Mass-conserving tempered fractional diffusion in a bounded interval

2019 ◽  
Vol 22 (6) ◽  
pp. 1561-1595 ◽  
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.

A 3D cylindrical PML/FDTD method for elastic waves in fluid‐filled pressurized boreholes in triaxially stressed formations

Geophysics ◽  
2003 ◽  
Vol 68 (5) ◽  
pp. 1731-1743 ◽  
Author(s):  
Qing Huo Liu ◽  
Bikash K. Sinha
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Time Domain ◽  
Elastic Waves ◽  
Numerical Solutions ◽  
Fdtd Method ◽  
Uniaxial Stress ◽  
Absorbing Boundary Conditions ◽  
Principle Of Superposition ◽  
Absorbing Boundary

A new 3D cylindrical perfectly matched layer (PML) formulation is developed for elastic wave propagation in a pressurized borehole surrounded by a triaxially stressed solid formation. The linear elastic formation is altered by overburden and tectonic stresses that cause significant changes in the wave propagation characteristics in a borehole. The 3D cylindrical problem with both radial and azimuthal heterogeneities is suitable for numerical solutions of the wave equations by finite‐difference time‐domain (FDTD) and pseudospectral time‐domain (PSTD) methods. Compared to the previous 2.5D formulation with other absorbing boundary conditions, this 3D cylindrical PML formulation allows modeling of a borehole‐conformal, full 3D description of borehole elastic waves in a stress‐induced heterogeneous formation. We have developed an FDTD method using this PML as an absorbing boundary condition. In addition to the ability to solve full 3D problems, this method is found to be advantageous over the previously reported 2.5D finite‐difference formulation because a borehole can now be adequately simulated with fewer grid points. Results from the new FDTD technique confirm the principle of superposition of the influence of various stress components on both the borehole monopole and dipole dispersions. In addition, we confirm that the increase in shear‐wave velocity caused by a uniaxial stress applied in the propagation direction is the same as that applied parallel to the radial polarization direction.

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

2009 ◽  
Vol 06 (02) ◽  
pp. 389-406 ◽  
Author(s):  
ERIK E. ENDRES ◽  
HELGE KRISTIAN JENSSEN
Keyword(s):  
Large Data ◽  
Euler System ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary ◽  
Scaling Invariance ◽  
Bounded Interval ◽  
Lagrangian Variables ◽  
The Cauchy Problem ◽  
Particle Velocities ◽  
The Right

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.

scholarly journals Entanglement wedge reconstruction and the information paradox

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Geoffrey Penington
Keyword(s):  
Black Hole ◽  
Hawking Radiation ◽  
Entanglement Entropy ◽  
State Dependence ◽  
Absorbing Boundary Conditions ◽  
Late Time ◽  
Mixed States ◽  
Initial State ◽  
Absorbing Boundary ◽  
The Past

Abstract When absorbing boundary conditions are used to evaporate a black hole in AdS/CFT, we show that there is a phase transition in the location of the quantum Ryu-Takayanagi surface, at precisely the Page time. The new RT surface lies slightly inside the event horizon, at an infalling time approximately the scrambling time β/2πlogSBH into the past. We can immediately derive the Page curve, using the Ryu-Takayanagi formula, and the Hayden-Preskill decoding criterion, using entanglement wedge reconstruction. Because part of the interior is now encoded in the early Hawking radiation, the decreasing entanglement entropy of the black hole is exactly consistent with the semiclassical bulk entanglement of the late-time Hawking modes, despite the absence of a firewall.By studying the entanglement wedge of highly mixed states, we can understand the state dependence of the interior reconstructions. A crucial role is played by the existence of tiny, non-perturbative errors in entanglement wedge reconstruction. Directly after the Page time, interior operators can only be reconstructed from the Hawking radiation if the initial state of the black hole is known. As the black hole continues to evaporate, reconstructions become possible that simultaneously work for a large class of initial states. Using similar techniques, we generalise Hayden-Preskill to show how the amount of Hawking radiation required to reconstruct a large diary, thrown into the black hole, depends on both the energy and the entropy of the diary. Finally we argue that, before the evaporation begins, a single, state-independent interior reconstruction exists for any code space of microstates with entropy strictly less than the Bekenstein-Hawking entropy, and show that this is sufficient state dependence to avoid the AMPSS typical-state firewall paradox.

An empirical study of instability and improvement of absorbing boundary conditions for the elastic wave equation

Geophysics ◽  
1986 ◽  
Vol 51 (7) ◽  
pp. 1499-1501 ◽  
Author(s):  
Kenneth D. Mahrer
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Incident Energy ◽  
Numerical Solutions ◽  
Finite Size ◽  
Computer Time ◽  
Absorbing Boundary Conditions ◽  
Elastic Wave Equation ◽  
Absorbing Boundary ◽  
Numerical Grid

One of the persistent problems with numerical solutions to the elastic wave equation is the finite size of the numerical grid. As with a physical body, the grid boundaries will reflect incident energy. If not eliminated or reduced substantially, these reflections will invade the grid interior and interfere with the desired solution. One method for eliminating reflections is creating a large and/or expanding grid. This method may be impractical since it can be quite costly in both computer time and memory. Another method is making the grid boundary “transparent” to outgoing energy. This method is ideally done by designing absorbing or nonreflecting boundaries which are mathematically equivalent to a one‐way, or outgoing, elastic wave equation only. In practice, an outgoing elastic wave equation is an approximation since the wave equation is not generally separable into outgoing and incoming parts. Two absorbing boundary condition approximations commonly used are those from Reynolds (Reynolds, 1978) and Clayton and Engquist, (Clayton and Engquist, 1977).

scholarly journals A Generating - Absorbing Boundary Condition Applied to Wave - Current Interactions Using the Method of Fundamental Solutions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammed Loukili ◽  
Kamila Kotrasova ◽  
Amine Bouaine
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
Fundamental Solutions ◽  
Absorbing Boundary Conditions ◽  
Method Of Fundamental Solutions ◽  
Computational Domain ◽  
Numerical Wave Tank ◽  
Absorbing Boundary ◽  
Wave Properties

Abstract The purpose of this work is to study the feasibility and efficiency of Generating Absorbing Boundary Conditions (GABCs), applied to wave-current interactions using the Method of Fundamental Solutions (MFS) as radial basis function, the problem is solved by collocation method. The objective is modeling wave-current interactions phenomena applied in a Numerical Wave Tank (NWT) where the flow is described within the potential theory, using a condition without resorting to the sponge layers on the boundaries. To check the feasibility and efficiency of GABCs presented in this paper, we verify accurately the numerical solutions by comparing the numerical solutions with the analytical ones. Further, we check the accuracy of numerical solutions by trying a different number of nodes. Thereafter, we evaluate the influence of different aspects of current (coplanar current, without current, and opposing current) on the wave properties. As an application, we take into account the generating-absorbing boundary conditions GABCs in a computational domain with a wavy downstream wall to confirm the efficiency of the adopted numerical boundary condition.

Toward perfectly absorbing boundary conditions for Euler equations

AIAA Journal ◽  
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

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

2020 ◽  
Vol 66 (4) ◽  
pp. 773-793 ◽  
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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