Asymptotic decay to relaxation shock fronts in two dimensions

2001 ◽  
Vol 131 (6) ◽  
pp. 1385-1410 ◽  
Author(s):  
Hailiang Liu
Keyword(s):  
Shock Front ◽  
Initial Perturbation ◽  
Two Dimensions ◽  
Decay Rates ◽  
Front Solution ◽  
Asymptotic Decay ◽  
Planar Shock ◽  
Spatial Dimensions ◽  
Shock Front Solution ◽  
Shock Fronts

We prove nonlinear stability of planar shock fronts for certain relaxation systems in two spatial dimensions. If the subcharacteristic condition is assumed and the initial perturbation is sufficiently small and the mass carried by the perturbations is not necessarily finite, then the solution converges to a shifted planar shock front solution as time t ↑ ∞. The asymptotic phase shift of shock fronts is, in general, non-zero and governed by a similarity solution to the heat equation. The asymptotic decay rate to the shock front is proved to be t−1/4 in L∞(R2) without imposing extra decay rates in space for the initial perturbations. The proofs are based on an elementary weighted energy analysis to the error equation.

GLOBAL EXISTENCE OF SHOCK FRONT SOLUTIONS TO THE AXIALLY SYMMETRIC PISTON PROBLEM FOR COMPRESSIBLE FLUIDS

2004 ◽  
Vol 01 (01) ◽  
pp. 51-84 ◽  
Author(s):  
SHUXING CHEN ◽  
ZEJUN WANG ◽  
YONGQIAN ZHANG
Keyword(s):  
Global Existence ◽  
Shock Front ◽  
Compressible Fluids ◽  
Axially Symmetric ◽  
Piston Problem ◽  
Front Solution ◽  
Glimm Scheme ◽  
Front Solutions ◽  
Shock Front Solution

In this paper, we study the axially symmetric piston problem for compressible fluids when the velocity of the piston is a perturbation of a constant. Under the assumptions that both the velocity of the piston and the density of the gas outside the piston are small, we prove the global existence of a shock front solution by using a modified Glimm scheme.

scholarly journals On Monotonic Pattern in Periodic Boundary Solutions of Cylindrical and Spherical Kortweg–De Vries–Burgers Equations

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 220
Author(s):  
Alexey Samokhin
Keyword(s):  
Periodic Boundary ◽  
Asymptotic Formulas ◽  
Burgers Equations ◽  
Spherical Waves ◽  
Constant Height ◽  
De Vries ◽  
Spatial Dimensions ◽  
Integral Characteristics ◽  
Boundary Solutions ◽  
Shock Fronts

We studied, for the Kortweg–de Vries–Burgers equations on cylindrical and spherical waves, the development of a regular profile starting from an equilibrium under a periodic perturbation at the boundary. The regular profile at the vicinity of perturbation looks like a periodical chain of shock fronts with decreasing amplitudes. Further on, shock fronts become decaying smooth quasi-periodic oscillations. After the oscillations cease, the wave develops as a monotonic convex wave, terminated by a head shock of a constant height and equal velocity. This velocity depends on integral characteristics of a boundary condition and on spatial dimensions. In this paper the explicit asymptotic formulas for the monotonic part, the head shock and a median of the oscillating part are found.

ASYMPTOTIC DECAY TOWARD THE PLANAR RAREFACTION WAVES FOR A MODEL SYSTEM OF THE RADIATING GAS IN TWO DIMENSIONS

2008 ◽  
Vol 18 (04) ◽  
pp. 511-541 ◽  
Author(s):  
WENLIANG GAO ◽  
CHANGJIANG ZHU
Keyword(s):  
Cauchy Problem ◽  
Maximum Principle ◽  
Coupled System ◽  
Model System ◽  
Two Dimensions ◽  
Rarefaction Waves ◽  
Asymptotic Decay ◽  
The Maximum Principle ◽  
The Cauchy Problem ◽  
Asymptotic Decay Rate

In this paper, we consider the asymptotic decay rate towards the planar rarefaction waves to the Cauchy problem for a hyperbolic–elliptic coupled system called as a model system of the radiating gas in two dimensions. The analysis based on the standard L2-energy method, L1-estimate and the monotonicity of profile obtained by the maximum principle.

scholarly journals Exact solution and the multidimensional Godunov scheme for the acoustic equations

