# Burgers' Equation (burger + equation)

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Distribution within Engineering

## Selected Abstracts

### Improved non-staggered central NT schemes for balance laws with geometrical source terms

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2004
rnjari
Abstract In this paper we extend the non-staggered version of the central NT (Nessyahu,Tadmor) scheme to the balance laws with geometrical source term. This extension is based on the source term evaluation that includes balancing between the flux gradient and the source term with an additional reformulation that depends on the source term discretization. The main property of the scheme obtained by the proposed reformulation is preservation of the particular set of the steady-state solutions. We verify the improved scheme on two types of balance laws with geometrical source term: the shallow water equations and the non-homogeneous Burger's equation. The presented results show good behaviour of the considered scheme when compared with the analytical or numerical results obtained by using other numerical schemes. Furthermore, comparison with the numerical results obtained by the classical central NT scheme where the source term is simply pointwise evaluated shows that the proposed reformulations are essential. Copyright © 2004 John Wiley & Sons, Ltd. [source]

### An accurate integral-based scheme for advection,diffusion equation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2001
Tung-Lin Tsai
Abstract This paper proposes an accurate integral-based scheme for solving the advection,diffusion equation. In the proposed scheme the advection,diffusion equation is integrated over a computational element using the quadratic polynomial interpolation function. Then elements are connected by the continuity of first derivative at boundary points of adjacent elements. The proposed scheme is unconditionally stable and results in a tridiagonal system of equations which can be solved efficiently by the Thomas algorithm. Using the method of fractional steps, the proposed scheme can be extended straightforwardly from one-dimensional to multi-dimensional problems without much difficulty and complication. To investigate the computational performances of the proposed scheme five numerical examples are considered: (i) dispersion of Gaussian concentration distribution in one-dimensional uniform flow; (ii) one-dimensional viscous Burgers equation; (iii) pure advection of Gaussian concentration distribution in two-dimensional uniform flow; (iv) pure advection of Gaussian concentration distribution in two-dimensional rigid-body rotating flow; and (v) three-dimensional diffusion in a shear flow. In comparison not only with the QUICKEST scheme, the fully time-centred implicit QUICK scheme and the fully time-centred implicit TCSD scheme for one-dimensional problem but also with the ADI-QUICK scheme, the ADI-TCSD scheme and the MOSQUITO scheme for two-dimensional problems, the proposed scheme shows convincing computational performances. Copyright © 2001 John Wiley & Sons, Ltd. [source]

### He's homotopy perturbation method for solving Korteweg-de Vries Burgers equation with initial condition

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2010
Mustafa Inc
Abstract In this article, we present the Homotopy Perturbation Method (Shortly HPM) for obtaining the numerical solutions of the Korteweg-de Vries Burgers (KdVB) equation. The series solutions are developed and the reccurance relations are given explicity. The initial approximation can be freely chosen with possibly unknown constants which can be determined by imposing the boundary and initial conditions. The results reveal that HPM is very simple and effective. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 [source]

### A stabilized Hermite spectral method for second-order differential equations in unbounded domains

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2007
Heping Ma
Abstract A stabilized Hermite spectral method, which uses the Hermite polynomials as trial functions, is presented for the heat equation and the generalized Burgers equation in unbounded domains. In order to overcome instability that may occur in direct Hermite spectral methods, a time-dependent scaling factor is employed in the Hermite expansions. The stability of the scheme is examined and optimal error estimates are derived. Numerical experiments are given to confirm the theoretical results.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source]

### Incomplete sensitivities and cost function reformulation leading to multi-criteria investigation of inverse problems

OPTIMAL CONTROL APPLICATIONS AND METHODS, Issue 2 2003
A. Cabot
Abstract This paper deals with the application of typical minimization methods based on dynamical systems to the solution of a characteristic inverse problem. The state equation is based on the Burgers equation. The control is meant to achieve a prescribed state distribution and a given shock location. We show how to use incomplete sensitivities during the minimization process. We also show through a redefinition of the cost function that a multi-criteria problem needs to be considered in inverse problems. This example shows that a correct definition of the minimization problem is crucial and needs to be studied before a direct application of brute force minimization approaches. Copyright © 2003 John Wiley & Sons, Ltd. [source]

### Longitudinal Dust Lattice Shock Wave in a Strongly Coupled Complex Dusty Plasma

CONTRIBUTIONS TO PLASMA PHYSICS, Issue 8 2008
S. Ghosh
Abstract The effect of hydrodynamical damping that arises due to the irreversible processes within the system have been studied on 1D nonlinear longitudinal dust lattice wave (LDLW) in homogeneous strongly coupled complex (dusty) plasma. Analytical investigation shows that the nonlinear wave is governed by Korteweg-de Vries Burgers' equation. This hydrodynamical damping induced dissipative effect is responsible for the Burgers' term that causes the generation of shock wave in dusty plasma crystal. Numerical investigation on the basis of the glow-discharge plasma parameters reveal that LDLW exhibits both oscillatory and monotonic shock. The shock is compressive in nature and its strength decreases (increases) with the increase of the shielding parameter , (characteristic length L). The effects of dust-neutral collision are also discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### An exponentially fitted method for solving Burgers' equation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2009
Turgut Özi
Abstract In this paper, an exponentially fitted method is used to numerically solve the one-dimensional Burgers' equation. The performance of the method is tested on the model involving moderately large Reynolds numbers. The obtained numerical results show that the method is efficient, stable and reliable for solving Burgers' equation accurately even involving high Reynolds numbers for which the exact solution fails. Copyright © 2009 John Wiley & Sons, Ltd. [source]

