B-spline Finite Element Method Research Articles

Stability, convergence and error analysis of B-spline collocation with Crank–Nicolson method and finite element methods for numerical solution of Schrodinger equation arises in quantum mechanics

In the present study, the complex-valued Schrodinger equation (CVSE) is solved numerically by a nonic B-spline finite element method (FEM) in quantum mechanics. The approach employed is based on the collocation of nonic B-splines over spatial finite elements, so that we have continuity of the dependent variable and its first eight derivatives throughout the solution range. For time discretization, the Crank-Nicolson scheme of second order based on FEM is employed. The method is shown to be unconditionally stable and accurate to order. Three problems are considered to validate the algorithm. Comparisons are made with existing methods and analytical solutions. The proposed method exhibits good conservation properties and performs well with regards to analytical solutions for different error norms and conservative constants related to parameters in quantum classes in Physics. The computational complexity of (2N+18) arithmetic operations with the help of a nonic-diagonal matrix is also tackled by the present scheme.

An efficient computational technique for higher order KdV equation arising in shallow water waves

This paper focuses on the computational study of seventh-order KdV equations via the quintic B-spline (QBS) finite element method (FEM). The KdV equations contain higher-order derivative terms. We transform the higher order into lower order and obtain a coupled system to ease the solution procedure. Forward difference combined with the theta-weighted scheme is used to discretize the temporal part. In addition, the quintic B-spline approximates the functions and their derivatives. The Von Neumann method is used to check stability of the scheme. The performance of the proposed method is evaluated using four test problems. The numerical results are presented in the tables to evaluate the accuracy and proficiency of the scheme at various collocation points. Furthermore, the approximate and exact solutions are compared graphically which match each other, showing validation of the proposed method.

Cubic B-spline finite element method for generalized reaction-diffusion equation with delay

Cubic B-spline finite element method for generalized reaction-diffusion equation with delay

Computational Study of Multiterm Time-Fractional Differential Equation Using Cubic B-Spline Finite Element Method

Due to the symmetry feature in nature, fractional differential equations precisely measure and describe biological and physical processes. Multiterm time-fractional order has been introduced to model complex processes in different physical phenomena. This article presents a numerical method based on the cubic B-spline finite element method for the solution of multiterm time-fractional differential equations. The temporal fractional part is defined in the Caputo sense while the B-spline finite element method is employed for space approximation. In addition, the four-point Gauss−Legendre quadrature is applied to evaluate the source term. The stability of the proposed scheme is discussed by the Von Neumann method, which verifies that the scheme is unconditionally stable. L 2 and L ∞ norms are used to check the efficiency and accuracy of the proposed scheme. Computed results are compared with the exact and available methods in the literature, which show the betterment of the proposed method.

Nonlinear Fokker-Planck collision operator in Rosenbluth form for gyrokinetic simulations using discontinuous Galerkin method

A gyroaveraged nonlinear collision operator is formulated based on the Fokker-Planck operator in the Rosenbluth-MacDonald-Judd (RMJ) potential form and implemented for the gyrokinetic simulations with the discontinuous Galerkin scheme. The divergence structure of the original RMJ form is carefully preserved throughout the formulation to guarantee the density conservation while neglecting the finite Larmor radius effect. The B-spline finite element method is used to calculate the Rosenbluth potentials for the nonlinear collision operator. In addition to the nonlinear collision operator, linear and Dougherty collision models are also implemented to assess the benefits and drawbacks of each model. For the conservation of the parallel momentum and energy, we adopt a simple advection-diffusion model which numerically enforces the conservation of physical quantities. From bump-on-tail relaxation tests, the monotonically increasing entropy in time and conservation properties are demonstrated for the developed collision operator. Also, a few theoretical predictions for the neoclassical physics such as the neoclassical heat flux, poloidal flow and collisional damping of zonal flow are successfully reproduced by numerical simulations.

An extensive analytical and numerical study of the generalized [formula omitted]-deformed Sinh-Gordon equation

In this paper, the q-deformed Sinh-Gordon equation is solved analytically using a new general form based on the extended tanh approach. The numerical solutions of the equation is obtained using a b-spline finite element method. Also, we present numerous figures to demonstrate the various solitons propagation patterns. This type of equation has not been previously dealt with in such ways, whether analytical or numerical. This study is very useful in studying several physical systems that have lost their symmetry.

