RT Journal Article
T1 Effect of the initial conditions on a one-dimensional model of small-amplitude wave propagation in shallow water: II: Blowup for nonsmooth conditions.
A1 Ramos-Sobrados, Juan Ignacio
A1 García-López, Carmen María
K1 Ecuación de onda
K1 Mecánica ondulatoria
K1 Diferencias finitas
K1 Distribución gaussiana
AB Abstract Purpose – The purpose of this paper is to analyze numerically the blowup in finite time of the solutions to a one-dimensional, bidirectional, nonlinear wave model equation for the propagation of small-amplitude waves in shallow water, as a function of the relaxation time, linear and nonlinear drift, power of the nonlinear advection flux, viscosity coefficient, viscous attenuation, and amplitude, smoothness and width of three types of initial conditions. Design/methodology/approach – An implicit, first-order accurate in time, finite difference method valid for semipositive relaxation times has been used to solve the equation in a truncated domain for three different initial conditions, a first-order time derivative initially equal to zero and several constant wave speeds. Findings – The numerical experiments show a very rapid transient from the initial conditions to the formation of a leading propagating wave, whose duration depends strongly on the shape, amplitude and width of the initial data as well as on the coefficients of the bidirectional equation. The blowup times for the triangular conditions have been found to be larger than those for the Gaussian ones, and the latter are larger than those for rectangular conditions, thus indicating that the blowup time decreases as the smoothness of the initial conditions decreases. The blowup time has also been found to decrease as the relaxation time, degree of nonlinearity, linear drift coefficient and amplitude of the initial conditions are increased, and as the width of the initial condition is decreased, but it increases as the viscosity coefficient is increased. No blowup has been observed for relaxation times smaller than one-hundredth, viscosity coefficients larger than ten-thousandths, quadratic and cubic nonlinearities, and initial Gaussian, triangular and rectangular conditions of unity amplitude.
PB Emerald
YR 2023
FD 2023-11-27
LK https://hdl.handle.net/10630/28153
UL https://hdl.handle.net/10630/28153
LA eng
NO Ramos, J.I. and García López, C.M. (2023), "Effect of initial conditions on a one-dimensional model of small-amplitude wave propagation in shallow water. II: Blowup for nonsmooth conditions", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/HFF-07-2023-0413
NO Funding for open access charge: Universidad de Málaga / CBUA
DS RIUMA. Repositorio Institucional de la Universidad de Málaga
RD 22 ene 2026