scholarly journals A Lawson-type exponential integrator for the Korteweg–de Vries equation

2019 ◽  
Vol 40 (4) ◽  
pp. 2399-2414
Author(s):  
Alexander Ostermann ◽  
Chunmei Su
Keyword(s):  
Vries Equation ◽  
Time Integration ◽  
Mesh Size ◽  
Convergence Result ◽  
Time Step ◽  
Numerical Examples ◽  
First Order ◽  
De Vries ◽  
Exponential Integrator ◽  
Explicit Numerical Method

Abstract We propose an explicit numerical method for the periodic Korteweg–de Vries equation. Our method is based on a Lawson-type exponential integrator for time integration and the Rusanov scheme for Burgers’ nonlinearity. We prove first-order convergence in both space and time under a mild Courant–Friedrichs–Lewy condition $\tau =O(h)$, where $\tau$ and $h$ represent the time step and mesh size for solutions in the Sobolev space $H^3((-\pi , \pi ))$, respectively. Numerical examples illustrating our convergence result are given.

Abundant rogue wave solutions for the (2 + 1)-dimensional generalized Korteweg–de Vries equation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huanhuan Lu ◽  
Yufeng Zhang
Keyword(s):  
Vries Equation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
Wave Solutions ◽  
De Vries ◽  
Rogue Wave Solutions ◽  
Bilinear Formulation ◽  
Hirota Bilinear

AbstractIn this paper, we analyse two types of rogue wave solutions generated from two improved ansatzs, to the (2 + 1)-dimensional generalized Korteweg–de Vries equation. With symbolic computation, the first-order rogue waves, second-order rogue waves, third-order rogue waves are generated directly from the first ansatz. Based on the Hirota bilinear formulation, another type of one-rogue waves and two-rogue waves can be obtained from the second ansatz. In addition, the dynamic behaviours of obtained rogue wave solutions are illustrated graphically.

scholarly journals Error estimates of finite difference schemes for the Korteweg–de Vries equation

2018 ◽  
Vol 40 (1) ◽  
pp. 628-685 ◽  
Author(s):  
Clémentine Courtès ◽  
Frédéric Lagoutière ◽  
Frédéric Rousset
Keyword(s):  
Cauchy Problem ◽  
Finite Difference ◽  
Vries Equation ◽  
Strong Solutions ◽  
Finite Difference Schemes ◽  
Order Convergence ◽  
First Order ◽  
De Vries ◽  
The Cauchy Problem ◽  
Dispersive Term

Abstract This article deals with the numerical analysis of the Cauchy problem for the Korteweg–de Vries equation with a finite difference scheme. We consider the explicit Rusanov scheme for the hyperbolic flux term and a 4-point $\theta $-scheme for the dispersive term. We prove the convergence under a hyperbolic Courant–Friedrichs–Lewy condition when $\theta \geq \frac{1}{2}$ and under an ‘Airy’ Courant–Friedrichs–Lewy condition when $\theta <\frac{1}{2}$. More precisely, we get a first-order convergence rate for strong solutions in the Sobolev space $H^s(\mathbb{R})$, $s \geq 6$ and extend this result to the nonsmooth case for initial data in $H^s(\mathbb{R})$, with $s\geq \frac{3}{4}$, at the price of a reduction in the convergence order. Numerical simulations indicate that the orders of convergence may be optimal when $s\geq 3$.

LINEARLY IMPLICIT ENERGY-PRESERVING FOURIER PSEUDOSPECTRAL SCHEMES FOR THE COMPLEX MODIFIED KORTEWEG–DE VRIES EQUATION

2020 ◽  
Vol 62 (3) ◽  
pp. 256-273
Author(s):  
J. L. YAN ◽  
L. H. ZHENG ◽  
L. ZHU ◽  
F. Q. LU
Keyword(s):  
Vries Equation ◽  
Pseudospectral Method ◽  
Time Discretization ◽  
Time Step ◽  
Fourier Pseudospectral Method ◽  
Fully Discrete ◽  
De Vries ◽  
Space Discretization ◽  
Invariant Energy Quadratization ◽  
Fourier Pseudospectral

