Shock waves analysis of planar and non planar nonlinear Burgers’ equation using Scale-2 Haar wavelets

2017 ◽  
Vol 27 (8) ◽  
pp. 1814-1850 ◽  
Author(s):  
Sapna Pandit ◽  
Manoj Kumar ◽  
R.N. Mohapatra ◽  
Ali Saleh Alshomrani
Keyword(s):  
Nonlinear Term ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Finite Difference Approximation ◽  
Haar Wavelets ◽  
Difference Approximation ◽  
Content Type ◽  
Gauss Elimination ◽  
Gauss Elimination Method ◽  
Better Than

Purpose This paper aims to find the numerical solution of planar and non-planar Burgers’ equation and analysis of the shock behave. Design/methodology/approach First, the authors discritize the time-dependent term using Crank–Nicholson finite difference approximation and use quasilinearization to linearize the nonlinear term then apply Scale-2 Haar wavelets for space integration. After applying this scheme on partial differential, the equation transforms into a system of algebraic equation. Then, the system of equation is solved using Gauss elimination method. Findings Present method is the extension of the method (Jiwari, 2012). The numerical solutions using Scale-2 Haar wavelets prove that the proposed method is reliable for planar and non-planar nonlinear Burgers’ equation and yields results better than other methods and compatible with the exact solutions. Originality/value The numerical results for non-planar Burgers’ equation are very sparse. In the present paper, the authors identify where the shock wave and discontinuity occur in planar and non-planar Burgers’' equation.

Unit Mobility Ratio Displacement Calculations for Pattern Floods in Homogeneous Medium

1966 ◽  
Vol 6 (03) ◽  
pp. 217-227 ◽  
Author(s):  
Hubert J. Morel-Seytoux
Keyword(s):  
Oil Recovery ◽  
Numerical Solutions ◽  
Mobility Ratio ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Numerical Techniques ◽  
Sweep Efficiency ◽  
Closed Form Solutions ◽  
Regular Patterns ◽  
Quantities Of Interest

Abstract The influence of pattern geometry on assisted oil recovery for a particular displacement mechanism is the object of investigation in this paper. The displacement is assumed to be of unit mobility ratio and piston-like. Fluids are assumed incompressible and gravity and capillary effects are neglected. With these assumptions it is possible to calculate by analytical methods the quantities of interest to the reservoir engineer for a great variety of patterns. Specifically, this paper presentsvery briefly, the methods and mathematical derivations required to obtain the results of engineering concern, andtypical results in the form of graphs or formulae that can be used readily without prior study of the methods. Results of this work provide checks for solutions obtained from programmed numerical techniques. They also reveal the effect of pattern geometry and, even though the assumptions of piston-like displacement and of unit mobility ratio are restrictive, they can nevertheless be used for rather crude but quick, cheap estimates. These estimates can be refined to account for non-unit mobility ratio and two-phase flow by correlating analytical results in the case M=1 and the numerical results for non-Piston, non-unit mobility ratio displacements. In an earlier paper1 it was also shown that from the knowledge of closed form solutions for unit mobility ratio, quantities called "scale factors" could be readily calculated, increasing considerably the flexibility of the numerical techniques. Many new closed form solutions are given in this paper. INTRODUCTION BACKGROUND Pattern geometry is a major factor in making water-flood recovery predictions. For this reason many numerical schemes have been devised to predict oil recovery in either regular patterns or arbitrary configurations. The numerical solutions, based on the method of finite difference approximation, are subject to errors often difficult to evaluate. An estimate of the error is possible by comparison with exact solutions. To provide a variety of checks on numerical solutions, a thorough study of the unit mobility ratio displacement process was undertaken. To calculate quantities of interest to the reservoir engineer (oil recovery, sweep efficiency, etc.), it is necessary to first know the pressure distribution in the pattern. Then analytical procedures are used to calculate, in order of increasing difficulty: injectivity, breakthrough areal sweep efficiency, normalized oil recovery and water-oil ratio as a function of normalized PV injected. BACKGROUND Pattern geometry is a major factor in making water-flood recovery predictions. For this reason many numerical schemes have been devised to predict oil recovery in either regular patterns or arbitrary configurations. The numerical solutions, based on the method of finite difference approximation, are subject to errors often difficult to evaluate. An estimate of the error is possible by comparison with exact solutions. To provide a variety of checks on numerical solutions, a thorough study of the unit mobility ratio displacement process was undertaken. To calculate quantities of interest to the reservoir engineer (oil recovery, sweep efficiency, etc.), it is necessary to first know the pressure distribution in the pattern. Then analytical procedures are used to calculate, in order of increasing difficulty: injectivity, breakthrough areal sweep efficiency, normalized oil recovery and water-oil ratio as a function of normalized PV injected.