2022 ◽  
Author(s):  
Wasilij Barsukow ◽  
Christian Klingenberg
Keyword(s):  
Exact Solution ◽  
Mach Number ◽  
Euler Equations ◽  
Two Dimensions ◽  
Finite Volume Scheme ◽  
Godunov Scheme ◽  
Low Mach Number ◽  
Low Mach Number Limit ◽  
Acoustic Equations ◽  
Spatial Dimensions

The acoustic equations derived as a linearization of the Euler equations are a valuable system for studies of multi-dimensional solutions. Additionally they possess a low Mach number limit analogous to that of the Euler equations. Aiming at understanding the behaviour of the multi-dimensional Godunov scheme in this limit, first the exact solution of the corresponding Cauchy problem in three spatial dimensions is derived. The appearance of logarithmic singularities in the exact solution of the 4-quadrant Riemann Problem in two dimensions is discussed. The solution formulae are then used to obtain the multidimensional Godunov finite volume scheme in two dimensions. It is shown to be superior to the dimensionally split upwind/Roe scheme concerning its domain of stability and ability to resolve multi-dimensional Riemann problems. It is shown experimentally and theoretically that despite taking into account multi-dimensional information it is, however, not able to resolve the low Mach number limit.

scholarly journals Periodic self-reformation of rippled perpendicular collisionless shocks in two dimensions

2018 ◽  
Vol 36 (4) ◽  
pp. 1047-1055 ◽  
Author(s):  
Takayuki Umeda ◽  
Yuki Daicho
Keyword(s):  
Magnetic Field ◽  
Shock Front ◽  
Large Scale ◽  
Large Mass ◽  
Small Mass ◽  
Two Dimensions ◽  
Electron Mass ◽  
The Reformation ◽  
Ion Cyclotron ◽  
Mass Ratios

Abstract. Large-scale two-dimensional (2-D) full particle-in-cell (PIC) simulations are carried out for studying periodic self-reformation of a supercritical collisionless perpendicular shock with an Alfvén–Mach number MA∼6. Previous self-consistent one-dimensional (1-D) hybrid and full PIC simulations have demonstrated that the periodic reflection of upstream ions at the shock front is responsible for the formation and vanishing of the shock-foot region on a timescale of the local ion cyclotron period, which was defined as the reformation of (quasi-)perpendicular shocks. The present 2-D full PIC simulations with different ion-to-electron mass ratios show that the dynamics at the shock front is strongly modified by large-amplitude ion-scale fluctuations at the shock overshoot, which are known as ripples. In the run with a small mass ratio, the simultaneous enhancement of the shock magnetic field and the reflected ions take place quasi-periodically, which is identified as the reformation. In the runs with large mass ratios, the simultaneous enhancement of the shock magnetic field and the reflected ions occur randomly in time, and the shock magnetic field is enhanced on a timescale much shorter than the ion cyclotron period. These results indicate a coupling between the shock-front ripples and electromagnetic microinstabilities in the foot region in the runs with large mass ratios. Keywords. Space plasma physics (wave–particle interactions)

scholarly journals Application of the Gravity Flow Theory to the Percolation of Melt Water Through Firn

1981 ◽  
Vol 27 (95) ◽  
pp. 67-75 ◽  
Author(s):  
W. Ambach ◽  
M. Blumthaler ◽  
P. Kirchlechner
Keyword(s):  
Experimental Data ◽  
Shock Front ◽  
Water Flux ◽  
Flow Theory ◽  
Gravity Flow ◽  
Total Thickness ◽  
Total Input ◽  
Melt Water ◽  
Shock Fronts ◽  
Good Agreement

Abstract Application of the gravity flow theory to the percolation of melt water through the firn in the accumulation area of a temperate glacier explains the occurrence of shock fronts in the melt-water flux. The time of propagation of a shock front moving from the surface through the entire firn was calculated under various assumptions. Various time input functions of melt-water flux at the surface with constant total input volumes yield only slight differences in the time of propagation of the shock front at greater depths. The dependence of the time of propagation of a shock front on the input volume, on snow parameters, and on the total thickness of the firn was calculated. An approximately linear relation was found to exist between the time of propagation of a shock front moving through the firn and the total thickness of the firn. The drainage of melt water from the firn after the summer ablation period is also quantitatively explained by the gravity flow theory. All results are in good agreement with experimental data.

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