Transport-Elliptic Estimates

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Elliptic Equation ◽  
Transport Equation ◽  
Transport Properties ◽  
Error Term ◽  
Main Lemma ◽  
Spatial Derivative ◽  
Material Derivative ◽  
Divergence Operator ◽  
Spatial Derivatives ◽  
Elliptic Estimates

This chapter solves the underdetermined, elliptic equation ∂ⱼQsuperscript jl = Usuperscript l and Qsuperscript jl = Qsuperscript lj (Equation 1069) in order to eliminate the error term in the parametrix. For the proof of the Main Lemma, estimates for Q and the material derivative as well as its spatial derivatives are derived. The chapter finds a solution to Equation (1069) with good transport properties by solving it via a Transport equation obtained by commuting the divergence operator with the material derivative. It concludes by showing the solutions, spatial derivative estimates, and material derivative estimates for the Transport-Elliptic equation, as well as cutting off the solution to the Transport-Elliptic equation.

Energy Approximation

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Main Lemma ◽  
The Other ◽  
Coarse Scale ◽  
Continuous Solutions ◽  
Material Derivative ◽  
Energy Variation ◽  
Energy Approximation ◽  
The Difference ◽  
Derivatives Of

This chapter presents the equations and calculations for energy approximation. It establishes the estimates (261) and (262) of the Main Lemma (10.1) for continuous solutions; these estimates state that we are able to accurately prescribe the energy that the correction adds to the solution, as well as bound the difference between the time derivatives of these two quantities. The chapter also introduces the proposition for prescribing energy, followed by the relevant computations. Each integral contributing to the other term can be estimated. Another proposition for estimating control over the rate of energy variation is given. Finally, the coarse scale material derivative is considered.

Bounds for the Corrections

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Phase Function ◽  
Universal Constant ◽  
Coarse Scale ◽  
Material Derivative ◽  
Velocity Correction ◽  
Individual Wave ◽  
Phase Direction ◽  
Spatial Derivatives ◽  
Correction Terms ◽  
Derivatives Of

This chapter derives the bounds for the correction terms, starting with bounds for the velocity correction. Based on V of the form V = Δ‎ x W, it introduces a proposition for estimating the spatial derivatives of W. Since the number of Wsubscript I supported at any given region of ℝ x ³ is bounded by a universal constant, it suffices to estimate Wsubscript I uniformly in I. For an individual wave, it is easy to see that the estimate will hold. During repeated differentiation, the derivative hits either the oscillatory factor, the phase direction, or the amplitude wsubscript I or one of its derivatives. In any case, the largest cost happens when differentiating the phase function. The chapter also gives estimates for derivatives of the coarse scale material derivative of W and concludes with bounds for the pressure correction.

Bounds for the Vector Amplitudes

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Material Derivative ◽  
Spatial Derivatives ◽  
Derivatives Of ◽  
Transport Part

This chapter estimates the bounds for the vector amplitudes vsubscript I and wsubscript I of the correction. It first presents the proposition about deriving the estimates for the transport part of vsubscript I and proceeds by computing estimates for vsubscript I, starting with the spatial derivatives. It then introduces a second proposition dealing with spatial derivatives of vector amplitudes and a third proposition for the first material derivative of vector amplitudes. The fourth proposition is concerned with the second material derivative of the vector amplitudes. The chapter shows that all the bounds for γ‎subscript I and α‎subscript I are identical until the second material derivative. It concludes with a corollary about estimates for corrected amplitudes.