Numerical study of non-linear waves for one-dimensional planar, cylindrical and spherical flow using B-spline finite element method

<abstract> <p>In a recent study, an evolution equation is found for waves' behavior at far-field with relaxation mode of molecules. An analytical technique was used to solve this evolution problem, which is a generalized Burger equation. The analytical approach has limitations and requires a very accurate initial guess by a trial method. In this paper, the evolution equation for one-dimensional planar, cylindrical, and spherical flow in the presence of relaxation mode is solved using a collocation approach with a cubic B-spline function. The numerical results are graphed and compared with the exact solution for planar flow. The obtained numerical results match the exact solution quite well and show that the technique is quite reliable and can deal with the nonlinearity involved in the present problem. Results have also been obtained for cylindrical and spherical flow at the far-field. The obtained numerical results show that the present approach with the cubic B-spline function works well and accurately. Fourier stability analysis is used to investigate the stability of the cubic B-spline collocation method.</p> </abstract>

Two efficient numerical methods for solving Rosenau-KdV-RLW equation

In this study, two efficient numerical schemes based on B-spline finite element method (FEM) and time-splitting methods for solving Rosenau-KdV-RLW equation are presented. In the first method, the equation is solved by cubic B-spline Galerkin FEM. For the second method, after splitting Rosenau-KdV-RLW equation in time, it is solved by Strang timesplitting technique using cubic B-spline Galerkin FEM. The differential equation system in the methods is solved by the fourth-order Runge-Kutta method. The stability analysis of the methods is performed. Both methods are applied to an example. The obtained numerical results are compared with some methods available in the literature via the error norms and , convergence rates, and mass and energy conservation constants. The present results are found to be consistent with the compared ones.

Numerical investigation of the solutions of Schrödinger equation with exponential cubic B-spline finite element method

Abstract In this paper, we investigate the numerical solutions of the cubic nonlinear Schrödinger equation via the exponential cubic B-spline collocation method. Crank–Nicolson formulas are used for time discretization of the target equation. A linearization technique is also employed for the numerical purpose. Four numerical examples related to single soliton, collision of two solitons that move in opposite directions, the birth of standing and mobile solitons and bound state solution are considered as the test problems. The accuracy and the efficiency of the purposed method are measured by max error norm and conserved constants. The obtained results are compared with the possible analytical values and those in some earlier studies.

Flow and heat transfer coupling analysis of oil-gas two-phase fluid in clearance of single screw compressor based on fuzzy curvelet finite element method

In order to analyze the flow and heat transfer coupling rules of oil-gas two-phase fluid in clearance of single screw compressor, the fuzzy curvelet finite element model is constructed based on fuzzy theory, curvelet transform and finite element method. Firstly, flow and heat transfer coupling model of oil-gas two-phase fluid in clearance of single screw compressor is constructed according to the real characteristics of single screw compressor, and then the flow and heat transfer simulation of oil-gas two-phase fluid in clearance of single screw compressor is carried out based on fuzzy curvelet finite element method, fuzzy B-spline finite element method and traditional finite element, and results show that the proposed fuzzy curvelet finite element method can improve the computing effectiveness of flow and heat transfer of oil-gas two-phase flow in clearance of single screw compressor. The changing rules of maximum gas phase volume fraction are obtained, and the effect of inlet end face angle on outlet temperature of oil gas two phase fluid is obtained. In addition the influence of various turbulent models on the computing precision has been carried out.

UMERICAL SOLUTION OF THE PROBLEM OF BEAM ANALYSIS WITH THE USE OF B-SPLINE FINITE ELEMENT METHOD

Numerical solution of the problem of beam analysis (bending analysis of the Bernoulli beam) with the use of B-spline finiteelement method is under consideration in the distinctive paper. The original continual and finiteelement formulations of the problem are given, some actual aspects of construction of normalized basis functions of a B-spline are considered, the corresponding local constructions for an arbitrary finiteelement are described, some information about the numerical implementation and an example of analysis are presented.

Solving time fractional Burgers\u2019 and Fisher\u2019s equations using cubic B-spline approximation method

This article presents a numerical algorithm for solving time fractional Burgers’ and Fisher’s equations using cubic B-spline finite element method. The L1 formula with Caputo derivative is used to discretized the time fractional derivative, whereas the Crank–Nicolson scheme based on cubic B-spline functions is used to interpolate the solution curve along the spatial grid. The numerical scheme has been implemented on three test problems. The obtained results indicate that the proposed method is a good option for solving nonlinear fractional Burgers’ and Fisher’s equations. The error norms L_{2} and L_{infty } have been calculated to validate the efficiency and accuracy of the presented algorithm.

