce a system of first order ordinary differential equations. We solve this system by using SSP-RK3 scheme. The numerical approximate solutions to the nonlinear parabolic partial differential equations have been computed without transforming the equation and without using the linearization. Four illustrative examples are included to demonstrate the validity and applicability of the technique. In numerical test problems, the performance of this method is shown by computing L-infinity and L-2 error norms for different time levels. Results shown by this method are found to be in good agreement with the known exact solutions. (C) 2012 Elsevier B. V. All rights reserved.

关键词： Nonlinear parabolic partial differential equation Neumann boundary conditions Cubic B-splines basis functions SSP-RK3 scheme thomas algorithm

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A hybrid thermal error modeling method of heavy machine tools in z-axis

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INTERNATIONAL JOURNAL OF ADVANCED MANUFACTURING TECHNOLOGY 2015年第1-4期80卷 389-400页

作者： Wang, Liping Wang, Haitong Li, Tiemin Li, Fengchun Tsinghua Univ Inst Mfg Engn Dept Mech Engn Beijing 100084 Peoples R China

Thermal error is one of the major contributors to dimensional errors of machine tools. In this paper, a hybrid thermal error modeling method is proposed to forecast the thermal expansion of a heavy boring and milling machine tool in z-axis. In this model, the thermal error in z-axis is calculated as the expansion of a one-dimensional rod. The inputs of the model are temperatures of spindle motor, hydrostatic bearing and environment, selected through fuzzy c-means (FCM) clustering method. The temperatures of the nodes in the rod are solved by thomas algorithm and the coefficients of the model are optimized by genetic algorithm (GA). The accuracy of the proposed model is compared with that of the output error model and stepwise regression model, and the performance of the proposed model is better in accuracy. In addition, a compensation system is developed based on the proposed model to reduce the thermal error in z-axis.

关键词： Thermal error modeling Heavy machine tools Fuzzy c-means clustering method thomas algorithm Genetic algorithm

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The numerical solution of advection-diffusion problems using new cubic trigonometric B-splines approach

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APPLIED MATHEMATICAL MODELLING 2016年第7-8期40卷 4586-4611页

作者： Nazir, Tahir Abbas, Muhammad Ismail, Ahmad Izani Md Abd Majid, Ahmad Rashid, Abdur Univ Sargodha Dept Math Sargodha 40100 Pakistan Univ Sains Malaysia Sch Math Sci George Town 11800 Malaysia Gomal Univ Dept Math Dera Ismail Khan 29050 Pakistan

A new cubic trigonometric B-spline collocation approach is developed for the numerical solution of the advection-diffusion equation with Dirichlet and Neumann's type boundary conditions. The approach is based on the usual finite difference scheme to discretize the time derivative while a cubic trigonometric B-spline is utilized as an interpolation function in the space dimension with the help of theta-weighted scheme. The present scheme stabilizes the oscillations that are normally displayed by the approximate solution of the transient advective-diffusive equation in the locality of sharp gradients produced by transient loads and boundary layers. The scheme is shown to be stable and the accuracy of the scheme is tested by application to various test problems. The proposed approach is numerically verified to second order and shown to work for the Peclet number <= 5. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Advection-diffusion equation Cubic trigonometric B-spline basis functions Cubic trigonometric B-spline collocation method Interpolation function thomas algorithm

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Parallel Dichotomy algorithm for solving tridiagonal system of linear equations with multiple right-hand sides

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PARALLEL COMPUTING 2010年第8期36卷 423-438页

作者： Terekhov, Andrew V. Inst Computat Math & Math Geophys Novosibirsk 630090 Russia Budker Inst Nucl Phys Novosibirsk 630090 Russia Novosibirsk State Univ Novosibirsk 630090 Russia

