NUMERICAL STUDY OF ELLIPTIC-HYPERBOLIC DAVEY–STEWARTSON SYSTEM: DROMIONS SIMULATION AND BLOW-UP

1998 ◽  
Vol 08 (08) ◽  
pp. 1363-1386 ◽  
Author(s):  
C. BESSE ◽  
C. H. BRUNEAU
Keyword(s):  
Finite Differences ◽  
Numerical Experiments ◽  
Numerical Approximation ◽  
Blow Up ◽  
Numerical Study ◽  
Phase Error ◽  
Qualitative Behaviour ◽  
Hyperbolic Form ◽  
Nicolson Scheme ◽  
Crank Nicolson

This paper is devoted to the numerical approximation of the elliptic-hyperbolic form of the Davey–Stewartson equations. A well-suited finite differences scheme that preserves the energy is derived. This scheme is tested to compute the famous dromion 1–1 and dromion 2–2 solutions. The accuracy of Crank–Nicolson scheme is discussed and it is shown that it induces a phase error. The qualitative behaviour of the solutions is then studied; in particular the influence of the initial datum and of the various parameters is pointed out. Finally, numerical experiments show the existence of blow-up solutions.

scholarly journals Numerical Study of Electro-Osmotic Fluid Flow and Vortex Formation

Micromachines ◽  
2019 ◽  
Vol 10 (12) ◽  
pp. 796 ◽  
Author(s):  
Wesley De Souza Bezerra ◽  
Antonio Castelo ◽  
Alexandre M. Afonso
Keyword(s):  
Fluid Flow ◽  
Finite Differences ◽  
Numerical Approximation ◽  
Numerical Study ◽  
Vortex Formation ◽  
Flow Perturbation ◽  
Osmotic Flow ◽  
Electro Osmosis ◽  
Abrupt Contraction ◽  
Electro Osmotic Flow

The phenomenon of electro-osmosis was studied by performing numerical simulations on the flow between parallel walls and at the nozzle microchannels. In this work, we propose a numerical approximation to perform simulations of vortex formation which occur after the passage of the fluid through an abrupt contraction at the microchannel. The motion of the charges in the solution is described by the Poisson–Nernst–Planck equations and used the generalized finite differences to solve the numerical problem. First, solutions for electro-osmotic flow were obtained for the Phan–Thien/Thanner model in a parallel walls channel. Later simulations for electro-osmotic flow were performed in a nozzle. The formation of vortices near the contraction within the nozzle was verified by taking into account a flow perturbation model.

scholarly journals Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Vincent Duchêne ◽  
Christian Klein
Keyword(s):  
Solitary Wave ◽  
Numerical Experiments ◽  
Numerical Scheme ◽  
Krylov Subspace ◽  
Blow Up ◽  
Numerical Study ◽  
Iterative Technique ◽  
Fourier Spectral Method ◽  
Solitary Wave Solutions ◽  
Wave Solutions

<p style='text-indent:20px;'>We perform numerical experiments on the Serre-Green-Naghdi (SGN) equations and a fully dispersive "Whitham-Green-Naghdi" (WGN) counterpart in dimension 1. In particular, solitary wave solutions of the WGN equations are constructed and their stability, along with the explicit ones of the SGN equations, is studied. Additionally, the emergence of modulated oscillations and the possibility of a blow-up of solutions in various situations is investigated.</p><p style='text-indent:20px;'>We argue that a simple numerical scheme based on a Fourier spectral method combined with the Krylov subspace iterative technique GMRES to address the elliptic problem and a fourth order explicit Runge-Kutta scheme in time allows to address efficiently even computationally challenging problems.</p>

Beam Propagation Method Based on the Iterated Crank–Nicolson Scheme for Solving Large-Scale Wave Propagation Problems

2015 ◽  
Vol 51 (3) ◽  
pp. 1-4 ◽  
Author(s):  
Dimitra A. Ketzaki ◽  
Ioannis T. Rekanos ◽  
Theodoros I. Kosmanis ◽  
Traianos V. Yioultsis
Keyword(s):  
Wave Propagation ◽  
Large Scale ◽  
Beam Propagation Method ◽  
Beam Propagation ◽  
Nicolson Scheme ◽  
Crank Nicolson

scholarly journals A Mixed Element Algorithm Based on the Modified L1 Crank–Nicolson Scheme for a Nonlinear Fourth-Order Fractional Diffusion-Wave Model

2021 ◽  
Vol 5 (4) ◽  
pp. 274
Author(s):  
Jinfeng Wang ◽  
Baoli Yin ◽  
Yang Liu ◽  
Hong Li ◽  
Zhichao Fang
Keyword(s):  
Fourth Order ◽  
Mixed Finite Element ◽  
Calculated Data ◽  
Coupled System ◽  
Wave Model ◽  
Fractional Diffusion ◽  
Second Order ◽  
Diffusion Wave ◽  
Nicolson Scheme ◽  
Crank Nicolson

In this article, a new mixed finite element (MFE) algorithm is presented and developed to find the numerical solution of a two-dimensional nonlinear fourth-order Riemann–Liouville fractional diffusion-wave equation. By introducing two auxiliary variables and using a particular technique, a new coupled system with three equations is constructed. Compared to the previous space–time high-order model, the derived system is a lower coupled equation with lower time derivatives and second-order space derivatives, which can be approximated by using many time discrete schemes. Here, the second-order Crank–Nicolson scheme with the modified L1-formula is used to approximate the time direction, while the space direction is approximated by the new MFE method. Analyses of the stability and optimal L2 error estimates are performed and the feasibility is validated by the calculated data.

scholarly journals Crank-Nicolson scheme for two-dimensional in space fractional diffusion equations with functional delay

2021 ◽  
Vol 57 ◽  
pp. 128-141
Author(s):  
M. Ibrahim ◽  
V.G. Pimenov
Keyword(s):  
Piecewise Linear ◽  
Linear Interpolation ◽  
Fractional Diffusion ◽  
Dimensional System ◽  
Two Dimensional ◽  
Piecewise Linear Interpolation ◽  
Functional Delay ◽  
General Difference ◽  
Nicolson Scheme ◽  
Crank Nicolson

A two-dimensional in space fractional diffusion equation with functional delay of a general form is considered. For this problem, the Crank-Nicolson method is constructed, based on shifted Grunwald-Letnikov formulas for approximating fractional derivatives with respect to each spatial variable and using piecewise linear interpolation of discrete history with continuation extrapolation to take into account the delay effect. The Douglas scheme is used to reduce the emerging high-dimensional system to tridiagonal systems. The residual of the method is investigated. To obtain the order of the method, we reduce the systems to constructions of the general difference scheme with heredity. A theorem on the second order of convergence of the method in time and space steps is proved. The results of numerical experiments are presented.

scholarly journals Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS

2021 ◽  
Vol 50 (6) ◽  
pp. 1799-1814
Author(s):  
Norazak Senu ◽  
Nur Amirah Ahmad ◽  
Zarina Bibi Ibrahim ◽  
Mohamed Othman
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Numerical Study ◽  
Kutta Method ◽  
Initial Value ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.

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