Journal Article

# Two meshless methods based on pseudo spectral delta-shaped basis functions and barycentric rational interpolation for numerical solution of modified Burgers equation

04 Mar 2021-International Journal of Computer Mathematics (Informa UK Limited)-Vol. 98, Iss: 3, pp 461-479
Abstract: In this paper, we solve modified Burgers equation numerically. Time discretization for modified Burgers equation is made by using finite difference approach along with a linearization technique. Fo...

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Open accessJournal Article
Abstract: We propose a polynomial-based numerical scheme for solving some important nonlinear partial differential equations (PDEs). In the proposed technique, the temporal part is discretized by finite difference method together with θ-weighted scheme. Then, for the approximation of spatial part of unknown function and its spatial derivatives, we use a mixed approach based on Lucas and Fibonacci polynomials. With the help of these approximations, we transform the nonlinear partial differential equation to a system of algebraic equations, which can be easily handled. We test the performance of the method on the generalized Burgers–Huxley and Burgers–Fisher equations, and one- and two-dimensional coupled Burgers equations. To compare the efficiency and accuracy of the proposed scheme, we computed $L_{\infty }$ , $L_{2}$ , and root mean square (RMS) error norms. Computations validate that the proposed method produces better results than other numerical methods. We also discussed and confirmed the stability of the technique.

10 Citations

Open access
04 Jun 2015-
Topics: Burgers' equation (71%)

7 Citations

Journal Article
Abstract: This paper proposes an efficient localized meshless technique for approximating the viscoelastic wave model. This model is a significant methodology to explain wave propagation in solids modeled with a wide collection of viscoelastic laws. In the first method, a difference scheme with the second-order accuracy is implemented to obtain a semi-discrete scheme. Then, a localized radial basis function partition of unity scheme is adopted to get a full-discrete scheme. This localization technique consists of decomposing the initial domain into several sub-domains and constructing a local radial basis function approximation over every sub-domain. A well-conditioned resulting linear system and a low computational burden are the main merits of this technique compared to global collocation methods. Further, the stability and convergence analysis of the temporal discretization scheme are deduced using discrete energy method. Numerical results are shown to validate the accuracy and effectiveness of the proposed method.

6 Citations

Journal Article
Abstract: This paper is dedicated to numerically solving the Sobolev equations that have several applications in physics and mechanical engineering. First, the temporal derivative is discretized by the Crank-Nicolson finite difference technique to obtain a semi-discrete scheme in the temporal direction. Afterward, the stability and convergence analysis of the time semi-discrete scheme are proven by applying the energy method. It also implies that the convergence order in the temporal direction is O ( d t 2 ) . Second, a fully discrete formula has been acquired by discretizing the spatial derivatives via Legendre spectral element method (LSEM). This method applies the Lagrange polynomial based on the Gauss-Legendre-Lobatto (GLL) points. Moreover, an error estimation is given for the obtained fully discrete scheme. Eventually, the two-dimensional Sobolev equations are solved by using the proposed procedure. The accuracy and efficiency of the mentioned procedure are demonstrated by several numerical examples.

5 Citations

Journal Article
01 Mar 2021-Wave Motion
Abstract: In this work, we examine generalized equal width (GEW) equation which is a highly nonlinear partial differential equation and describes plasma waves and shallow water waves. Nonlinearity of the equation is tackled by a linearization technique and finite difference approach is utilized for time derivatives. For spatial derivatives we first introduce delta-shaped basis functions which are relatively less studied in literature. Then, by using delta-shaped basis functions, pseudospectral differentiation matrices are constructed for spatial derivatives. Therefore combining finite difference approach for time derivatives and pseudospectral differentiation matrices for spatial derivatives, we obtain a system of linear equations. Solution of this system of equations gives nodal values of numerical solution of the GEW equation for each time step. Stability of the proposed method is given by using linear matrix stability analysis. To measure performance of the proposed method, four classic test problems are chosen namely the propagation of a single solitary wave, interaction of two solitary waves, Maxwellian initial condition and collision of solitons. Also, conservations of mass, momentum and energy are monitored during simulations. The results of numerical computations are compared with exact results if available and with previous studies in the literature such as Petrov–Galerkin, B-spline Galerkin and some collocation methods. From the comparison we can deduce that the proposed method gives reliable and accurate results in less computational cost.

4 Citations

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