scholarly journals B-Spline Method of Lines for Simulation of Contaminant Transport in Groundwater

Water ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1607
Author(s):  
Ersin Bahar ◽  
Gurhan Gurarslan
Keyword(s):  
Contaminant Transport ◽  
Numerical Solutions ◽  
Method Of Lines ◽  
Time Integration ◽  
Good Alternative ◽  
Spline Functions ◽  
B Spline ◽  
Advection Dispersion ◽  
The Method Of Lines ◽  
Transport Problems

In this study, we propose a new numerical method, which can be effectively applied to the advection-dispersion equation, based on B-spline functions and method of lines approach. In the proposed approach, spatial derivatives are calculated using quintic B-spline functions. Thanks to the method of lines approach, the partial differential equation governing the contaminant transport in groundwater is converted into time-dependent ordinary differential equations. After this transformation, the time-integration of this system is realized by using an adaptive Runge–Kutta formula. In order to test the accuracy of the proposed method, four numerical examples were solved and the obtained results compared with various analytical and numerical solutions given in the literature. It is proven that the proposed method is faster and more reliable than other methods referenced herein and is a good alternative for simulation of contaminant transport problems as a result of these comparisons.

An efficient Bi-cubic B-spline ADI method for numerical solutions of two-dimensional unsteady advection diffusion equations

2018 ◽  
Vol 28 (11) ◽  
pp. 2620-2649 ◽  
Author(s):  
Rajni Rohila ◽  
R.C. Mittal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Diffusion Equations ◽  
Spline Functions ◽  
Two Dimensional ◽  
Partial Differential ◽  
Content Type ◽  
B Spline ◽  
Advection Diffusion

Purpose This paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method. Design/methodology/approach A bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids. Findings Bi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also. Originality/value ADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.

scholarly journals Exact and Numerical Solitary Wave Structures to the Variant Boussinesq System

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1473 ◽  
Author(s):  
Abdulghani Alharbi ◽  
Mohammed B. Almatrafi
Keyword(s):  
Solitary Wave ◽  
Evolution Equations ◽  
Numerical Solutions ◽  
Nonlinear Evolution ◽  
Numerical Technique ◽  
Method Of Lines ◽  
Nonlinear Evolution Equations ◽  
Boussinesq System ◽  
Solitary Wave Solutions ◽  
The Method Of Lines

Solutions such as symmetric, periodic, and solitary wave solutions play a significant role in the field of partial differential equations (PDEs), and they can be utilized to explain several phenomena in physics and engineering. Therefore, constructing such solutions is significantly essential. This article concentrates on employing the improved exp(−ϕ(η))-expansion approach and the method of lines on the variant Boussinesq system to establish its exact and numerical solutions. Novel solutions based on the solitary wave structures are obtained. We present a comprehensible comparison between the accomplished exact and numerical results to testify the accuracy of the used numerical technique. Some 3D and 2D diagrams are sketched for some solutions. We also investigate the L2 error and the CPU time of the used numerical method. The used mathematical tools can be comfortably invoked to handle more nonlinear evolution equations.

THE BOUNDARY FACE METHOD WITH VARIABLE APPROXIMATION BY B-SPLINE BASIS FUNCTION

2012 ◽  
Vol 09 (01) ◽  
pp. 1240009 ◽  
Author(s):  
JINLIANG GU ◽  
JIANMING ZHANG ◽  
XIAOMIN SHENG
Keyword(s):  
Numerical Solutions ◽  
Basis Functions ◽  
Spline Functions ◽  
Well Performance ◽  
Geometric Information ◽  
B Spline ◽  
Numerical Tests ◽  
Boundary Integration ◽  
Spline Basis ◽  
Boundary Face

B-spline basis functions as a new approximation method is introduced in the boundary face method (BFM) to obtain numerical solutions of 3D potential problems. In the BFM, both boundary integration and variable approximation are performed in the parametric spaces of the boundary surfaces, therefore, keeps the exact geometric information of a body in which the problem is defined. In this paper, local bivariate B-spline functions are proposed to alleviate the influence of B-spline tensor product that will deteriorate the exactness of numerical results. Numerical tests show that the new method has well performance in both exactness and convergence.

scholarly journals Black hole initial data by numerical integration of the parabolic–hyperbolic form of the constraints

2021 ◽  
Author(s):  
Anna Nakonieczna ◽  
Łukasz Nakonieczny ◽  
István Rácz
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Initial Data ◽  
Fourth Order ◽  
Main Part ◽  
Strong Field ◽  
Method Of Lines ◽  
Time Integration ◽  
The Method Of Lines ◽  
Hyperbolic Form

The parabolic–hyperbolic form of the constraints is integrated numerically. The applied numerical stencil is fourth-order accurate (in the spatial directions) while “time”-integration is made by using the method of lines with a fourth-order order accurate Runge–Kutta scheme. The proper implementation of the applied numerical method is verified by convergence tests and monitoring the relative and absolute errors is determined by comparing numerically and analytically known solutions of the constraints involving boosted and spinning vacuum single black hole configurations. The main part of our investigations is, however, centered on the construction of initial data for distorted black holes which, in certain cases, have non-negligible gravitational wave content. Remarkably, the applied new method is unprecedented in that it allows to construct initial data for highly boosted and spinning black holes, essentially for the full physical allowed ranges of these parameters. In addition, the use of the evolutionary form of the constraints is free from applying any sort of boundary conditions in the strong field regime.