AbstractWe propose two linearly implicit energy-preserving schemes for the complex modified Korteweg–de Vries equation, based on the invariant energy quadratization method. First, a new variable is introduced and a new Hamiltonian system is constructed for this equation. Then the Fourier pseudospectral method is used for the space discretization and the Crank–Nicolson leap-frog schemes for the time discretization. The proposed schemes are linearly implicit, which is only needed to solve a linear system at each time step. The fully discrete schemes can be shown to conserve both mass and energy in the discrete setting. Some numerical examples are also presented to validate the effectiveness of the proposed schemes.

Linearly implicit energy-preserving Fourier pseudospectral schemes for the complex modified Korteweg-de Vries equation

2021 ◽  
Vol 62 ◽  
pp. 256-273
Author(s):  
J. L. Yan ◽  
L. H. Zheng ◽  
L. Zhu ◽  
F. Q. Lu
Keyword(s):  
Vries Equation ◽  
Pseudospectral Method ◽  
Time Discretization ◽  
Time Step ◽  
Fourier Pseudospectral Method ◽  
Fully Discrete ◽  
De Vries ◽  
Space Discretization ◽  
Invariant Energy Quadratization ◽  
Fourier Pseudospectral

We propose two linearly implicit energy-preserving schemes for the complex modified Korteweg–de Vries equation, based on the invariant energy quadratization method. First, a new variable is introduced and a new Hamiltonian system is constructed for this equation. Then the Fourier pseudospectral method is used for the space discretization and the Crank–Nicolson leap-frog schemes for the time discretization. The proposed schemes are linearly implicit, which is only needed to solve a linear system at each time step. The fully discrete schemes can be shown to conserve both mass and energy in the discrete setting. Some numerical examples are also presented to validate the effectiveness of the proposed schemes.   doi:10.1017/S1446181120000218

scholarly journals Bilinear Form, Soliton, Breather, Hybrid and Periodic-Wave Solutions for a (3+1)-Dimensional Korteweg-De Vries Equation in a Fluid

2021 ◽  
Author(s):  
Cheng Chong-Dong ◽  
Tian Bo ◽  
Zhang Chen-Rong ◽  
Zhao Xin
Keyword(s):  
Bilinear Form ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Second Order ◽  
Periodic Wave Solutions ◽  
First Order ◽  
Wave Solutions ◽  
De Vries ◽  
Limiting Condition

Abstract Fluids are studied in such disciplines as atmospheric science, oceanography and astrophysics. In this paper, we investigate a (3+1)-dimensional Korteweg-de Vries equation in a fluid. Bilinear form and N-soliton solutions are obtained, where N is a positive integer. Via the N-soliton solutions, we derive the higher-order breather solutions. We observe that the interaction between two perpendicular first-order breathers on the x-y and x-z planes and the periodic line wave interacts with the first-order breather on the y-z plane, where x, y and z are the independent variables in the equation. Furthermore, we discuss the effects of α, β, γ and δ on the amplitudes of the second-order breathers, where α, β, γ and δ are the constant coefficients in the equation: Amplitude of the second-order breather decreases as α increases; Amplitude of the second-order breather increases as β increases; Amplitude of the second-order breather keeps invariant as γ and δ increase. Via the N-soliton solutions, hybrid solutions comprising the breathers and solitons are derived. Based on the Riemann theta function, we obtain the periodic-wave solutions. Furthermore, we find that the periodic-wave solutions approach to the one-soliton solutions under certain limiting condition.

scholarly journals A Compact-Type CIP Method for General Korteweg-de Vries Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
YuFeng Shi ◽  
Biao XU ◽  
Yan Guo
Keyword(s):  
High Resolution ◽  
Vries Equation ◽  
High Order ◽  
Cip Method ◽  
Compact Type ◽  
Constrained Interpolation ◽  
Numerical Examples ◽  
The Third ◽  
De Vries ◽  
High Order Compact Scheme

