scholarly journals Numerical solutions of the fractional KdV-Burgers-Kuramoto equation

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 153-158 ◽  
Author(s):  
Dogan Kaya ◽  
Sema Gulbahar ◽  
Asıf Yokus
Keyword(s):  
Stability Analysis ◽  
Finite Difference ◽  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Von Neumann ◽  
Von Neumann Stability ◽  
Difference Methods ◽  
Von Neumann Stability Analysis ◽  
Methods Numerical ◽  
Linearization Techniques

Non-linear terms of the time-fractional KdV-Burgers-Kuramoto equation are linearized using by some linearization techniques. Numerical solutions of this equation are obtained with the help of the finite difference methods. Numerical solutions and corresponding analytical solutions are compared. The L2 and L? error norms are computed. Stability of given method is investigated by using the Von Neumann stability analysis.

scholarly journals Numerical Solutions of the Modified Burgers’ Equation by Finite Difference Methods

2017 ◽  
Vol 13 (1) ◽  
pp. 19-30 ◽  
Author(s):  
Yusuf Ucar ◽  
Nuri Murat Yagmurlu ◽  
Orkun Tasbozan
Keyword(s):  
Stability Analysis ◽  
Finite Difference ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Finite Difference Methods ◽  
Solution Process ◽  
Modified Burgers Equation ◽  
Finite Difference Equations ◽  
Difference Methods ◽  
Linearization Techniques

Abstract In this study, a numerical solution of the modified Burgers’ equation is obtained by the finite difference methods. For the solution process, two linearization techniques have been applied to get over the non-linear term existing in the equation. Then, some comparisons have been made between the obtained results and those available in the literature. Furthermore, the error norms L2 and L∞ are computed and found to be sufficiently small and compatible with others in the literature. The stability analysis of the linearized finite difference equations obtained by two different linearization techniques has been separately conducted via Fourier stability analysis method.

Mimetic seismic wave modeling including topography on deformed staggered grids

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. T125-T141 ◽  
Author(s):  
Josep de la Puente ◽  
Miguel Ferrer ◽  
Mauricio Hanzich ◽  
José E. Castillo ◽  
José M. Cela
Keyword(s):  
Finite Difference ◽  
Seismic Wave ◽  
Seismic Waves ◽  
Numerical Solutions ◽  
Surface Condition ◽  
Finite Difference Methods ◽  
Staggered Grid ◽  
Grid Deformation ◽  
Large Dispersion ◽  
Difference Methods

Finite-difference methods for modeling seismic waves are known to be inaccurate when including a realistic topography, due to the large dispersion errors that appear in the modelled surface waves and the scattering introduced by the staircase approximation to the topography. As a consequence, alternatives to finite-difference methods have been proposed to circumvent these issues. We present a new numerical scheme for 3D elastic wave propagation in the presence of strong topography. This finite-difference scheme is based upon a staggered grid of the Lebedev type, or fully staggered grid (FSG). It uses a grid deformation strategy to make a regular Cartesian grid conform to a topographic surface. In addition, the scheme uses a mimetic approach to accurately solve the free-surface condition and hence allows for a less restrictive grid spacing criterion in the computations. The scheme can use high-order operators for the spatial derivatives and obtain low-dispersion results with as few as six points per minimum wavelength. A series of tests in 2D and 3D scenarios, in which our results are compared to analytical and numerical solutions obtained with other numerical approaches, validate the accuracy of our scheme. The resulting FSG mimetic scheme allows for accurate and efficient seismic wave modelling in the presence of very rough topographies with the advantage of using a structured staggered grid.

Stability of discrete schemes of Biot’s poroelastic equations

2020 ◽  
Author(s):  
Y Alkhimenkov ◽  
L Khakimova ◽  
Y Y Podladchikov
Keyword(s):  
Stability Analysis ◽  
Three Dimensions ◽  
Stability Conditions ◽  
Von Neumann ◽  
Explicit Schemes ◽  
Numerical Stability Analysis ◽  
Von Neumann Stability ◽  
Von Neumann Stability Analysis ◽  
Biot’S Equations ◽  
Implicit And Explicit

Summary The efficient and accurate numerical modeling of Biot’s equations of poroelasticity requires the knowledge of the exact stability conditions for a given set of input parameters. Up to now, a numerical stability analysis of the discretized elastodynamic Biot’s equations has been performed only for a few numerical schemes. We perform the von Neumann stability analysis of the discretized Biot’s equations. We use an explicit scheme for the wave propagation and different implicit and explicit schemes for Darcy’s flux. We derive the exact stability conditions for all the considered schemes. The obtained stability conditions for the discretized Biot’s equations were verified numerically in one-, two- and three-dimensions. Additionally, we present von Neumann stability analysis of the discretized linear damped wave equation considering different implicit and explicit schemes. We provide both the Matlab and symbolic Maple routines for the full reproducibility of the presented results. The routines can be used to obtain exact stability conditions for any given set of input material and numerical parameters.

NUMERICAL INTERFACES IN FINITE DIFFERENCE METHODS FOR HYPERBOLIC EQUATIONS WITH DISCONTINUOUS COEFFICIENTS

1993 ◽  
Vol 01 (02) ◽  
pp. 151-184 ◽  
Author(s):  
TAO LIN
Keyword(s):  
Stability Analysis ◽  
Finite Difference ◽  
Convergence Rates ◽  
Hyperbolic Equations ◽  
Finite Difference Methods ◽  
Discontinuous Coefficients ◽  
Interface Problems ◽  
Artificial Boundary ◽  
Numerical Examples ◽  
Difference Methods

In this paper, we discuss the interface problems arising in using finite difference methods to solve hyperbolic equations with discontinuous coefficients. The schemes developed here can be used to handle four important types of numerical interfaces due to: (1) the discontinuity of the coefficients of the PDE, (2) using artificial boundary, (3) using different finite difference formulae in different areas, and (4) using different grid sizes in different areas. Stability analysis for these schemes is carried out in terms of conventional l1, l2, and l∞ norms so that the convergence rates of these schemes are obtained. Several numerical examples are supplied to demonstrate properties of these schemes.

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