scholarly journals Viscoelastic Fluid over a Stretching Sheet with Electromagnetic Effects and Nonuniform Heat Source/Sink

2010 ◽  
Vol 2010 ◽  
pp. 1-14 ◽  
Author(s):  
Kai-Long Hsiao
Keyword(s):  
Heat Transfer ◽  
Heat Source ◽  
Viscoelastic Fluid ◽  
Stretching Sheet ◽  
Numerical Solutions ◽  
Transfer Effects ◽  
Gauss Elimination ◽  
Heat Transfer Effects ◽  
Source Sink ◽  
Gauss Elimination Method

A magnetic hydrodynamic (MHD) of an incompressible viscoelastic fluid over a stretching sheet with electric and magnetic dissipation and nonuniform heat source/sink has been studied. The buoyant effect and the electric numberE1couple with magnetic parameterMto represent the dominance of the electric and magnetic effects, and adding the specific item of nonuniform heat source/sink is presented in governing equations which are the main contribution of this study. The similarity transformation, the finite-difference method, Newton method, and Gauss elimination method have been used to analyze the present problem. The numerical solutions of the flow velocity distributions, temperature profiles, and the important wall unknown values off''(0)andθ'(0)have been carried out. The parameter Pr,E1, orEccan increase the heat transfer effects, but the parameterMorA*may decrease the heat transfer effects.

A numerical algorithm based on scale-3 Haar wavelets for fractional advection dispersion equation

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Sapna Pandit ◽  
R.C. Mittal
Keyword(s):  
Haar Wavelet ◽  
Haar Wavelets ◽  
System Control ◽  
Partial Differential ◽  
Content Type ◽  
Advection Dispersion ◽  
Novel Approach ◽  
Gauss Elimination ◽  
Time Variables ◽  
First Time

Purpose This paper aims to propose a novel approach based on uniform scale-3 Haar wavelets for unsteady state space fractional advection-dispersion partial differential equation which arises in complex network, fluid dynamics in porous media, biology, chemistry and biochemistry, electrode – electrolyte polarization, finance, system control, etc. Design/methodology/approach Scale-3 Haar wavelets are used to approximate the space and time variables. Scale-3 Haar wavelets converts the problems into linear system. After that Gauss elimination is used to find the wavelet coefficients. Findings A novel algorithm based on Haar wavelet for two-dimensional fractional partial differential equations is established. Error estimation has been derived by use of property of compactly supported orthonormality. The correctness and effectiveness of the theoretical arguments by numerical tests are confirmed. Originality/value Scale-3 Haar wavelets are used first time for these types of problems. Second, error analysis in new work in this direction.

General Deformations of Neo-Hookean Membranes

1973 ◽  
Vol 40 (1) ◽  
pp. 7-12 ◽  
Author(s):  
W. H. Yang ◽  
C. H. Lu
Keyword(s):  
Differential Equations ◽  
Strain Energy ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Strain Energy Function ◽  
Finite Deformations ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Special Cases ◽  
Convergent Algorithm

A set of three nonlinear partial-differential equations is derived for general finite deformations of a thin membrane. The material that composes the membrane is assumed to be hyperelastic. Its mechanical property is represented by the neo-Hookean strain-energy function. The equations reduce to special cases known in the literature. A fast convergent algorithm is developed. The numerical solutions to the finite-difference approximation of the differential equations are computed iteratively with a trivial initial iterant. As an example, the problem of inflating a rectangular membrane with fixed edges by a uniform pressure applied on one side is presented. The solutions and their convergence are displayed and discussed.

Application of a Hybrid Method to the Solution of the Nonlinear Burgers’ Equation

2003 ◽  
Vol 70 (6) ◽  
pp. 926-929 ◽  
Author(s):  
Bor-Lih Kuo ◽  
Chao-Kuang Chen
Keyword(s):  
Reynolds Number ◽  
Numerical Methods ◽  
Finite Difference ◽  
Hybrid Method ◽  
Burgers Equation ◽  
Initial Conditions ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Approximation Techniques ◽  
Good Agreement

This paper presents the use of a hybrid method which combines differential transformation and finite difference approximation techniques in the solution of the nonlinear Burgers’ equation for various values of Reynolds number including high values. In order to demonstrate the accuracy and validity of the proposed method, it is used to solve several examples of Burgers’ equation, with each example having different initial conditions and boundary conditions. It is found that the results obtained are in good agreement with the analytical solutions, and that the results are more accurate than those provided by other approximate numerical methods.