We proposed a hybrid compact-CIP scheme to solve the Korteweg-de Vries equation. The algorithm is based on classical constrained interpolation profile (CIP) method, which is coupled with high-order compact scheme for the third derivatives in Korteweg-de Vries equation. Several numerical examples are presented to confirm the high resolution of the proposed scheme.

scholarly journals Symmetry properties of a generalized Korteweg-de Vries equation and some explicit solutions

2005 ◽  
Vol 2005 (13) ◽  
pp. 2159-2173 ◽  
Author(s):  
Paul Bracken
Keyword(s):  
Vries Equation ◽  
First Integral ◽  
Generalized Equation ◽  
Group Method ◽  
Differential Constraint ◽  
Group Invariant ◽  
First Order ◽  
Symmetry Properties ◽  
De Vries ◽  
Group Invariant Solutions

The symmetry group method is applied to a generalized Korteweg-de Vries equation and several classes of group invariant solutions for it are obtained by means of this technique. Polynomial, trigonometric, and elliptic function solutions can be calculated. It is shown that this generalized equation can be reduced to a first-order equation under a particular second-order differential constraint which resembles a Schrödinger equation. For a particular instance in which the constraint is satisfied, the generalized equation is reduced to a quadrature. A condition which ensures that the reciprocal of a solution is also a solution is given, and a first integral to this constraint is found.

Integration of the nonlinear modified Korteweg-de Vries equation with a loaded term

2020 ◽  
Vol 2020 (2) ◽  
pp. 85-98
Author(s):  
A.B. Khasanov ◽  
T.J. Allanazarova
Keyword(s):  
Vries Equation ◽  
De Vries ◽  
Loaded Term

Convergence of the Rosenau-Korteweg-de Vries Equation to the Korteweg-de Vries One

2020 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
A Priori Estimates ◽  
Vries Equation ◽  
A Priori ◽  
Diffusion Parameter ◽  
Priori Estimates ◽  
De Vries ◽  
Wall Interactions

The Rosenau-Korteweg-de Vries equation describes the wave-wave and wave-wall interactions. In this paper, we prove that, as the diffusion parameter is near zero, it coincides with the Korteweg-de Vries equation. The proof relies on deriving suitable a priori estimates together with an application of the Aubin-Lions Lemma.

scholarly journals A Fully Implicit Finite Volume Scheme for a Seawater Intrusion Problem in Coastal Aquifers

Water ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1639
Author(s):  
Abdelkrim Aharmouch ◽  
Brahim Amaziane ◽  
Mustapha El Ossmani ◽  
Khadija Talali
Keyword(s):  
Finite Volume ◽  
Coastal Aquifers ◽  
Nonlinear Partial Differential Equations ◽  
Time Integration ◽  
Mass Conservation ◽  
Coupled System ◽  
Finite Volume Scheme ◽  
Time Step ◽  
Volume Scheme ◽  
Fully Implicit

We present a numerical framework for efficiently simulating seawater flow in coastal aquifers using a finite volume method. The mathematical model consists of coupled and nonlinear partial differential equations. Difficulties arise from the nonlinear structure of the system and the complexity of natural fields, which results in complex aquifer geometries and heterogeneity in the hydraulic parameters. When numerically solving such a model, due to the mentioned feature, attempts to explicitly perform the time integration result in an excessively restricted stability condition on time step. An implicit method, which calculates the flow dynamics at each time step, is needed to overcome the stability problem of the time integration and mass conservation. A fully implicit finite volume scheme is developed to discretize the coupled system that allows the use of much longer time steps than explicit schemes. We have developed and implemented this scheme in a new module in the context of the open source platform DuMu X . The accuracy and effectiveness of this new module are demonstrated through numerical investigation for simulating the displacement of the sharp interface between saltwater and freshwater in groundwater flow. Lastly, numerical results of a realistic test case are presented to prove the efficiency and the performance of the method.

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