COMPRESSIBLE 1D EULER EQUATIONS WITH LARGE DATA: A CASE STUDY

Consider 1D flow of a compressible, ideal, and polytropic gas on a bounded interval in Lagrangian variables. We study the Cauchy problem when the initial data consist of four constant states that yield two contact waves bounding an interval of lower density, together with an admissible shock between them. To render the solution tractable for direct calculations, we also impose absorbing boundary conditions, at fixed locations (in Lagrangian coordinates) to the left and to the right of the two contacts. By estimating the wave strengths in shock–contact interactions, we show that the resulting flow is defined for all times. In particular, the pressure, density, particle velocities, and shock speeds are all uniformly bounded in time. We also record a scaling invariance of the system and comment on its relevance to large data solutions of the Euler system.

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Mass-conserving tempered fractional diffusion in a bounded interval

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2019-0081 ◽

2019 ◽

Vol 22 (6) ◽

pp. 1561-1595 ◽

Cited By ~ 1

Author(s):

Anna Lischke ◽

James F. Kelly ◽

Mark M. Meerschaert

Keyword(s):

Numerical Solutions ◽

Mass Conservation ◽

Fractional Diffusion ◽

Absorbing Boundary Conditions ◽

Late Time ◽

Conditional Stability ◽

Absorbing Boundary ◽

Bounded Interval ◽

Reflecting Boundaries ◽

Euler Methods

Abstract Transient anomalous diffusion may be modeled by a tempered fractional diffusion equation. A reflecting boundary condition enforces mass conservation on a bounded interval. In this work, explicit and implicit Euler schemes for tempered fractional diffusion with discrete reflecting or absorbing boundary conditions are constructed. Discrete reflecting boundaries are formulated such that the Euler schemes conserve mass. Conditional stability of the explicit Euler methods and unconditional stability of the implicit Euler methods are established. Analytical steady-state solutions involving the Mittag-Leffler function are derived and shown to be consistent with late-time numerical solutions. Several numerical examples are presented to demonstrate the accuracy and usefulness of the proposed numerical schemes.

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On a Sensor Placement Methodology for Monitoring the Vibrations of Horizontally Excited Ground

Sensors ◽

10.3390/s20071938 ◽

2020 ◽

Vol 20 (7) ◽

pp. 1938 ◽

Cited By ~ 2

Author(s):

Aneta Herbut ◽

Jarosław Rybak ◽

Włodzimierz Brząkała

Keyword(s):

Equations Of Motion ◽

Boundary Surface ◽

Ground Surface ◽

Reduction Factor ◽

Absorbing Boundary Conditions ◽

Vibration Monitoring ◽

Absorbing Boundary ◽

Soil Medium ◽

Sensor Arrangement ◽

Particle Velocities

In this paper, the problem of optimal sensor arrangement during vibration monitoring is analysed. The wave propagation caused by horizontal excitation is investigated to predict the areas of the largest ground and structure response. The equations of motion for a transversally isotropic elastic medium with appropriate absorbing boundary conditions are solved using the finite element method (FlexPDE software). The possibility of an amplified soil medium response is examined for points located on the ground surface and at various depths. The results are presented in the form of a dimensionless vibration reduction factor, defined as the ratio of the peak particle velocity observed at the selected depth to the corresponding value observed at the ground surface. Significant amplifications (≈50%) can be observed below the ground surface, especially in the case of a weak layer below a stiff layer. The effect of vibration amplification is most significant near the boundary surface of two layers. For the points located on the ground surface, the greatest peak particle velocities are observed in the direction perpendicular to the load direction. However, the greatest vertical velocity component at the ground surface is observed in front of the applied force.

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Toward perfectly absorbing boundary conditions for Euler equations

AIAA Journal ◽

10.2514/3.14263 ◽

1999 ◽

Vol 37 ◽

pp. 912-918

Author(s):

M. E. Hayder ◽

Fang Q. Hu ◽

M. Y. Hussaini

Keyword(s):

Boundary Conditions ◽

Euler Equations ◽

Absorbing Boundary Conditions ◽

Absorbing Boundary

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Dirichlet absorbing boundary conditions for classical and peridynamic diffusion-type models

Computational Mechanics ◽

10.1007/s00466-020-01879-1 ◽

2020 ◽

Vol 66 (4) ◽

pp. 773-793 ◽

Cited By ~ 1

Author(s):

Arman Shojaei ◽

Alexander Hermann ◽

Pablo Seleson ◽

Christian J. Cyron

Keyword(s):

Boundary Conditions ◽

Materials Science ◽

Unbounded Domains ◽

Diffusion Models ◽

Absorbing Boundary Conditions ◽

The Finite Element Method ◽

Absorbing Boundary ◽

Diffusion Type ◽

2D And 3D ◽

Accuracy And Stability

Abstract Diffusion-type problems in (nearly) unbounded domains play important roles in various fields of fluid dynamics, biology, and materials science. The aim of this paper is to construct accurate absorbing boundary conditions (ABCs) suitable for classical (local) as well as nonlocal peridynamic (PD) diffusion models. The main focus of the present study is on the PD diffusion formulation. The majority of the PD diffusion models proposed so far are applied to bounded domains only. In this study, we propose an effective way to handle unbounded domains both with PD and classical diffusion models. For the former, we employ a meshfree discretization, whereas for the latter the finite element method (FEM) is employed. The proposed ABCs are time-dependent and Dirichlet-type, making the approach easy to implement in the available models. The performance of the approach, in terms of accuracy and stability, is illustrated by numerical examples in 1D, 2D, and 3D.

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