A numerical method based on Haar wavelets for the Hadamard-type fractional differential equations

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Zain ul Abdeen ◽  
Mujeeb ur Rehman
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Fractional Differential Equations ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
Haar Wavelets ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Content Type ◽  
Fractional Differential

PurposeThe purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear Hadamard-type fractional differential equations.Design/methodology/approachThe aim of this paper is to develop a numerical scheme for numerical solutions of Hadamard-type fractional differential equations. The classical Haar wavelets are modified to align them with Hadamard-type operators. Operational matrices are derived and used to convert differential equations to systems of algebraic equations.FindingsThe upper bound for error is estimated. With the help of quasilinearization, nonlinear problems are converted to sequences of linear problems and operational matrices for modified Haar wavelets are used to get their numerical solution. Several numerical examples are presented to demonstrate the applicability and validity of the proposed method.Originality/valueThe numerical method is purposed for solving Hadamard-type fractional differential equations.

Analysis on the steady state performance of a multi pad externally adjustable fluid film bearing

2019 ◽  
Vol 71 (6) ◽  
pp. 803-809 ◽  
Author(s):  
Girish Hariharan ◽  
Raghuvir Pai
Keyword(s):  
Steady State ◽  
Reynolds Equation ◽  
Load Capacity ◽  
Fluid Film ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Content Type ◽  
Bearing Load ◽  
Performance Capability ◽  
Film Profile

Purpose This study aims to investigate the performance characteristics of an externally adjustable bearing with multiple pads in steady state conditions. The proposed adjustable bearing geometry can effectively control the hydrodynamic operation in bearing clearances by adjusting the pads in radial and tilt directions. These pad adjustments have a significant role in improving the bearing characteristics such as load capacity, attitude angle, side leakage, friction variable and Sommerfeld number, which will be analysed in this paper. Design/methodology/approach The adjustable bearing is designed with circumferentially spaced four bearing pads subjected to similar radial and tilt adjustments. Tilt angles are applied along the leading edges of bearing pads. A modified film thickness equation is used to incorporate the pad adjustments and accurately predict the variation in film profile. Finite difference approximation is adopted to solve the Reynolds equation and discretize the fluid film domain. Findings For negative radial and tilt adjustments, higher hydrodynamic pressures are generated in bearing clearances, which increases the bearing load capacity at different eccentricity ratios. From comparative analysis for different pad adjustments, superior bearing performance is observed for bearing pads under negative radial and negative tilt adjustments. Originality/value This research presents a detailed theoretical approach to analyse the performance capability of a four pad adjustable bearing geometry, which is not available in literatures. Improved bearing performances with negative pad adjustments can attract bearing designers to implement the proposed adjustability-bearing concept in rotating machineries.

A quartic trigonometric tension b-spline algorithm for nonlinear partial differential equation system

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Özlem Ersoy Hepson
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Higher Order ◽  
Spline Collocation ◽  
Partial Differential ◽  
Content Type ◽  
B Spline ◽  
B Splines ◽  
Coupled Burgers Equation

Purpose The purpose of this study is to construct quartic trigonometric tension (QTT) B-spline collocation algorithms for the numerical solutions of the Coupled Burgers’ equation. Design/methodology/approach The finite elements method (FEM) is a numerical method for obtaining an approximate solution of partial differential equations (PDEs). The development of high-speed computers enables to development FEM to solve PDEs on both complex domain and complicated boundary conditions. It also provides higher-order approximation which consists of a vector of coefficients multiplied by a set of basis functions. FEM with the B-splines is efficient due both to giving a smaller system of algebraic equations that has lower computational complexity and providing higher-order continuous approximation depending on using the B-splines of high degree. Findings The result of the test problems indicates the reliability of the method to get solutions to the CBE. QTT B-spline collocation approach has convergence order 3 in space and order 1 in time. So that nonpolynomial splines provide smooth solutions during the run of the program. Originality/value There are few numerical methods build-up using the trigonometric tension spline for solving differential equations. The tension B-spline collocation method is used for finding the solution of Coupled Burgers’ equation.

