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.

A new high-order energy-preserving scheme for the modified Korteweg-de Vries equation

2016 ◽  
Vol 74 (3) ◽  
pp. 659-674 ◽  
Author(s):  
Jin-Liang Yan ◽  
Qian Zhang ◽  
Zhi-Yue Zhang ◽  
Dong Liang
Keyword(s):  
Vries Equation ◽  
High Order ◽  
Order Energy ◽  
De Vries

scholarly journals On optimal high-order in time approximations for the Korteweg-de Vries equation

1990 ◽  
Vol 55 (192) ◽  
pp. 473-473 ◽  
Author(s):  
Ohannes Karakashian ◽  
William McKinney
Keyword(s):  
Vries Equation ◽  
High Order ◽  
De Vries

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.

High-orders methods and high-precision arithmetics make direct scattering transform for the Korteweg-De Vries equation robust.

2021 ◽  
Author(s):  
Aleksandr Gudko ◽  
Andrey Gelash ◽  
Rustam Mullyadzhanov
Keyword(s):  
High Precision ◽  
Vries Equation ◽  
High Order ◽  
Scattering Data ◽  
Computational Domain ◽  
Wave Fields ◽  
Scattering Coefficients ◽  
Direct Scattering ◽  
Scattering Transform ◽  
De Vries

<p>Similar to the theory of direct scattering transform for nonlinear wave fields containing solitons within the focusing one-dimensional nonlinear Schrödinger equation [1], we revisit the theory associated with the Korteweg–De Vries equation. We study a crucial fundamental property of the scattering problem for multisoliton potentials demonstrating that in many cases position parameters of solitons cannot be identified with standard machine precision arithmetics making solitons in some sense “uncatchable”. Using the dressing method we find the landscape of soliton scattering coefficients in the plane of the complex spectral parameter for multisoliton wave fields truncated within a finite domain, allowing us to capture the nature of such anomalous numerical errors. They depend on the size of the computational domain L leading to a counterintuitive exponential divergence when increasing L in the presence of a small uncertainty in soliton eigenvalues. Then we demonstrate how one of the scattering coefficients loses its analytical properties due to the lack of the wave-field compact support in case of L→∞. Finally, we show that despite this inherent direct scattering transform feature, the wave fields of arbitrary complexity can be reliably analyzed using high-precision arithmetics and high-order algorithms based on the Magnus expansion [2, 3] providing accurate information about soliton amplitudes, velocities<span>, positions</span> and intensity of the radiation. This procedure is robust even in the presence of noise opening broad perspectives in analyzing experimental data on propagation of surface waves on shallow water.</p><p>The work is partially funded by Russian Science Foundation grant No 19-79-30075.</p><p>[1] Gelash A., Mullyadzhanov R. Anomalous errors of direct scattering transform // Physical Review E 101 (5), 052206, 2020.</p><p>[2] Mullyadzhanov R., Gelash A. Direct scattering transform of large wave packets // Optics Letters 44 (21), 5298-5301, 2019.</p><p>[3] Gudko A., Gelash A., Mullyadzhanov R. High-order numerical method for scattering data of the Korteweg—De Vries equation // Journal of Physics: Conference Series 1677 (1), 012011, 2020.</p><p> </p><p> </p>

Sign in / Sign up

Export Citation Format

Share Document