$L_\infty $ Estimates of Optimal Orders for Galerkin Methods for One-Dimensional Second Order Parabolic and Hyperbolic Equations

1973 ◽  
Vol 10 (5) ◽  
pp. 908-913 ◽  
Author(s):  
Mary Fanett Wheeler
Keyword(s):  
Hyperbolic Equations ◽  
Second Order ◽  
Galerkin Methods ◽  
One Dimensional

Effect of the initial conditions on a one-dimensional model of small-amplitude wave propagation in shallow water

2020 ◽  
Vol 30 (11) ◽  
pp. 4979-5014
Author(s):  
J.I. Ramos ◽  
Carmen María García López
Keyword(s):  
Small Amplitude ◽  
Hyperbolic Equations ◽  
Initial Conditions ◽  
Time Derivative ◽  
Second Order ◽  
Shallow Waters ◽  
Content Type ◽  
One Dimensional ◽  
Pulse Type ◽  
Dimensional Equation

Purpose The purpose of this paper is to determine both analytically and numerically the solution to a new one-dimensional equation for the propagation of small-amplitude waves in shallow waters that accounts for linear and nonlinear drift, diffusive attenuation, viscosity and dispersion, its dependence on the initial conditions, and its linear stability. Design/methodology/approach An implicit, finite difference method valid for both parabolic and second-order hyperbolic equations has been used to solve the equation in a truncated domain for five different initial conditions, a nil initial first-order time derivative and relaxation times linearly proportional to the viscosity coefficient. Findings A fast transition that depends on the coefficient of the linear drift, the diffusive attenuation and the power of the nonlinear drift are found for initial conditions corresponding to the exact solution of the generalized regularized long-wave equation. For initial Gaussian, rectangular and triangular conditions, the wave’s amplitude and speed increase as both the amplitude and the width of these conditions increase and decrease, respectively; wide initial conditions evolve into a narrow leading traveling wave of the pulse type and a train of slower oscillatory secondary ones. For the same initial mass and amplitude, rectangular initial conditions result in larger amplitude and velocity waves of the pulse type than Gaussian and triangular ones. The wave’s kinetic, potential and stretching energies undergo large changes in an initial layer whose thickness is on the order of the diffusive attenuation coefficient. Originality/value A new, one-dimensional equation for the propagation of small-amplitude waves in shallow waters is proposed and studied analytically and numerically. The equation may also be used to study the displacement of porous media subject to seismic effects, the dispersion of sound in tunnels, the attenuation of sound because of viscosity and/or heat and mass diffusion, the dynamics of second-order, viscoelastic fluids, etc., by appropriate choices of the parameters that appear in it.

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