scholarly journals Capturing of solitons collisions and reflections in nonlinear Schrödinger type equations by a conservative scheme based on MOL

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohamed M. Mousa ◽  
Praveen Agarwal ◽  
Fahad Alsharari ◽  
Shaher Momani
Keyword(s):  
Method Of Lines ◽  
Optical Solitons ◽  
Dimensional System ◽  
Interesting Phenomenon ◽  
Von Neumann ◽  
Conservative Scheme ◽  
Nonlinear Schrödinger ◽  
The Method Of Lines ◽  
Von Neumann Stability ◽  
Von Neumann Stability Analysis

AbstractIn this work, we develop an efficient numerical scheme based on the method of lines (MOL) to investigate the interesting phenomenon of collisions and reflections of optical solitons. The established scheme is of second order in space and of fourth order in time with an explicit nature. We deduce stability restrictions using the von Neumann stability analysis. We consider a $(2+ 1)$ ( 2 + 1 ) -dimensional system of a coupled nonlinear Schrödinger equation as a general model of nonlinear Schrödinger-type equations. We consider several numerical experiments to demonstrate the robustness of the scheme in capturing many scenarios of collisions and reflections of the optical solitons related to nonlinear Schrödinger-type equations. We verify the scheme accuracy through computing the conserved invariants and comparing the present results with some existing ones in the literature.

scholarly journals Runge-Kutta Integration of the Equal Width Wave Equation Using the Method of Lines

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
M. A. Banaja ◽  
H. O. Bakodah
Keyword(s):  
Wave Motion ◽  
Method Of Lines ◽  
Unconditionally Stable ◽  
Von Neumann ◽  
The Method Of Lines ◽  
Von Neumann Stability ◽  
Runge Kutta ◽  
Von Neumann Stability Analysis ◽  
Wave Phenomena ◽  
Good Agreement

The equal width (EW) equation governs nonlinear wave phenomena like waves in shallow water. Numerical solution of the (EW) equation is obtained by using the method of lines (MOL) based on Runge-Kutta integration. Using von Neumann stability analysis, the scheme is found to be unconditionally stable. Solitary wave motion and interaction of two solitary waves are studied using the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Accuracy of the proposed method is discussed by computing theL2andL∞error norms. The results are found in good agreement with exact solution.

Efficient modeling of shallow water equations using method of lines and artificial viscosity

2019 ◽  
Vol 34 (04) ◽  
pp. 2050051
Author(s):  
Mohamed M. Mousa ◽  
Wen-Xiu Ma
Keyword(s):  
Shallow Water ◽  
Numerical Solutions ◽  
Method Of Lines ◽  
Artificial Viscosity ◽  
Water Model ◽  
Numerical Schemes ◽  
Von Neumann ◽  
The Method Of Lines ◽  
Shallow Water Models ◽  
Von Neumann Stability

In this work, two numerical schemes were developed to overcome the problem of shock waves that appear in the solutions of one/two-layer shallow water models. The proposed numerical schemes were based on the method of lines and artificial viscosity concept. The robustness and efficiency of the proposed schemes are validated on many applications such as dam-break problem and the problem of interface propagation of two-layer shallow water model. The von Neumann stability of proposed schemes is studied and hence, the sufficient condition for stability is deduced. The results were presented graphically. The verification of the obtained results is achieved by comparing them with exact solutions or another numerical solutions founded in literature. The results are satisfactory and in much have a close agreement with existing results.

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.

scholarly journals A Numerical Solution for Hirota-Satsuma Coupled KdV Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
M. S. Ismail ◽  
H. A. Ashi
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Approximation Technique ◽  
Conserved Quantities ◽  
Test Functions ◽  
Order Accuracy ◽  
Implicit Midpoint Rule ◽  
Von Neumann ◽  
Von Neumann Stability ◽  
Von Neumann Stability Analysis

A Petrov-Galerkin method and product approximation technique are used to solve numerically the Hirota-Satsuma coupled Korteweg-de Vries equation, using cubicB-splines as test functions and a linearB-spline as trial functions. The implicit midpoint rule is used to advance the solution in time. Newton’s method is used to solve the block nonlinear pentadiagonal system we have obtained. The resulting schemes are of second order accuracy in both directions, space and time. The von Neumann stability analysis of the schemes shows that the two schemes are unconditionally stable. The single soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness of the schemes. The interaction of two solitons, three solitons, and birth of solitons is also discussed.

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