Numerical Solution Of Burgers Research Articles

A novel approach based on minimization for numerical solutions of Burgers’ equation

This paper deals with the numerical solution of Burgers’ equation. By using radial basis functions (RBFs), an optimization approach is used to derive the approximated solution. Furthermore, we use the regularization term to avoid the growth of RBF coefficients. Also, the stability of the proposed method is verified, and finally we present some numerical examples to confirm our given approach.

A stabilized FEM formulation with discontinuity-capturing for solving Burgers’-type equations at high Reynolds numbers

This computational study is concerned with the numerical solutions of Burgers’-type equations at high Reynolds numbers. The high Reynolds numbers drive the nonlinearity to play an essential role and the equations to become more convection-dominated, which causes the solutions obtained with the standard numerical methods to involve spurious oscillations. To overcome this challenge, the Galerkin finite element formulation is stabilized by using the streamline-upwind/Petrov–Galerkin method. The stabilized formulation is further supplemented with YZβ shock-capturing to achieve better solution profiles around strong gradients. The nonlinear equation systems arising from the space and time discretizations are solved by using the Newton–Raphson (N–R) method at each time step. The resulting linearized equation systems are solved with the BiCGStab technique combined with ILU preconditioning at each N–R iteration. A comprehensive set of test examples is provided to demonstrate the robustness of the proposed formulation and the techniques used.

Numerical Solution of Burgers’ Equation Based on Mixed Finite Volume Element Methods

In this article, mixed finite volume element (MFVE) methods are proposed for solving the numerical solution of Burgers’ equation. By introducing a transfer operator, semidiscrete and fully discrete MFVE schemes are constructed. The existence, uniqueness, and stability analyses for semidiscrete and fully discrete MFVE schemes are given in detail. The optimal a priori error estimates for the unknown and auxiliary variables in the L2Ω norm are derived by using the stability results. Finally, numerical results are given to verify the feasibility and effectiveness.

Numerical solution of Burgers’ equation using Biorthogonal wavelet-based full approximation scheme

Recently, wavelet-based numerical methods have been newly developed in the areas of science and engineering. In this paper, we proposed a full-approximation scheme for the numerical solution of Burgers’ equation using biorthogonal wavelet filter coefficients as prolongation and restriction operators. The presented scheme gives higher accuracy in terms of higher convergence in less CPU time, which has been illustrated through the test problem.

Computer modeling system for the numerical solution of the one-dimensional non-stationary Burgers’ equation

The computer modeling system for numerical solution of the nonlinear one-dimensional non-stationary Burgers’ equation is described. The numerical solution of the Burgers’ equation is obtained by a meshless scheme using the method of partial solutions and radial basis functions. Time discretization of the one-dimensional Burgers’ equation is obtained by the generalized trapezoidal method (θ-scheme). The inverse multiquadric function is used as radial basis functions in the computer modeling system. The computer modeling system allows setting the initial conditions and boundary conditions as well as setting the source function as a coordinate- and time-dependent function for solving partial differential equation. A computer modeling system allows setting such parameters as the domain of the boundary-value problem, number of interpolation nodes, the time interval of non-stationary boundary-value problem, the time step size, the shape parameter of the radial basis function, and coefficients in the Burgers’ equation. The solution of the nonlinear one-dimensional non-stationary Burgers’ equation is visualized as a three-dimensional surface plot in the computer modeling system. The computer modeling system allows visualizing the solution of the boundary-value problem at chosen time steps as three-dimensional plots. The computational effectiveness of the computer modeling system is demonstrated by solving two benchmark problems. For solved benchmark problems, the average relative error, the average absolute error, and the maximum error have been calculated.

