16,208 research outputs found
Variation Iteration Method for The Approximate Solution of Nonlinear Burgers Equation
In this study, we considered the numerical solution of the nonlinear Burgers equation using the Variational Iteration Method (VIM). The method seeks to examine the convergence of solutions of the Burgers equation at the expense of the parameters x and t of which the amount of errors depends. Numerical experimentation was carried out on the Burgers equation with the Variational Iteration Method (VIM). The resulting solution showed that the rate of convergence decreases with increase in the values of the parameters x and t at each iterate level. However, as the number of iterations increases, there is a rapid rate of convergence of the approximate solution to the analytic solution. Results obtained with the Variational Iteration Method (VIM) on the Burgers equation were compared with the exact found in literature. All computational framework of the research were performed with the aid of Maple 18 software.Keywords: Variational Iteration Method, Burgers Equation, Partial Differential Equations, Approximate Solution, Mean Value Theorem, Schwartz Inequality
A spectral-based numerical method for Kolmogorov equations in Hilbert spaces
We propose a numerical solution for the solution of the
Fokker-Planck-Kolmogorov (FPK) equations associated with stochastic partial
differential equations in Hilbert spaces.
The method is based on the spectral decomposition of the Ornstein-Uhlenbeck
semigroup associated to the Kolmogorov equation. This allows us to write the
solution of the Kolmogorov equation as a deterministic version of the
Wiener-Chaos Expansion. By using this expansion we reformulate the Kolmogorov
equation as a infinite system of ordinary differential equations, and by
truncation it we set a linear finite system of differential equations. The
solution of such system allow us to build an approximation to the solution of
the Kolmogorov equations. We test the numerical method with the Kolmogorov
equations associated with a stochastic diffusion equation, a Fisher-KPP
stochastic equation and a stochastic Burgers Eq. in dimension 1.Comment: 28 pages, 10 figure
A result of convergence for a mono-dimensional two-velocities lattice Boltzmann scheme
We consider a mono-dimensional two-velocities scheme used to approximate the
solutions of a scalar hyperbolic conservative partial differential equation. We
prove the convergence of the discrete solution toward the unique entropy
solution by first estimating the supremum norm and the total variation of the
discrete solution, and second by constructing a discrete kinetic
entropy-entropy flux pair being given a continuous entropy-entropy flux pair of
the hyperbolic system. We finally illustrate our results with numerical
simulations of the advection equation and the Burgers equation
A comparison of the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method for solutions of partial differential equations
We compare numerical experiments from the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method, applied to three benchmark problems based on two different partial differential equations. Both methods are described in detail and we highlight some strengths and weaknesses of each method via the numerical comparisons. The two equations used in the benchmark problems are the viscous Burgers' equation and the porous medium equation, both in one dimension. Simulations are made for the two methods for: a) a travelling wave solution for the viscous Burgers' equation, b) the Barenblatt selfsimilar analytical solution of the porous medium equation, and c) a waiting-time solution for the porous medium equation. Simulations are carried out for varying mesh sizes, and the numerical solutions are compared by computing errors in two ways. In the case of an analytic solution being available, the errors in the numerical solutions are computed directly from the analytic solution. In the case of no availability of an analytic solution, an approximation to the error is computed using a very fine mesh numerical solution as the reference solution
A bounded upwinding scheme for computing convection-dominated transport problems
A practical high resolution upwind differencing scheme for the numerical solution of convection-dominated transport problems is presented. The scheme is based on TVD and CBC stability criteria and is implemented in the context of the finite difference methodology. The performance of the scheme is investigated by solving the 1D/2D scalar advection equations, 1D inviscid Burgers’ equation, 1D scalar convection–diffusion equation, 1D/2D compressible Euler’s equations, and 2D incompressible Navier–Stokes equations. The numerical results displayed good agreement with other existing numerical and experimental data
Playing with Burgers's equation
International audienceThe 1D Burgers equation is used as a toy model to mimick the resulting behaviour of numerical schemes when replacing a conservation law by a form which is equivalent for smooth solutions, such as the total energy by the internal energy balance in the Euler equations. If the initial Burgers equation is replaced by a balance equation for one of its entropies (the square of the unknown) and discretized by a standard scheme, the numerical solution converges, as expected, to a function which is not a weak solution to the initial problem. However, if we first add to Burgers' equation a diffusion term scaled by a small positive parameter ǫ before deriving the entropy balance (this yields a non conservative diffusion term in the resulting equation), and then choose ǫ and the discretization parameters adequately and let them tend to zero, we observe that we recover a convergence to the correct solution
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