Interpretation of TDEM data using first and second spatial derivatives and time decay analysis

1989 ◽  
Vol 20 (2) ◽  
pp. 57 ◽  
Author(s):  
J. Silic
Keyword(s):  
Decay Curve ◽  
Fault Zones ◽  
Curve Analysis ◽  
High Grade ◽  
Time Decay ◽  
Spatial Derivative ◽  
Electromagnetic Data ◽  
Ore Bodies ◽  
Spatial Derivatives

Current gathering in fixed loop electromagnetic data often dominates responses from large high-grade ore bodies as well as responses from less desirable features such as fault zones, weathering troughs and regional conductors. Through decay curve analysis, current gathering can now be unambiguously recognised.Many widely used EM interpretation techniques are not applicable to current gathering (channelling) responses. An effective method of deriving the location and shape of the causative source is to study the second spatial derivative, as is shown in several examples.

scholarly journals FLOW WITH DENSITY AND TRANSPORT EQUATION IN

2019 ◽  
Vol 7 ◽  
Author(s):  
RENJIN JIANG ◽  
KANGWEI LI ◽  
JIE XIAO
Keyword(s):  
Cauchy Problem ◽  
Transport Equation ◽  
Quantitative Estimate ◽  
Well Posedness ◽  
Spatial Derivative

We show that, if $b\in L^{1}(0,T;L_{\operatorname{loc}}^{1}(\mathbb{R}))$ has a spatial derivative in the John–Nirenberg space $\operatorname{BMO}(\mathbb{R})$ , then it generates a unique flow $\unicode[STIX]{x1D719}(t,\cdot )$ which has an $A_{\infty }(\mathbb{R})$ density for each time $t\in [0,T]$ . Our condition on the map $b$ is not only optimal but also produces a sharp quantitative estimate for the density. As a killer application we achieve the well-posedness for a Cauchy problem of the transport equation in $\operatorname{BMO}(\mathbb{R})$ .

Solving the Time-Dependent Transport Equation Using Time-Dependent Method of Characteristics and Rosenbrock Method

2010 ◽  
Author(s):  
Xue Yang ◽  
Tatjana Jevremovic
Keyword(s):  
Transport Equation ◽  
Method Of Characteristics ◽  
Time Dependent ◽  
Spatial Derivative ◽  
Step Size ◽  
Delayed Neutrons ◽  
Rosenbrock Method ◽  
Dependent Transport ◽  
Cell Average ◽  
Time Dependent Transport Equation

A new approach based on the method of characteristics (MOC) and Rosenbrock method is developed to solve the time-dependent transport equation in one-dimensional (1D) geometry without any approximation and considering delayed neutrons. Within the MOC methodology, the leakage term in time-dependent transport equation can be simplified to spatial derivative of the angular flux along the characteristics lines. For 1D geometry, the proposed exponential correlation derived from the steady-state MOC equations provides the correlation between the cell outgoing angular flux and the cell average angular flux. Thus, the spatial derivative term can be further substituted by the relation containing only the cell average angular flux that represents the unknowns. Therefore, the 1D time-dependent transport equation is decomposed into a series of locally coupled ordinary differential equations (ODE). Rosenbrock method was chosen to solve the system of ODEs. It is a fourth order explicit method with automatic step size control feature developed for stiff ODEs. The FORTRAN90 numerical program is developed to thus solve the time-dependent transport equation considering delayed neutrons in 1D geometry with both vacuum and reflective boundary conditions. The step perturbation is currently supported. The method presented in this paper was verified in comparison to 1D fast reactor benchmark showing good accuracy and efficiency.

scholarly journals Thermodynamic Transport Properties in Dense Stars

1980 ◽  
Vol 58 ◽  
pp. 627-633
Author(s):  
Terry W. Edwards
Keyword(s):  
Transport Equation ◽  
Transport Properties ◽  
Momentum Distribution ◽  
Boltzmann Transport Equation ◽  
Distribution Functions ◽  
Boltzmann Transport ◽  
Thermal Fields ◽  
Momentum Distribution Function ◽  
Dense Stars ◽  
Imperfect Fluids

AbstractThe thermodynamic transport properties of special relativistic imperfect fluids, as found in dense stars, are investigated. These properties, which include thermal and electrical conductivities, electrothermal coefficients, and bulk and shear viscosities may be formulated in terms of the momentum distribution functions obtained from the solution of the Boltzmann transport equation. Spherical harmonic solutions of the relaxation form of the relativistic magnetic Boltzmann transport equation have also been obtained which give the non-equilibrium momentum distribution function perturbation f-f(0) = Δf(p) in terms of electromagnetic and thermal fields.

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