Numerical Solutions of the Equal Width Wave Equation Using an Adaptive Method of Lines

2001 ◽  
Author(s):  
S Hamdi ◽  
J Gottlieb ◽  
J Hansen
Keyword(s):  
Wave Equation ◽  
Numerical Solutions ◽  
Method Of Lines ◽  
Adaptive Method

scholarly journals NUMERICAL SIMULATION OF MAGNETIC DROPLET DYNAMICS IN A ROTATING FIELD

2013 ◽  
Vol 18 (1) ◽  
pp. 80-96
Author(s):  
Andrejs Cebers ◽  
Harijs Kalis
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Partial Differential Equation ◽  
Method Of Lines ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Continuous Functions ◽  
Rotating Magnetic Field ◽  
Droplet Dynamics ◽  
Ill Posed ◽  
Initial Boundary Value

Dynamics and hysteresis of an elongated droplet under the action of a rotating magnetic field is considered for mathematical modelling. The shape of droplet is found by regularization of the ill-posed initial–boundary value problem for nonlinear partial differential equation (PDE). It is shown that two methods of the regularization – introduction of small viscous bending torques and construction of monotonous continuous functions are equivalent. Their connection with the regularization of the ill-posed reverse problems for the parabolic equation of heat conduction is remarked. Spatial discretization is carried out by the finite difference scheme (FDS). Time evolution of numerical solutions is obtained using method of lines for solving a large system of ordinary differential equations (ODE).

scholarly journals Exact and Numerical Solitary Wave Structures to the Variant Boussinesq System

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1473 ◽  
Author(s):  
Abdulghani Alharbi ◽  
Mohammed B. Almatrafi
Keyword(s):  
Solitary Wave ◽  
Evolution Equations ◽  
Numerical Solutions ◽  
Nonlinear Evolution ◽  
Numerical Technique ◽  
Method Of Lines ◽  
Nonlinear Evolution Equations ◽  
Boussinesq System ◽  
Solitary Wave Solutions ◽  
The Method Of Lines

Solutions such as symmetric, periodic, and solitary wave solutions play a significant role in the field of partial differential equations (PDEs), and they can be utilized to explain several phenomena in physics and engineering. Therefore, constructing such solutions is significantly essential. This article concentrates on employing the improved exp(−ϕ(η))-expansion approach and the method of lines on the variant Boussinesq system to establish its exact and numerical solutions. Novel solutions based on the solitary wave structures are obtained. We present a comprehensible comparison between the accomplished exact and numerical results to testify the accuracy of the used numerical technique. Some 3D and 2D diagrams are sketched for some solutions. We also investigate the L2 error and the CPU time of the used numerical method. The used mathematical tools can be comfortably invoked to handle more nonlinear evolution equations.

scholarly journals Adaptive method of lines for multi-component aerosol condensational growth and cloud droplet activation

2010 ◽  
Vol 3 (3) ◽  
pp. 1271-1315
Author(s):  
S. Arabas ◽  
H. Pawlowska
Keyword(s):  
Method Of Lines ◽  
Cloud Droplet ◽  
Adaptive Method ◽  
Size Spectrum ◽  
Model Formulation ◽  
Cloud Droplets ◽  
The Method Of Lines ◽  
Model Set ◽  
Set Up ◽  
Condensational Growth

Abstract. The process of formation of cloud droplets on an ensemble of aerosol particles is modelled by numerous investigators using the method of lines (MOL). The method involves discretization of the aerosol size spectrum into bins whose position and width evolve with time. One of the drawbacks of the method is its poor representation of the aerosol spectrum shape in the region between the unactivated aerosol mode and the activated cloud-droplet mode. An adaptive spectrum refinement procedure that improves the performance of the method is introduced and tested. A model of drop formation on multi-component aerosol is formulated for the purpose of the study. Model formulation includes explicit treatment of the drop temperature evolution. Several examples of the model set-up are used to demonstrate model capabilities. Model results are compared to those without adaptivity, and are compared to the Twomey's formul\\ae. A C++ implementation of the model is available as an electronic supplement of the paper.

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