A parallel algorithm for solving a series of matrix equations with a constant tridiagonal matrix and different right-hand sides is proposed and studied. The process of solving the problem is represented in two steps. The first preliminary step is calculating some rows of the inverse matrix of system of linear algebraic equations. The second step consists in calculating solutions for all right-hand sides. For reducing the communication interactions, based on the formulated and proved the main Gaussian Parallel Elimination Theorem for tridiagonal system of equations, we propose an original algorithm for calculating share components of the solution vector. Theoretical estimates validating the efficiency of the approach for both the common- and distributed-memory supercomputers are obtained. Direct and iterative methods of solving a 2D Poisson equation, which include procedures of tridiagonal matrix inversion, are realized using the MPI paradigm. Results of computational experiments on a multicomputer demonstrate a high efficiency and scalability of the parallel Dichotomy algorithm. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Parallel algorithm Tridiagonal matrix algorithm (TDMA) thomas algorithm Poisson equation Alternating direction method Fourier method

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BCYCLIC: A parallel block tridiagonal matrix cyclic solver

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JOURNAL OF COMPUTATIONAL PHYSICS 2010年第18期229卷 6392-6404页

作者： Hirshman, S. P. Perumalla, K. S. Lynch, V. E. Sanchez, R. Oak Ridge Natl Lab Oak Ridge TN 37830 USA

A block tridiagonal matrix is factored with minimal fill-in using a cyclic reduction algorithm that is easily parallelized. Storage of the factored blocks allows the application of the inverse to multiple right-hand sides which may not be known at factorization time. Scalability with the number of block rows is achieved with cyclic reduction, while scalability with the block size is achieved using multithreaded routines (OpenMP, GotoBLAS) for block matrix manipulation. This dual scalability is a noteworthy feature of this new solver, as well as its ability to efficiently handle arbitrary (non-powers-of-2) block row and processor numbers. Comparison with a state-of-the art parallel sparse solver is presented. It is expected that this new solver will allow many physical applications to optimally use the parallel resources on current supercomputers. Example usage of the solver in magneto-hydrodynamic (MHD), three-dimensional equilibrium solvers for high-temperature fusion plasmas is cited. (C) 2010 Elsevier Inc. All rights reserved.

关键词： Cyclic reduction Block matrix Dense blocks Tridiagonal matrix thomas algorithm Parallel computing

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Higher order B-spline differential quadrature rule to approximate generalized Rosenau-RLW equation

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2020年第11期43卷 6812-6822页

作者： Mustahsan, Muhammad Kiran, Ayesha Singh, Jagdev Nisar, Kottakkaran Sooppy Kumar, Devendra Islamia Univ Bahawalpur Dept Math Bahawalpur 63100 Pakistan JECRC Univ Dept Math Jaipur 303905 Rajasthan India Prince Sattam Bin Abdulaziz Univ Coll Arts & Sci Dept Math Wadi Aldawaser 11991 Saudi Arabia Univ Rajasthan Dept Math Jaipur 302004 Rajasthan India

In this article, B-spline-based collocation method is employed to approximate the usual and modified Rosenau-RLW nonlinear equations. The weighted extended B-spline (WEB-spline) is used as the modified form of B-spline as the usual B-splines fail to obey the Dirichlet boundary conditions. The WEB method is more general method that allows to discretize the domain into finite number of elements not necessarily start from the boundary points of the domain. Our method omits the linearization process of the nonlinear partial differential equation (PDE). Different cases are discussed by setting the parameter p=2,3,4, and 6 that appears in Rosenau-RLW equations. The error estimation is calculated, which gives good agreement of the exact solution.

关键词： collocation method Rosenau-RLW equation SSP-RK54 scheme thomas algorithm WEB-spline

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Counting solutions of polynomial systems via iterated fibrations

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ARCHIV DER MATHEMATIK 2009年第1期92卷 44-56页

作者： Plesken, Wilhelm Univ Aachen Rhein Westfal TH Aachen Lehrstuhl Mathemat B D-52062 Aachen Germany

The paper introduces a new polynomial to count the solutions of a system of polynomial equations and inequations over an algebraically closed field of characteristic zero based on the triangular decomposition algorithm by J. M. thomas of the nineteen-thirties. In the special case of projective varieties examples indicate that it is a finer invariant than the Hilbert polynomial.

