TY - JOUR

T1 - A simple and robust boundary treatment for the forced Korteweg-de Vries equation

AU - Lee, Hyun Geun

AU - Kim, Junseok

N1 - Funding Information:
The first author (Hyun Geun Lee) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( 2009-0093827 ). The authors thank the reviewers for the constructive and helpful comments on the revision of this article. The corresponding author (J.S. Kim) also thanks Professor Jeong-Whan Choi for suggesting this problem and for valuable discussions.

PY - 2014/7

Y1 - 2014/7

N2 - In this paper, we propose a simple and robust numerical method for the forced Korteweg-de Vries (fKdV) equation which models free surface waves of an incompressible and inviscid fluid flow over a bump. The fKdV equation is defined in an infinite domain. However, to solve the equation numerically we must truncate the infinite domain to a bounded domain by introducing an artificial boundary and imposing boundary conditions there. Due to unsuitable artificial boundary conditions, most wave propagation problems have numerical difficulties (e.g., the truncated computational domain must be large enough or the numerical simulation must be terminated before the wave approaches the artificial boundary for the quality of the numerical solution). To solve this boundary problem, we develop an absorbing non-reflecting boundary treatment which uses outward wave velocity. The basic idea of the proposing algorithm is that we first calculate an outward wave velocity from the solutions at the previous and present time steps and then we obtain a solution at the next time step on the artificial boundary by moving the solution at the present time step with the velocity. And then we update solutions at the next time step inside the domain using the calculated solution on the artificial boundary. Numerical experiments with various initial conditions for the KdV and fKdV equations are presented to illustrate the accuracy and efficiency of our method.

AB - In this paper, we propose a simple and robust numerical method for the forced Korteweg-de Vries (fKdV) equation which models free surface waves of an incompressible and inviscid fluid flow over a bump. The fKdV equation is defined in an infinite domain. However, to solve the equation numerically we must truncate the infinite domain to a bounded domain by introducing an artificial boundary and imposing boundary conditions there. Due to unsuitable artificial boundary conditions, most wave propagation problems have numerical difficulties (e.g., the truncated computational domain must be large enough or the numerical simulation must be terminated before the wave approaches the artificial boundary for the quality of the numerical solution). To solve this boundary problem, we develop an absorbing non-reflecting boundary treatment which uses outward wave velocity. The basic idea of the proposing algorithm is that we first calculate an outward wave velocity from the solutions at the previous and present time steps and then we obtain a solution at the next time step on the artificial boundary by moving the solution at the present time step with the velocity. And then we update solutions at the next time step inside the domain using the calculated solution on the artificial boundary. Numerical experiments with various initial conditions for the KdV and fKdV equations are presented to illustrate the accuracy and efficiency of our method.

KW - Absorbing non-reflecting boundary treatment

KW - Forced Korteweg-de Vries equation

KW - Free surface waves

KW - Semi-implicit finite difference method

UR - http://www.scopus.com/inward/record.url?scp=84892976610&partnerID=8YFLogxK

U2 - 10.1016/j.cnsns.2013.12.019

DO - 10.1016/j.cnsns.2013.12.019

M3 - Article

AN - SCOPUS:84892976610

SN - 1007-5704

VL - 19

SP - 2262

EP - 2271

JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

IS - 7

ER -