A B-spline finite element method for solving a class of nonlinear parabolic equations modeling epitaxial thin-film growth with variable coefficient

In this paper, we propose an efficient B-spline finite element method for a class of fourth order nonlinear differential equations with variable coefficient. For the temporal discretization, we choose the Crank–Nicolson scheme. Boundedness and error estimates are rigorously derived for both semi-discrete and fully discrete schemes. A numerical experiment confirms our theoretical analysis.

B-Spline FEM for Time-Harmonic Acoustic Scattering and Propagation

We study the application of a B-splines Finite Element Method (FEM) to time-harmonic scattering acoustic problems. The infinite space is truncated by a fictitious boundary and second-order Absorbing Boundary Conditions (ABCs) are applied. The truncation error is included in the exact solution so that the reported error is an indicator of the performance of the numerical method, in particular of the size of the pollution error. Numerical results performed with high-order basis functions (third or fourth order) showed no visible pollution error even for very high frequencies. To prove the ability of the method to increase its accuracy in the high frequency regime, we show how to implement a high-order Padé-type ABC on the fictitious outer boundary. The above-mentioned properties combined with exact geometrical representation make B-Spline FEM a very promising platform to solve high-frequency acoustic problems.

A B-spline finite element method for nonlinear differential equations describing crystal surface growth with variable coefficient

In this paper, an efficient finite element scheme is presented for a class of fourth-order nonlinear parabolic problems with variable coefficient. To deal with second-order term in weak formulation, we choose the cubic B-spline function as a trial function. Rigorous error estimates are derived for both semi-discrete and fully-discrete schemes. We provide a numerical example to confirm our theoretical results.

Approximate solution of linear Volterra integro-differential equation by using cubic B-spline finite element method in the complex plane

So far, there are no any publications for solving and obtaining a numerical solution of Volterra integro-differential equations in the complex plane by using the finite element method. In this work, we use the linear B-spline finite element method (LBS-FEM) and cubic B-spline finite element method (CBS-FEM) for solving this equation in the complex plane. We also discuss the error and convergence of the method. Furthermore, we give some numerical examples to substantiate efficiency of the proposed method.

Quartic trigonometric B-spline algorithm for numerical solution of the regularized long wave equation

In this paper, an application of the quartic trigonometric B-spline finite element method is used to solve the regularized long wave equation numerically. This approach involves a Galerkin method based on the quartic trigonometric B-spline function in space discretization together with second and fourth order schemes in time discretization. The accuracy of the proposed methods are demonstrated by test problems and numerical results are compared with the exact solution and some previous results.

Quadratic B-spline finite element method for a rotating nonuniform Euler–Bernoulli beam

In this article, we solve the free vibration problem of a rotating non-uniform Euler-Bernoulli beam using the quadratic B-spline finite element method. The Galerkin method is used to obtain the weak form of the problem. The quadratic B-spline approximation provides the required continuity. This approximation yields the mass and stiffness matrices, which are half the size of the matrices obtained by the conventional finite element approximation. The resulting polynomials are quadratic while the Hermite polynomials are cubic. The mass and the stiffness matrix are derived. Results are matched with the published literature and compared with the conventional finite element results. A non-uniform approximation is used as well to observe an impact of the centrifugal force in numerical solution. Results are found to be more accurate with the non-uniform B-spline approximation for the first three natural frequencies of a rotating beam. Since the centrifugal force makes a difference to the first few modes, it is a desirable outcome.

Finite element based hybrid techniques for advection-diffusion-reaction processes

In this paper, numerical solutions of the advection-diffusion-reaction (ADR) equation are investigated using the Galerkin, collocation and Taylor-Galerkin cubic B-spline finite element method in strong form of spatial elements using an ?-family optimization approach for time variation. The main objective of this article is to capture effective results of the finite element techniques with B-spline basis functions under the consideration of the ADR processes. All produced results are compared with the exact solution and the literature for various versions of problems including pure advection, pure diffusion, advection-diffusion, and advection-diffusion-reaction equations. It is proved that the present methods have good agreement with the exact solution and the literature.

An application of cubic B-Spline finite element method for the Burgers` equation

It is difficult to achieve exact solution of non-linear PDE, directly. Sometimes, it is possible to convert non-linear PDE into equivalent linear PDE by applying a convenient transformation. Hence, Burgers? equation replaces with heat equation by means of the Hope-Cole transformation. In this study, Burgers? equation was converted to a set of non-linear ODE by keeping non-linear structure of Burgers? equation. In this case, solutions for each of the non-linear ODE were obtained by the help of the cubic B-spline finite element method. Model problems were considered to verify the efficiency of this method. Agreement of the solutions was shown with graphics and tables.