Effective numerical technique applied for Burgers' equation of (1 + 1)‐, (2 + 1)‐dimensional, and coupled forms

2021 ◽  
Author(s):  
Norhan A. Mohamed ◽  
Ahmed S. Rashed ◽  
Ammar Melaibari ◽  
Hamid M. Sedighi ◽  
Mohamed A. Eltaher
Keyword(s):  
Burgers Equation ◽  
Numerical Technique

scholarly journals Numerical Techniques for Unsteady Nonlinear Burgers Equation Based on Backward Differentiation Formulas

2018 ◽  
Vol 7 (3) ◽  
pp. 171-181 ◽  
Author(s):  
Vijitha Mukundan ◽  
Ashish Awasthi
Keyword(s):  
Numerical Solutions ◽  
Burgers Equation ◽  
Numerical Technique ◽  
System Stability ◽  
Test Problems ◽  
Numerical Techniques ◽  
Ode System ◽  
Backward Differentiation Formulas ◽  
Nonlinear Ode ◽  
Differentiation Formulas

AbstractWe introduce new numerical techniques for solving nonlinear unsteady Burgers equation. The numerical technique involves discretization of all variables except the time variable which converts nonlinear PDE into nonlinear ODE system. Stability of the nonlinear system is verified using Lyapunov’s stability criteria. Implicit stiff solvers backward differentiation formula of order one, two and three are used to solve the nonlinear ODE system. Four test problems are included to show the applicability of introduced numerical techniques. Numerical solutions so obtained are compared with solutions of existing schemes in literature. The proposed numerical schemes are found to be simple, accurate, fast, practical and superior to some existing methods.

scholarly journals An efficient numerical technique for solving time fractional Burgers equation

2020 ◽  
Vol 59 (4) ◽  
pp. 2201-2220 ◽  
Author(s):  
Tayyaba Akram ◽  
Muhammad Abbas ◽  
Muhammad Bilal Riaz ◽  
Ahmad Izani Ismail ◽  
Norhashidah Mohd. Ali
Keyword(s):  
Burgers Equation ◽  
Numerical Technique

scholarly journals Spectrally accurate approximate solutions and convergence analysis of fractional Burgers’ equation

2020 ◽  
Vol 9 (3) ◽  
pp. 633-644
Author(s):  
A. K. Mittal
Keyword(s):  
Numerical Solutions ◽  
Burgers Equation ◽  
Numerical Technique ◽  
Time Derivative ◽  
Pseudospectral Method ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Time And Space ◽  
Time Space ◽  
Coupled Burgers Equation

Abstract In this paper, a new numerical technique implements on the time-space pseudospectral method to approximate the numerical solutions of nonlinear time- and space-fractional coupled Burgers’ equation. This technique is based on orthogonal Chebyshev polynomial function and discretizes using Chebyshev–Gauss–Lobbato (CGL) points. Caputo–Riemann–Liouville fractional derivative formula is used to illustrate the fractional derivatives matrix at CGL points. Using the derivatives matrices, the given problem is reduced to a system of nonlinear algebraic equations. These equations can be solved using Newton–Raphson method. Two model examples of time- and space-fractional coupled Burgers’ equation are tested for a set of fractional space and time derivative order. The figures and tables show the significant features, effectiveness, and good accuracy of the proposed method.

scholarly journals Explicit Exponential Finite Difference Scheme for 1D Navier-Stokes Equation with Time Dependent Pressure Gradient

2017 ◽  
Vol 36 ◽  
pp. 79-90
Author(s):  
MAK Azad ◽  
LS Andallah
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Stokes Equation ◽  
Finite Difference Scheme ◽  
Burgers Equation ◽  
Numerical Technique ◽  
Numerical Results ◽  
Time Dependent ◽  
Navier Stokes ◽  
Navier Stokes Equation

In this paper, numerical technique for solving the one-dimensional (1D) unsteady, incompressible Navier-Stokes equation (NSE) is presented. The governing time dependent non-linear partial equation is reduced to non-linear partial differential equation named as viscous Burgers’ equation by introducing Orlowski and Sobczyk transformation (OST). An explicit exponential finite difference scheme (Expo FDS) has been used for solving reduced 1D NSE. The accuracy of the method has been illustrated by taking two numerical examples. Results are compared with the analytical solutions and those obtained based on the numerical results of reduced 1D NSE as Burgers’ equation. The accuracy and numerical feature of convergence of the Expo FDS is presented by estimating their error norms. Excellent numerical results indicate that the proposed numerical technique is efficient admissible with efficient accuracy for the numerical solutions of the NSE.GANIT J. Bangladesh Math. Soc.Vol. 36 (2016) 79-90

HIGHER ORDER BURGERS EQUATION

1986 ◽  
Vol 6 (3) ◽  
pp. 353-360 ◽  
Author(s):  
Mingliang Wang
Keyword(s):  
Burgers Equation ◽  
Higher Order

A Novel Approach for Thermomechanical Analysis of Stationary Rolling Tires within an ALE–Kinematic Framework

2013 ◽  
Vol 41 (3) ◽  
pp. 174-195 ◽  
Author(s):  
Anuwat Suwannachit ◽  
Udo Nackenhorst
Keyword(s):  
Constitutive Model ◽  
Numerical Solutions ◽  
Numerical Technique ◽  
Thermomechanical Analysis ◽  
Rolling Contact ◽  
Computational Technique ◽  
Heat Equations ◽  
Arbitrary Lagrangian Eulerian ◽  
Material Particle ◽  
Operator Split

ABSTRACT A new computational technique for the thermomechanical analysis of tires in stationary rolling contact is suggested. Different from the existing approaches, the proposed method uses the constitutive description of tire rubber components, such as large deformations, viscous hysteresis, dynamic stiffening, internal heating, and temperature dependency. A thermoviscoelastic constitutive model, which incorporates all the mentioned effects and their numerical aspects, is presented. An isentropic operator-split algorithm, which ensures numerical stability, was chosen for solving the coupled mechanical and energy balance equations. For the stationary rolling-contact analysis, the constitutive model presented and the operator-split algorithm are embedded into the Arbitrary Lagrangian Eulerian (ALE)–relative kinematic framework. The flow of material particles and their inelastic history within the spatially fixed mesh is described by using the recently developed numerical technique based on the Time Discontinuous Galerkin (TDG) method. For the efficient numerical solutions, a three-phase, staggered scheme is introduced. First, the nonlinear, mechanical subproblem is solved using inelastic constitutive equations. Next, deformations are transferred to the subsequent thermal phase for the solution of the heat equations concerning the internal dissipation as a source term. In the third step, the history of each material particle, i.e., each internal variable, is transported through the fixed mesh corresponding to the convective velocities. Finally, some numerical tests with an inelastic rubber wheel and a car tire model are presented.

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