Numerical solution of Burgers’ equation by B-spline collocation

In this paper, the Burgers’ equation which is two-dimensional in space, time dependent parabolic differential equation was solved by b-spline collocation algorithms for solving two-dimensional parabolic partial differential equation. At first b-spline interpolation is introduced moreover, the numerical solution is represented as a bi-variate piecewise polynomial with unknown time-dependent coefficients are determined by requiring the numerical solution to satisfy the PDE at a number of points within the spatial domain i.e. we collocate simultaneously in both spatial dimensions. The accuracy of the proposed method is demonstrates by some test problems. The numerical results are found good agreement with exact solution.

Qualitative analysis and numerical solution of burgers’ equation via B-spline collocation with implicit euler method on piecewise uniform mesh

AbstractIn the present paper, a numerical method is proposed for solving one-dimensional time dependent Burgers’ equation for various values of Reynolds number on a rectangular domain in the

An efficient scheme for numerical solution of Burgers’ equation using quintic Hermite interpolating polynomials

A numerical scheme combining the features of quintic Hermite interpolating polynomials and orthogonal collocation method has been presented to solve the well-known non-linear Burgers’ equation. The quintic Hermite collocation method (QHCM) solves the non-linear Burgers’ equation directly without converting it into linear form using Hopf–Cole transformation. Stability of the QHCM has been checked using Eucledian and Supremum norms. Numerical values obtained from QHCM are compared with the values obtained from other techniques such as orthogonal collocation method, orthogonal collocation on finite elements and pdepe solver. Numerical values have been plotted using plane and surface plots to demonstrate the results graphically.

Numerical solution of Burgers’ equation with high order splitting methods

In this work, high order splitting methods have been used for calculating the numerical solutions of Burgers’ equation in one space dimension with periodic, Dirichlet, Neumann and Robin boundary conditions. However, splitting methods with real coefficients of order higher than two necessarily have negative coefficients and cannot be used for time-irreversible systems, such as Burgers’ equations, due to the time-irreversibility of the Laplacian operator. Therefore, the splitting methods with complex coefficients and extrapolation methods with real and positive coefficients have been employed. If we consider the system as the perturbation of an exactly solvable problem (or one that can be easily approximated numerically), it is possible to employ highly efficient methods to approximate Burgers’ equation. The numerical results show that both the methods with complex time steps having one set of coefficients real and positive, say ai∈R+ and bi∈C+, and high order extrapolation methods derived from a lower order splitting method produce very accurate solutions of Burgers’ equation.

On the Solution of Burgers’ Equation with the New Fractional Derivative

AbstractFirstly in this article, the exact solution of a time fractional Burgers’ equation, where the derivative is conformable fractional derivative, with dirichlet and initial conditions is found byHopf-Cole transform. Thereafter the approximate analytical solution of the time conformable fractional Burger’s equation is determined by using a Homotopy Analysis Method(HAM). This solution involves an auxiliary parameter ~ which we also determine. The numerical solution of Burgers’ equation with the analytical solution obtained by using the Hopf-Cole transform is compared.

Numerical solution of Burgers’ equation by cubic Hermite collocation method

Abstract In this paper, numerical solution of the non-linear Burgers’ equation are obtained by using cubic Hermite collocation method (CHCM). The advantage of the method is continuity of the dependent variable and its derivative throughout the solution range. A linear stability analysis shows that the numerical scheme based on Crank–Nicolson approximation in time is unconditionally stable. This method is applied on some test problems, with different choice of collocation points to validate the accuracy of the method. The obtained numerical results show that the method is efficient, robust and reliable even for high Reynolds numbers, for which the exact solution fails. Moreover, the method can be applied to a wide class of nonlinear partial differential equations.

Numerical solution of Burgers’ equation with modified cubic B-spline differential quadrature method

In this paper, a new numerical method, “modified cubic-B-spline differential quadrature method (MCB-DQM)” is proposed to find the approximate solution of the Burgers’ equation. The modified cubic-B-spline basis functions are used in differential quadrature to determine the weighting coefficients. The MCB-DQM is used in space, and the optimal four-stage, order three strong stability-preserving time-stepping Runge–Kutta (SSP-RK43) scheme is used in time for solving the resulting system of ordinary differential equations. To check the efficiency and accuracy of the method, four examples of Burgers’ equation are included with their numerical solutions, L2 and L∞ errors and comparisons are done with the results given in the literature. The proposed method produces better results as compared to the results obtained by almost all the schemes available in the literature, and approaching to the exact solutions. The presented method is seen to be easy, powerful, efficient and economical to implement as compared to the existing techniques for finding the numerical solutions for various kinds of linear/nonlinear physical models.