### A compact finite difference method for solving Burgers' equation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2010
Shusen Xie
Abstract In this paper, a high-order accurate compact finite difference method using the Hopf,Cole transformation is introduced for solving one-dimensional Burgers' equation numerically. The stability and convergence analyses for the proposed method are given, and this method is shown to be unconditionally stable. To demonstrate efficiency, numerical results obtained by the proposed scheme are compared with the exact solutions and the results obtained by some other methods. The proposed method is second- and fourth-order accurate in time and space, respectively. Copyright © 2009 John Wiley & Sons, Ltd. [source]

### On the practical importance of the SSP property for Runge,Kutta time integrators for some common Godunov-type schemes

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2005
David I. Ketcheson
Abstract We investigate through analysis and computational experiment explicit second and third-order strong-stability preserving (SSP) Runge,Kutta time discretization methods in order to gain perspective on the practical necessity of the SSP property. We consider general theoretical SSP limits for these schemes and present a new optimal third-order low-storage SSP method that is SSP at a CFL number of 0.838. We compare results of practical preservation of the TVD property using SSP and non-SSP time integrators to integrate a class of semi-discrete Godunov-type spatial discretizations. Our examples involve numerical solutions to Burgers' equation and the Euler equations. We observe that ,well-designed' non-SSP and non-optimal SSP schemes with SSP coefficients less than one provide comparable stability when used with time steps below the standard CFL limit. Results using a third-order non-TVD CWENO scheme are also presented. We verify that the documented SSP methods with the number of stages greater than the order provide a useful enhanced stability region. We show by analysis and by numerical experiment that the non-oscillatory third-order reconstructions used in (Liu and Tadmor Numer. Math. 1998; 79:397,425, Kurganov and Petrova Numer. Math. 2001; 88:683,729) are in general only second- and first-order accurate, respectively. Copyright © 2005 John Wiley & Sons, Ltd. [source]

### Implicit schemes with large time step for non-linear equations: application to river flow hydraulics

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2004
J. Burguete
Abstract In this work, first-order upwind implicit schemes are considered. The traditional tridiagonal scheme is rewritten as a sum of two bidiagonal schemes in order to produce a simpler method better suited for unsteady transcritical flows. On the other hand, the origin of the instabilities associated to the use of upwind implicit methods for shock propagations is identified and a new stability condition for non-linear problems is proposed. This modification produces a robust, simple and accurate upwind semi-explicit scheme suitable for discontinuous flows with high Courant,Friedrichs,Lewy (CFL) numbers. The discretization at the boundaries is based on the condition of global mass conservation thus enabling a fully conservative solution for all kind of boundary conditions. The performance of the proposed technique will be shown in the solution of the inviscid Burgers' equation, in an ideal dambreak test case, in some steady open channel flow test cases with analytical solution and in a realistic flood routing problem, where stable and accurate solutions will be presented using CFL values up to 100. Copyright © 2004 John Wiley & Sons, Ltd. [source]

### Adaptive control of Burgers' equation with unknown viscosity

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 7 2001
Wei-Jiu Liu
Abstract In this paper, we propose a fortified boundary control law and an adaptation law for Burgers' equation with unknown viscosity, where no a priori knowledge of a lower bound on viscosity is needed. This control law is decentralized, i.e., implementable without the need for central computer and wiring. Using the Lyapunov method, we prove that the closed-loop system, including the parameter estimator as a dynamic component, is globally H1 stable and well posed. Furthermore, we show that the state of the system is regulated to zero by developing an alternative to Barbalat's Lemma which cannot be used in the present situation. Copyright © 2001 John Wiley & Sons, Ltd. [source]

### Energy properties preserving schemes for Burgers' equation,

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2008
R. Anguelov
Abstract The Burgers' equation, a simplification of the Navier,Stokes equations, is one of the fundamental model equations in gas dynamics, hydrodynamics, and acoustics that illustrates the coupling between convection/advection and diffusion. The kinetic energy enjoys boundedness and monotone decreasing properties that are useful in the study of the asymptotic behavior of the solution. We construct a family of non-standard finite difference schemes, which replicate the energy equality and the properties of the kinetic energy. Our approach is based on Mickens' rule [Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994.] of nonlocal approximation of nonlinear terms. More precisely, we propose a systematic nonlocal way of generating approximations that ensure that the trilinear form is identically zero for repeated arguments. We provide numerical experiments that support the theory and demonstrate the power of the non-standard schemes over the classical ones. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source]

### A marker method for the solution of the damped Burgers' equation

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2006
Jerome L. V. Lewandowski
Abstract A new method for the solution of the damped Burgers' equation is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details. The marker method is applicable to a general class of nonlinear dispersive partial differential equations. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 [source]

### Arbitrary discontinuities in space,time finite elements by level sets and X-FEM

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2004
Jack Chessa
Abstract An enriched finite element method with arbitrary discontinuities in space,time is presented. The discontinuities are treated by the extended finite element method (X-FEM), which uses a local partition of unity enrichment to introduce discontinuities along a moving hyper-surface which is described by level sets. A space,time weak form for conservation laws is developed where the Rankine,Hugoniot jump conditions are natural conditions of the weak form. The method is illustrated in the solution of first order hyperbolic equations and applied to linear first order wave and non-linear Burgers' equations. By capturing the discontinuity in time as well as space, results are improved over capturing the discontinuity in space alone and the method is remarkably accurate. Implications to standard semi-discretization X-FEM formulations are also discussed. Copyright © 2004 John Wiley & Sons, Ltd. [source]