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Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation

Jin-Liang Yan, Liang-Hong Zheng
2017 International Journal of Applied Mathematics and Computer Science  
scheme and the implicit midpoint scheme, for the complex modified Kortewegde Vries equation.  ...  The aim of this paper is to build and validate a class of energy-preserving schemes for simulating a complex modified Kortewegde Vries equation.  ...  The authors are grateful to reviewers for their careful reading of the paper as well as their comments and suggestions.  ... 
doi:10.1515/amcs-2017-0036 fatcat:wp46utymhfbihhufgg6o6j4ubq

Contents

2021 Zeitschrift für angewandte Mathematik und Mechanik  
nonlinear dispersive equation: the regularized long wave-Korteweg de Vries (RLW-KdV) equation.  ...  The finite volume method is used to discretize the equation while the finite element method is applied to estimate the gradient quantities at cell faces.  ... 
doi:10.1002/zamm.202102013 fatcat:5yvkzxz64vg5tnm7qb3nlwgixu

Finite volume schemes for Boussinesq type equations [article]

Denys Dutykh
2011 arXiv   pre-print
Finite volume schemes are commonly used to construct approximate solutions to conservation laws.  ...  In this study we extend the framework of the finite volume methods to dispersive water wave models, in particular to Boussinesq type systems.  ...  In the water wave theory dispersive equations have been well known since the pioneering work of J. Boussinesq [6] and Korteweg-de Vries [13] .  ... 
arXiv:1101.1728v1 fatcat:l2jpsjugpvdzzafweus2uqs7am

On time relaxed schemes and formulations for dispersive wave equations

Jean-Paul Chehab, Denys Dutykh
2019 AIMS Mathematics  
In this article we illustrate the application of the new relaxed scheme on the classical Korteweg-de Vries equation as a prototype of stiff dispersive PDEs.  ...  Simple and robust solvers are needed for numerical studies of water waves as well.  ...  Other types of dispersive equations In this Appendix we provide two other examples of relaxation for some widely used dispersive wave equations (scalar and system case).  ... 
doi:10.3934/math.2019.2.254 fatcat:yl2hfvqgt5gnfiwscv7ag5w7ve

On time relaxed schemes and formulations for dispersive wave equations [article]

Jean-Paul Chehab
2019 arXiv   pre-print
In this article we illustrate the application of the new relaxed scheme on the classical Korteweg-de Vries equation as a prototype of stiff dispersive PDEs.  ...  Simple and robust solvers are needed for numerical studies of water waves as well.  ...  Other types of dispersive equations In this Appendix we provide two other examples of relaxation for some widely used dispersive wave equations (scalar and system case).  ... 
arXiv:1903.02212v1 fatcat:itvciphrrrcszmbnqvsb6a7g6u

Dispersion Effects in the Falkner-Skan Problem and in the Kinetic Theory

Oleg Galaev, Evelina Prozorova
2017 Journal of Applied Mathematics and Physics  
The equations for gas are calculated from the modified Boltzmann equation and the phenomenological theory.  ...  The conservation laws of continuum mechanics and of the kinetic theory with the influence of the angular momentum and associated with its rotation of the elementary volume are considered, the variant of  ...  The conservation laws are obtained by writing the balance equations for the volume element, located in the infinite space. For every conservation law, we have its own chosen elementary volume.  ... 
doi:10.4236/jamp.2017.52045 fatcat:hjc7lytlxfbuxexw4npftohrde

The design of conservative finite element discretisations for the vectorial modified KdV equation [article]

James Jackaman, Georgios Papamikos, Tristan Pryer
2017 arXiv   pre-print
We design a consistent Galerkin scheme for the approximation of the vectorial modified Korteweg-de Vries equation. We demonstrate that the scheme conserves energy up to machine precision.  ...  This energy balance ensures there is no numerical dissipation allowing for extremely accurate long time simulations free from numerical artifacts.  ...  We describe some known results and history of the vectorial modified Korteweg-de Vries (vmKdV) equation, highlighting the Hamiltonian structure of the equation.  ... 
arXiv:1710.03527v1 fatcat:7y2nh53bivhitjw7uhv6c6olsa

Numerical solution of complex modified Korteweg–de Vries equation by mesh-free collocation method

Marjan Uddin, Sirajul Haq, Siraj-ul-Islam
2009 Computers and Mathematics with Applications  
A mesh-free method based on radial basis functions (RBFs) is proposed for the solution of complex modified Korteweg-de Vries (CMKdV) equation.  ...  A set of scattered nodes provided by initial data is used for solution of the problem.  ...  Acknowledgements The first author is thankful to HEC Pakistan for financial help through Grant no. 063-281079-Ps3-169. The authors are also thankful to the reviewers for their constructive comments.  ... 
doi:10.1016/j.camwa.2009.03.104 fatcat:mhppnppgabbqzk4zaudiyjnpwu

A REVIEW ON NUMERICAL METHODS FOR NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS

