scholarly journals Oscillation of damped second order quasilinear wave equations with mixed arguments

2021 ◽  
Vol 117 ◽  
pp. 107060
Author(s):  
Ying Sui ◽  
Huimin Yu
Keyword(s):  
Wave Equations ◽  
Second Order ◽  
Quasilinear Wave Equations ◽  
Mixed Arguments

Periodic solutions of second-order wave equations. IV

1989 ◽  
Vol 40 (6) ◽  
pp. 639-644
Author(s):  
Yu. A. Mitropol'skii ◽  
G. P. Khoma
Keyword(s):  
Periodic Solutions ◽  
Wave Equations ◽  
Second Order

Nonsplit complex-frequency shifted perfectly matched layer combined with symplectic methods for solving second-order seismic wave equations — Part 1: Method

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. T301-T311 ◽  
Author(s):  
Xiao Ma ◽  
Dinghui Yang ◽  
Xueyuan Huang ◽  
Yanjie Zhou
Keyword(s):  
Boundary Condition ◽  
Seismic Wave ◽  
Wave Equations ◽  
Perfectly Matched Layer ◽  
Absorbing Boundary Conditions ◽  
Second Order ◽  
Complex Frequency ◽  
Absorbing Boundary ◽  
Time Layer ◽  
The Time Domain

The absorbing boundary condition plays an important role in seismic wave modeling. The perfectly matched layer (PML) boundary condition has been established as one of the most effective and prevalent absorbing boundary conditions. Among the existing PML-type conditions, the complex frequency shift (CFS) PML attracts considerable attention because it can handle the evanescent and grazing waves better. For solving the resultant CFS-PML equation in the time domain, one effective technique is to apply convolution operations, which forms the so-called convolutional PML (CPML). We have developed the corresponding CPML conditions with nonconstant grid compression parameter, and used its combination algorithms specifically with the symplectic partitioned Runge-Kutta and the nearly analytic SPRK methods for solving second-order seismic wave equations. This involves evaluating second-order spatial derivatives with respect to the complex stretching coordinates at the noninteger time layer. Meanwhile, two kinds of simplification algorithms are proposed to compute the composite convolutions terms contained therein.

scholarly journals Small-data shock formation in solutions to 3D quasilinear wave equations: An overview

2016 ◽  
Vol 13 (01) ◽  
pp. 1-105 ◽  
Author(s):  
Gustav Holzegel ◽  
Sergiu Klainerman ◽  
Jared Speck ◽  
Willie Wai-Yeung Wong
Keyword(s):  
Wave Equations ◽  
Initial Conditions ◽  
Large Body ◽  
Small Data ◽  
Shock Formation ◽  
Time Behavior ◽  
Long Time ◽  
Remarkable Result ◽  
Quasilinear Wave Equations ◽  
Three Space

In his 2007 monograph, Christodoulou proved a remarkable result giving a detailed description of shock formation, for small [Formula: see text]-initial conditions (with [Formula: see text] sufficiently large), in solutions to the relativistic Euler equations in three space dimensions. His work provided a significant advancement over a large body of prior work concerning the long-time behavior of solutions to higher-dimensional quasilinear wave equations, initiated by John in the mid 1970’s and continued by Klainerman, Sideris, Hörmander, Lindblad, Alinhac, and others. Our goal in this paper is to give an overview of his result, outline its main new ideas, and place it in the context of the above mentioned earlier work. We also introduce the recent work of Speck, which extends Christodoulou’s result to show that for two important classes of quasilinear wave equations in three space dimensions, small-data shock formation occurs precisely when the quadratic nonlinear terms fail to satisfy the classic null condition.

Sign in / Sign up

Export Citation Format

Share Document