Numerical solution of Burgers’ equation by lattice Boltzmann method

In this paper, a lattice Boltzmann model for Burgers’ equation is proposed through selecting equilibrium distribution function properly. The model has been verified by three test examples. The numerical results are found in good agreement with analytical solutions and numerical solutions reported in previous studies. The advantages of the resulting method are algorithmic simplicity, parallel computation, easy handling of complicated boundary conditions, so it is very easy to implement and deal with more complex problems.

Nonlinear Functional Analysis of Boundary Value Problems: Novel Theory, Methods, and Applications

and Applied Analysis 3 1-homogeneous operator in a Banach space and then demonstrate its application in establishing the existence of positive solutions for p-Laplacian boundary value problems under certain conditions. (xi) In the paper titled “Existence of solutions for nonhomogeneous A-harmonic equations with variable growth,” the authors establish a theorem for the existence of weak solutions for nonhomogeneous A-harmonic equations in subspace and then give three examples to demonstrate its application. (xii) In the paper titled “Multiple solutions for degenerate elliptic systems near resonance at higher eigenvalues,” the authors study the degenerate semilinear elliptic system in an open bounded domain with smooth boundary, and some multiplicity results of solutions are obtained for the system near resonance at certain eigenvalues by the classical saddle point theorem and a local saddle point theorem in critical point theory. (xiii) In the paper titled “A regularity criterion for the Navier-Stokes equations in the multiplier spaces,” the authors establish a regularity criterion in terms of the pressure gradient for weak solutions to the NavierStokes equations in a special class. The third set of papers, including four papers, deal with several boundary value problems for highly nonlinear ordinary differential equations. (i) In the paper titled “Positive solutions for second-order singular semipositone differential equations involving Stieltjes integral conditions,” the authors investigate the existence of positive solutions for second-order singular differential equations with a negatively perturbed term, by means of the fixed-point theory in cones. (ii) In the paper titled “Positive solutions for Sturm-Liouville boundary value problems in a Banach Space,” the sufficient conditions for the existence of single and multiple positive solutions for a second-order SturmLiouville boundary value problem are established in a Banach space, by using the fixed-point theorem of strict set contraction operators in the frame of the ODE technique. (iii) In the paper titled “Positive solutions of a nonlinear fourth-order dynamic eigenvalue problem on time scales,” the authors study a nonlinear fourth-order dynamic eigenvalue problem on time scales and obtain the existence and nonexistence of positive solutions when 0 λ, respectively, for some λ, by using the Schauder fixed-point theorem and the upper and lower solution method. (iv) In the paper titled “Bifurcation analysis for a predatorprey model with time delay and delay-dependent parameters,” a class of stage-structured predator-prey model with time delay and delay-dependent parameters is considered. By using the normal form theory and center manifold theory, some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifur-cating from Hopf bifurcations are obtained. The fourth set of papers focus on finding the approximate and numerical solutions of various complex nonlinear boundary value problems. (i) In the paper titled “On spectral homotopy analysis method for solving linear Volterra and Fredholm integrodifferential equations,” a spectral homotopy analysis method (SHAM) is proposed to solve linear Volterra integrodifferential equations, and some examples are given to test the efficiency and the accuracy of the proposed method. (ii) In the paper titled “The solution of a class of singularly perturbed two-point boundary value problems by the iterative reproducing kernel method,” the authors establish an iterative reproducing kernel method (IRKM) for solving singular perturbation problems with boundary layers and give two numerical examples to demonstrate the effectiveness of the method. (iii) In the paper titled “A Galerkin solution for Burgers’ equation using cubic B-spline finite elements,” a Galerkin method using cubic B-splines is set up to find the numerical solutions of Burgers’ equation, and the method is shown to be capable of solving Burgers’ equation accurately for values of viscosity ranging from very small to very large. (iv) In the paper titled “Forward-backward splitting methods for accretive operators in Banach spaces,” the authors introduce two iterative forward-backward splitting methods with relaxations to find zeros of the sum of two accretive operators in Banach spaces and prove the weak and strong convergence of these methods under mild conditions, and also discuss applications of these methods to variational inequalities, the split feasibility problem, and a constrained convex minimization problem. Yong Hong Wu Lishan Liu Benchawan Wiwatanapataphee Shaoyong Lai Submit your manuscripts at http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering Hindawi Publishing Corporation http://www.hindawi.com Differential Equations International Journal of Volume 2014 Applied Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Probability and Statistics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Physics Advances in Complex Analysis Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Optimization Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Operations Research Advances in