Department of Mathematics, Chandigarh University, Gharuan., Dr. Jatinder kaur, Department of Mathematics, Chandigarh University, Gharuan.
2021 Journal of University of Shanghai for Science and Technology  
The range of application of partial differential equations comprises of recreation, calculation age, and investigation of higher request PDE and wave equations.  ...  Our work centres' around the survey of various numerical methods to settle Non-linear differential equations based on exactness and effectiveness, in order to diminish the emphases.  ...  Bothyana, S.H. (2011) used Adomian Decomposition Method in his research article to solve generalized Korteweg De Vries equations having boundary conditions. [25] The arrangement can be found in the series  ... 
doi:10.51201/jusst/21/07230 fatcat:qtqvu2hjh5bbxoa64fcc4aewaa

Data-driven discretization: machine learning for coarse graining of partial differential equations [article]

Yohai Bar-Sinai, Stephan Hoyer, Jason Hickey, Michael P. Brenner
2019 arXiv   pre-print
standard finite difference methods.  ...  Here we introduce data driven discretization, a method for learning optimized approximations to PDEs based on actual solutions to the known underlying equations.  ...  (a) Particular realization of the solution for the Korteweg-de Vries (KdV) equation at varying resolution solved by the baseline 1st order finite volume method (top row), optimal constant coefficients  ... 
arXiv:1808.04930v3 fatcat:xw4xiuatxfckvaeltuvcs6olie

Isogeometric analysis of the isothermal Navier–Stokes–Korteweg equations

Hector Gomez, Thomas J.R. Hughes, Xesús Nogueira, Victor M. Calo
2010 Computer Methods in Applied Mechanics and Engineering  
This paper is devoted to the numerical simulation of the Navier-Stokes-Korteweg equations, a phase-field model for water/water-vapor two-phase flows.  ...  We introduce a new refinement methodology that desensitizes the numerical solution to the computational mesh and achieves mesh invariant solutions.  ...  by Xunta de Galicia (grants # 09REM005118PR and #09MDS00718PR), Ministerio de Educación y Ciencia (grants #DPI2007-61214 and #DPI2009-14546-C02-01) cofinanced with FEDER funds, and Universidade da Coruña  ... 
doi:10.1016/j.cma.2010.02.010 fatcat:onxpsq6spraalme5he5p2e2aqi

Editorial: Recent Trends in Computational Fluid Dynamics

M. M. Bhatti, M. Marin, A. Zeeshan, Sara I. Abdelsalam
2020 Frontiers in Physics  
Due to the recent advancement in computer technology, numerical simulation for physically and geometrically complex systems can also be evaluated using PC clusters.  ...  The three basic principles that can determine the physical aspects of any fluid are the i) energy conservation, ii) Newton's second law, and the iii) mass conservation.  ...  [12] contemplated the Mohand decomposition scheme to examine the Kortewege-De Vries equations. The fractional derivatives are expressed by Caputo fractional derivative operator.  ... 
doi:10.3389/fphy.2020.593111 fatcat:6gbjv5y3sngsrdect2hpkagzt4

Hydrodynamic equations for the Ablowitz-Ladik discretization of the nonlinear Schroedinger equation [article]

Herbert Spohn
2021 arXiv   pre-print
Studied is also the discretized modified Korteweg-de-Vries equation which turns out to be related to the beta Jacobi log gas.  ...  The resulting hydrodynamic equations follow the pattern already known from other integrable many-body systems.  ...  I am most grateful for the splendid hospitality.  ... 
arXiv:2107.04866v3 fatcat:lo3kubuv4jdxzdgb3cybegf6qa

Solitary Wave Solutions of the Generalized Rosenau-KdV-RLW Equation

Zakieh Avazzadeh, Omid Nikan, José A. Tenreiro Machado
2020 Mathematics  
This model is obtained by coupling the Rosenau–Korteweg-de Vries and Rosenau-regularized-long wave equations.  ...  This paper investigates the solitary wave solutions of the generalized Rosenau–Korteweg-de Vries-regularized-long wave equation.  ...  Acknowledgments: The authors are thankful to the respected reviewers for their valuable comments and constructive suggestions towards the improvement of the original paper.  ... 
doi:10.3390/math8091601 doaj:cc92533ef7d74d618adb996967703308 fatcat:likv3a6ukjg7reaay4ccae44lu

High Order $$C^0$$ C 0 -Continuous Galerkin Schemes for High Order PDEs, Conservation of Quadratic Invariants and Application to the Korteweg-de Vries Model

Sebastian Minjeaud, Richard Pasquetti
2017 Journal of Scientific Computing  
We address the Korteweg-de Vries equation as an interesting model of high order partial differential equation, and show that it is possible to develop reliable and effective schemes, in terms of accuracy  ...  , computational efficiency, simplicity of implementation and, if required, conservation of the lower invariants, on the basis of a (only) H 1 -conformal Galerkin approximation, namely the Spectral Element  ...  The views and opinions expressed herein do not necessarily reflect those of the European Commission. We also thank our Colleagues D. Clamond and A. Galligo for fruitful discussions.  ... 
doi:10.1007/s10915-017-0455-2 fatcat:v45zip6d4fds3og7gazbxkw4mu
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