关键词： Polynomial equations polynomial inequations solutions of polynomial systems thomas algorithm counting polynomial simple systems triangular decomposition

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Numerical modeling of the electrical properties plasma argon in a RF magnetron sputtering and with Einstein's relation of electron diffusivity

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JOURNAL OF KING SAUD UNIVERSITY SCIENCE 2020年第1期32卷 620-627页

作者： Ballah, Z. Khelfaoui, F. Univ Ouargla Fac Math & Sci Mat Lab Rayonnement & Plasmas & Phys Surfaces Ouargla 30000 Algeria

Numerical modeling of RF magnetron sputtering discharge of argon plasma properties, using a one-dimensional time-dependent fluid model in presence of the magnetic field, has been developed. The model is based on continuity equation and electron temperature equation coupled with Poisson's equation. The electron mobility depends on magnetic field and the electron diffusivity is assumed to be dependent of electron energy by Einstein's relation. The flux is calculated by Scharfetter and Gummel schemes. Numerical simulations were resolved by using the Finite Volume Method (FVM) and the thomas algorithm. The obtained results for the electrical properties, electron and ion densities, electrical potential, electric field and electron temperature, are in good agreement with previous works. A parametric study varying the magnetic field intensity on the discharge properties is done. (C) 2018 Production and hosting by Elsevier B.V. on behalf of King Saud University.

关键词： Magnetron sputtering Electrical properties Fluid model Electron diffusivity Einstein's relation SG Scheme thomas algorithm Argon gas

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Numerical Modeling of Charring Material Ablation with Considering Chemical-Reaction Effects, Mass Transfer and Surface Heat Transfer

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ARABIAN JOURNAL FOR SCIENCE AND ENGINEERING 2013年第9期38卷 2533-2543页

作者： Mohammadiun, Hamid Mohammadiun, Mohammad Islamic Azad Univ Shahrood Branch Dept Mech Engn Shahrood Iran

The numerical solution of governing equations for charring material ablation for anticipating the recession rate and the heat response of charring heat shields is presented. The governing equations are nonlinear because of changing thermodynamic properties rho, C (p), k according to temperature. The nonlinear equations are linearized by the new method which is developed in this research and this method is based on Newton-Raphson method, and the TDMA algorithm (thomas algorithm) is used for solving these new linear equation systems. Using Newton-Raphson method is one of the advantages of the solving method, because it is very simple and it can be easily generalized to more difficult problems. The obtained results are compared with reliable sources to examine the accuracy of writing code.

关键词： Charring material ablation Heat response Newton-Raphson method Recession rate thomas algorithm

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An Efficient Recursive Approach for Electromagnetic Scattering by Arbitrary-Shaped Multilayer Cylinders

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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS 2020年第8期19卷 1375-1379页

作者： Gurbuz, Tolga Ulas Aslanyurek, Birol Gaziantep Univ Dept Comp Engn TR-27310 Gaziantep Turkey Yildiz Tech Univ Dept Math Engn TR-34210 Istanbul Turkey

We present a recursive version of the previously proposed fast algorithm for computation of the field scattered from arbitrary-shaped multilayer objects. As in the original version, the field at each layer is represented by a series of cylindrical functions with unknown coefficients. In order to determine these coefficients, a linear system of equations is obtained through a procedure based on the boundary conditions between the layers. Instead of solving this large system by applying conventional linear solvers as done in the original version, through an approach based on thomas algorithm, we derive recursive expressions to compute the scattered field. While the accuracy of the results for both versions are almost the same, the recursive version has lower time and memory complexity. Hence, it supersedes the original version.

关键词： Nonhomogeneous media Linear systems Boundary conditions Dielectrics Electromagnetic scattering Complexity theory Transforms Computational electromagnetics electromagne-tic scattering multilayer objects thomas algorithm

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