Numerical Solution of Burgers' Equation by Petrov-Galerkin Method with Adaptive Weighting Functions

Numerical Solution of Burgers' Equation by Petrov-Galerkin Method with Adaptive Weighting Functions

Numerical solution of Burgers’ equation by the Sobolev gradient method

Burgers’ equation is solved numerically with Sobolev gradient methods. A comparison is shown with other numerical schemes presented in this journal, such as modified Adomian method (MAM) [1] and by a variational method (VM) which is based on the method of discretization in time [2]. It is shown that the Sobolev gradient methods are highly efficient while at the same time retaining the simplicity of steepest descent algorithms.

Numerical solution of Burgers' equation with factorized diagonal Padé approximation

Purpose – The purpose of this paper is to present an approach capable of solving Burgers' equation. Diagonal Padé approximation with a factorization scheme is applied to find numerical solutions of the one‐dimensional Burgers' equation by presenting explicit factoring the polynomials of the approximation. The numerical results obtained by this approach, for various values of viscosity, have been compared with the exact solution and are found to be in good agreement with each other.

Numerical solution of Burgers’ equation by cubic B-spline quasi-interpolation

In this paper, numerical solution of the Burgers’ equation is presented based on the cubic B-spline quasi-interpolation. At first the cubic B-spline quasi-interpolation is introduced. Moreover, the numerical scheme is presented, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable. The accuracy of the proposed method is demonstrated by some test problems. The numerical results are found in good agreement with exact solutions. The advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement.

On the class of high order time stepping schemes based on Padé approximations for the numerical solution of Burgers’ equation

Numerical solution of Burgers’ equation is presented using finite difference methods in space and positivity preserving Padé approximations in time. A class of high order time stepping schemes is introduced. For practical purposes, first, second, third, and fourth order schemes are constructed. Efficient parallel versions of these schemes are given using a splitting technique of rational functions. Accuracy of the schemes is demonstrated by solving a test problem and comparing numerical results with the exact solution. Time evolution graphs show the physical phenomenon of the problem. Convergence tables are given to verify the theoretical order of convergence.

On numerical solution of Burgers' equation by homotopy analysis method

In this Letter, we present the Homotopy Analysis Method (shortly HAM) for obtaining the numerical solution of the one-dimensional nonlinear Burgers' equation. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Convergence of the solution and effects for the method is discussed. The comparison of the HAM results with the Homotopy Perturbation Method (HPM) and the results of [E.N. Aksan, Appl. Math. Comput. 174 (2006) 884; S. Kutluay, A. Esen, Int. J. Comput. Math. 81 (2004) 1433; S. Abbasbandy, M.T. Darvishi, Appl. Math. Comput. 163 (2005) 1265] are made. The results reveal that HAM is very simple and effective. The HAM contains the auxiliary parameter ℏ, which provides us with a simple way to adjust and control the convergence region of solution series. The numerical solutions are compared with the known analytical and some numerical solutions.