Numerical solutions of two-dimensional Burgers’ equations using modified Bi-cubic B-spline finite elements

2015 ◽  
Vol 32 (5) ◽  
pp. 1275-1306 ◽  
Author(s):  
R C Mittal ◽  
Amit Tripathi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Spline Functions ◽  
Parabolic Partial Differential Equations ◽  
Two Dimensional ◽  
Partial Differential ◽  
Content Type ◽  
B Spline ◽  
Burgers Equations

Purpose – The purpose of this paper is to develop an efficient numerical scheme for non-linear two-dimensional (2D) parabolic partial differential equations using modified bi-cubic B-spline functions. As a test case, method has been applied successfully to 2D Burgers equations. Design/methodology/approach – The scheme is based on collocation of modified bi-cubic B-Spline functions. The authors used these functions for space variable and for its derivatives. Collocation form of the partial differential equation results into system of first-order ordinary differential equations (ODEs). The obtained system of ODEs has been solved by strong stability preserving Runge-Kutta method. The computational complexity of the method is O(p log(p)), where p denotes total number of mesh points. Findings – Obtained numerical solutions are better than those available in literature. Ease of implementation and very small size of computational work are two major advantages of the present method. Moreover, this method provides approximate solutions not only at the grid points but also at any point in the solution domain. Originality/value – First time, modified bi-cubic B-spline functions have been applied to non-linear 2D parabolic partial differential equations. Efficiency of the proposed method has been confirmed with numerical experiments. The authors conclude that the method provides convergent approximations and handles the equations very well in different cases.

Efficient modeling of shallow water equations using method of lines and artificial viscosity

2019 ◽  
Vol 34 (04) ◽  
pp. 2050051
Author(s):  
Mohamed M. Mousa ◽  
Wen-Xiu Ma
Keyword(s):  
Shallow Water ◽  
Numerical Solutions ◽  
Method Of Lines ◽  
Artificial Viscosity ◽  
Water Model ◽  
Numerical Schemes ◽  
Von Neumann ◽  
The Method Of Lines ◽  
Shallow Water Models ◽  
Von Neumann Stability

In this work, two numerical schemes were developed to overcome the problem of shock waves that appear in the solutions of one/two-layer shallow water models. The proposed numerical schemes were based on the method of lines and artificial viscosity concept. The robustness and efficiency of the proposed schemes are validated on many applications such as dam-break problem and the problem of interface propagation of two-layer shallow water model. The von Neumann stability of proposed schemes is studied and hence, the sufficient condition for stability is deduced. The results were presented graphically. The verification of the obtained results is achieved by comparing them with exact solutions or another numerical solutions founded in literature. The results are satisfactory and in much have a close agreement with existing results.

scholarly journals Study of 2D contaminant transport with depth varying input source in a groundwater reservoir

2021 ◽  
Author(s):  
Mritunjay Kumar Singh ◽  
Sohini Rajput ◽  
Rakesh Kumar Singh
Keyword(s):  
Numerical Solution ◽  
Contaminant Transport ◽  
Numerical Solutions ◽  
Disposal Site ◽  
Biological Degradation ◽  
Transform Method ◽  
Soil Medium ◽  
Advection Dispersion ◽  
Groundwater Reservoir ◽  
Input Source

Abstract This study deals with a two-dimensional (2D) contaminant transport problem subject to depth varying input source in a finite homogeneous groundwater reservoir. A depth varying input source at the upstream boundary is assumed as the location of disposal site of the pollutant from where the contaminant enters the soil medium and ultimately to the groundwater reservoir. At the extreme boundary of the flow site, the concentration gradient of the contaminant is assumed to be zero. Contaminant dispersion is considered along the horizontal and vertical directions of the groundwater flow. The governing transport equation is the advection–dispersion equation (ADE) associated with linear sorption and first-order biological degradation. The ADE is solved analytically by adopting Laplace transform method. Crank–Nicolson scheme is also adopted for the numerical simulation of the modelled problem. In the graphical comparison of the analytical and numerical solutions, the numerical solution follows very closely with the analytical solution. Also, Root Mean Square (RMS) error and CPU run time are obtained to account for the performance of the numerical solution.

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