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.

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.

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.

scholarly journals PROPERTIES OF FOUR NUMERICAL SCHEMES APPLIED TO A NONLINEAR SCALAR WAVE EQUATION WITH A GR-TYPE NONLINEARITY

2004 ◽  
Vol 13 (05) ◽  
pp. 961-982 ◽  
Author(s):  
JAKOB HANSEN ◽  
ALEXEI KHOKHLOV ◽  
IGOR NOVIKOV
Keyword(s):  
Wave Equation ◽  
Method Of Lines ◽  
Linear Wave ◽  
Numerical Schemes ◽  
Linear Wave Equation ◽  
Von Neumann ◽  
Numerical Testing ◽  
The Method Of Lines ◽  
Von Neumann Analysis ◽  
Runge Kutta

We study stability, dispersion and dissipation properties of four numerical schemes (Itera-tive Crank–Nicolson, 3rd and 4th order Runge–Kutta and Courant–Fredrichs–Levy Nonlinear). By use of a Von Neumann analysis we study the schemes applied to a scalar linear wave equation as well as a scalar nonlinear wave equation with a type of nonlinearity present in GR-equations. Numerical testing is done to verify analytic results. We find that the method of lines (MOL) schemes are the most dispersive and dissipative schemes. The Courant–Fredrichs–Levy Nonlinear (CFLN) scheme is most accurate and least dispersive and dissipative, but the absence of dissipation at Nyquist frequency, if fact, puts it at a disadvantage in numerical simulation. Overall, the 4th order Runge–Kutta scheme, which has the least amount of dissipation among the MOL schemes, seems to be the most suitable compromise between the overall accuracy and damping at short wavelengths.

scholarly journals An explicit unconditionally stable scheme: application to diffusive Covid-19 epidemic model

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yasir Nawaz ◽  
Muhammad Shoaib Arif ◽  
Kamaleldin Abodayeh ◽  
Wasfi Shatanawi
Keyword(s):  
Epidemic Model ◽  
General Type ◽  
Source Term ◽  
Time Dependent ◽  
Unconditionally Stable ◽  
Von Neumann ◽  
First Order ◽  
Von Neumann Stability ◽  
Von Neumann Stability Analysis ◽  
Stable Scheme

AbstractAn explicit unconditionally stable scheme is proposed for solving time-dependent partial differential equations. The application of the proposed scheme is given to solve the COVID-19 epidemic model. This scheme is first-order accurate in time and second-order accurate in space and provides the conditions to get a positive solution for the considered type of epidemic model. Furthermore, the scheme’s stability for the general type of parabolic equation with source term is proved by employing von Neumann stability analysis. Furthermore, the consistency of the scheme is verified for the category of susceptible individuals. In addition to this, the convergence of the proposed scheme is discussed for the considered mathematical model.

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.

Sign in / Sign up

Export Citation Format

Share Document