Numerical solution of the Burgers equation with Neumann boundary noise

Abstract

In this paper we investigate the numerical solution of the one-dimensional Burgers equation with Neumann boundary noise. For the discretization scheme we use the Galerkin approximation in space and the exponential Euler method in time. The impact of the boundary noise on the solution is discussed in several numerical examples. Moreover, we analyze and illustrate some properties of the stochastic term and study the convergence numerically.

An implicit tailored finite point method for the Burgers’ equation: leveraging the Cole-Hopf transformation

“…Since its origination, a variety of numerical and analytical methods have been employed to address the one-dimensional Burgers' equation. In the past two decades, progress in finding solutions has been made through various approaches such as finite element method [2], variational approach utilizing temporal discretization [3], leastsquares quadratic B-spline finite element method [4], modified Adomianʼs decomposition method [5], variational iteration method [6], collocation of cubic B-splines over finite elements [7], fourth-order finite difference method [8], modified extended tanh-function method [9], Crank-Nicolson finite difference method [10], fourth-order compact finite difference method [11], cubic B-spline quasi-interpolation method [12], differential transformation method [13], uniform Haar wavelet quasilinearization approach [14], modified cubic B-splines collocation method [15], modified cubic-B-spline differential quadrature method [16], weighted average differential quadrature method [17], hybrid scheme [18], high order splitting methods [19], fifth-order finite volume weighted compact scheme [20], exponential modified cubic B-spline differential quadrature method [21], hybrid trigonometric differential quadrature method [22], polynomial based differential quadrature method [23], hybrid Galerkin approximation exponential Euler method [24], fully implicit finite difference method [25], space-time kernel-based numerical approach [26], time symmetric splitting method [27], unified Mahgoub transform and homotopy perturbation method [28], non-standard finite difference scheme [29], spectral solutions [30], undetermined coefficient method [31], modified quartic hyperbolic B-spline DQM [32], and an hybridized non-standard numerical scheme integrating a compact finite difference approach…”

A numerical study of the Burgers’ and Fisher’s equations using barycentric interpolation method

“…Recently, Galerkin approximation with exponential Euler method has been applied by Ghayebi et al (2016) to equation (1.1). Tamsir et al (2016) presented modified exponential B spline method to study Burgers' equation.…”