Analytical-Numerical Method in Waterflooding Predictions

1965 ◽  
Vol 5 (03) ◽  
pp. 247-258 ◽  
Author(s):  
Hubert J. Morel-Seytoux
Keyword(s):  
Finite Difference ◽  
Exact Solutions ◽  
Numerical Method ◽  
Analytical Method ◽  
Numerical Solutions ◽  
Numerical Technique ◽  
Scale Factor ◽  
Mobility Ratio ◽  
Finite Difference Approximation ◽  
Difference Approximation

Abstract Methods of predicting the influence of pattern geometry and mobility ratio on waterflooding recovery predictions are discussed. Two methods of calculation are used separately or concurrently. The analytical method yields exact solutions in a convenient form for a unit mobility ratio piston-like displacement. A few typical pressure distributions, sweep efficiencies and oil recoveries are presented for various patterns. For non-unit mobility ratio, one may resort to a numerical method, such as that of Sheldon and Dougherty. Because the domains of applicability of the analytical and numerical techniques overlap, the exact solutions provide estimates of the errors in the numerical procedures. The advantages of the analytical and numerical methods can be combined. To develop a numerical technique as independent of geometry as possible, the physical space is transformed into a standard rectangle. The entire effect of geometry is rendered through one term, the "scale-factor", derived from mapping relations. The scale factor can be calculated from the exact unit-mobility ratio solution for the particular pattern of interest. By this means recovery performances for arbitrary mobility ratio can be obtained for many patterns. A sample of results obtained in this manner is presented. Introduction Pattern geometry and mobility ratio are two major factors in making a waterflood recovery prediction. Because assisted recovery has become increasingly important to the oil industry, pattern configuration and mobility ratio also assume a greater significance in the assessment of the economic value of recovery projects. The influence of pattern geometry and mobility ratio in shaping a recovery curve and on the other quantities of interest to the reservoir engineer is the main subject of this paper. Much effort has already been spent on estimating quantitatively the influence of either pattern or mobility ratio or both on oil recovery. The literature reports many investigations of this nature. However, many results or methods of recovery prediction presented in the literature cannot be considered fully satisfactory. Even for unit mobility ratio and piston-like displacement, where analytical solutions are available, the literature shows discrepancies. For non-unit mobility ratio, the divergence in the results is extreme. For infinite mobility ratio in a repeated five-spot, depending on the investigator, the sweep efficiency ranges from 0 per cent to 60 per cent. With respect to the influence of pattern on recovery, only the repeated five-spot has received much attention. Other confined patterns and pilot configurations have received very little attention. Two calculation methods are presented in this paper, either separately or concurrently: the analytical method of potential theory and the numerical method of finite-difference approximation. The analytical method is more restricted in scope than the finite-difference method, but it has the definite advantage of providing exact solutions within its range of applicability. If a unit-mobility ratio piston-like displacement is assumed, the analytical approach is possible. A few typical results are reported in this paper; the detailed description of the general method and of a great variety of results will be the subject of other articles. For non-unity mobility ratio, we must resort to a numerical scheme. The numerical technique is that which was described by Sheldon and Dougherty. It is not limited to piston-like displacement. However, mainly single interface results will be presented here. Because the respective domains of applicability of the analytical and the numerical method overlap, useful comparisons of exact and numerical solutions can be made for a variety of patterns. The advantages of the analytical and numerical approaches can be combined. SPEJ P. 247ˆ

Quasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynamical systems

2018 ◽  
Vol 35 (5) ◽  
pp. 1907-1931 ◽  
Author(s):  
R.C. Mittal ◽  
Sapna Pandit
Keyword(s):  
Dynamical System ◽  
Numerical Simulation ◽  
Haar Wavelet ◽  
Haar Wavelets ◽  
Content Type ◽  
System A ◽  
Resolution Level ◽  
Gauss Elimination ◽  
First Time ◽  
Fractional Dynamical System

Purpose The main purpose of this work is to develop a novel algorithm based on Scale-3 Haar wavelets (S-3 HW) and quasilinearization for numerical simulation of dynamical system of ordinary differential equations. Design/methodology/approach The first step in the development of the algorithm is quasilinearization process to linearize the problem, and then Scale-3 Haar wavelets are used for space discretization. Finally, the obtained system is solved by Gauss elimination method. Findings Some numerical examples of fractional dynamical system are considered to check the accuracy of the algorithm. Numerical results show that quasilinearization with Scale-3 Haar wavelet converges fast even for small number of collocation points as compared of classical Scale-2 Haar wavelet (S-2 HW) method. The convergence analysis of the proposed algorithm has been shown that as we increase the resolution level of Scale-3 Haar wavelet error goes to zero rapidly. Originality/value To the best of authors’ knowledge, this is the first time that new Haar wavelets Scale-3 have been used in fractional system. A new scheme is developed for dynamical system based on new Scale-3 Haar wavelets. These wavelets take less time than Scale-2 Haar wavelets. This approach extends the idea of Jiwari (2015, 2012) via translation and dilation of Haar function at Scale-3.

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