Advanced Mathematical Methods for Scientists and Engineers I
Asymptotic Methods and Perturbation Theory
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Article
Discrete Rotational Symmetry, Moment Isotropy, and Higher Order Lattice Boltzmann Models
Conventional lattice Boltzmann models only satisfy moment isotropy up to fourth order. In order to accurately describe important physical effects beyond the isothermal Navier-Stokes fluid regime, higher-order...
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Article
Turbulence Effects on Kinetic Equations
It is well known that fluid dynamics can be derived from a kinetic (Boltzmann equation) framework. Here we propose that the variance of a fluctuating kinetic relaxation time be linked to turbulent time scales....
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Article
‘Perfectly’ Scalable Data I/O
This paper describes and analyzes a new architecture for file systems in which ‘metadata’, lock control, etc., are distributed among diverse resources. The basic data structure is a segment, viz. a logical gro...
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Article
A New Time-Based Iterative Solver for Linear Standing-Wave Problems
We explore time-based solvers for linear standing-wave problems, especially the oscillatory Helmholtz equation. Here, we show how to accelerate the convergence properties of timestep**. We introduce a new ti...
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Article
Analysis of the ∈-Expansion in Turbulence Theory: Approximate Renormalization Group for Diffusion of a Passive Scalar in a Random Velocity Field
The validity of the ∈-expansion in the turbulence problem is discussed using the example of diffusion of a passive scalar in a random velocity field. A generalization of Wilson's rule for calculating a diagram...
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Book
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Chapter
WKB Theory
WKB theory is a powerful tool for obtaining a global approximation to the solution of a linear differential equation whose highest derivative is multiplied by a small parameter ε; it contains boundary-layer theor...
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Chapter
Difference Equations
This chapter is a summary of the elementary methods available for solving difference equations. Difference equations are used to compute quantities which may be defined recursively, such as the nth coefficient...
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Chapter
Approximate Solution of Nonlinear Differential Equations
One cannot hope to obtain exact solutions to most nonlinear differential equations. As we saw in Chap. 1, there are only a limited number of systematic procedures for solving them, and these apply to a very re...
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Chapter
Asymptotic Expansion of Integrals
The analysis of differential and difference equations in Chaps. 3 to 5 is pure local analysis; there we predict the behavior of solutions near one point, but we do not incorporate initial-value or boundary-val...
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Chapter
Summation of Series
When perturbation methods such as those introduced in Chap. 7 are used to solve a problem, the answer emerges as an infinite series, usually involving powers of the perturbation parameter ε. In practice, only the...
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Chapter
Multiple-Scale Analysis
Multiple-scale analysis is a very general collection of perturbation techniques that embodies the ideas of both boundary-layer theory and WKB theory. Multiple-scale analysis is particularly useful for construc...
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Chapter
Ordinary Differential Equations
An nth-order differential equation has the form $$ {y^{\left( n \right)}}\left( x \right) = F\left[ {x,y\left( x \right),y'\left( x \right), \...
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Chapter
Approximate Solution of Linear Differential Equations
The theory of linear differential equations is so powerful that one can usually predict the local behavior of the solutions near a point x 0 without knowing how to solve the differential equation. I...
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Chapter
Approximate Solution of Difference Equations
Difference equations (recursion relations) occur so frequently in applied mathematics that we allot a full chapter to a discussion of the behavior of their solutions. We will study the problem of determining t...
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Chapter
Perturbation Series
Perturbation theory is a large collection of iterative methods for obtaining approximate solutions to problems involving a small parameter ε. These methods are so powerful that sometimes it is actually advisable ...
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Chapter
Boundary-Layer Theory
In this and the next chapter we discuss perturbative methods for solving a differential equation whose highest derivative is multiplied by the perturbing parameter ε. The most elementary of these methods is calle...
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Article
Two-Equation RNG Transport Modeling of High Reynolds Number Pipe Flow
New solutions of two-equation RNG turbulence transport model are used to calculate high Reynolds number pipe flows. The results are compared with experimental “Superpipe” data of Zagarola et al. (1996) up to Reyn...
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Article
Renormalization Group-Based Transport Modeling of Premixed Turbulent Combustion. II. Finite Density Gradient and Direct Heat Release
A new method for numerical simulation of flame propagation in turbulent premixed combustible gaseous mixtures is proposed and tested. The method combines (I) a modified eikonal equation, employed to model the ...
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Article
Renormalization Group-Based Transport Modeling of Premixed Turbulent Combustion: I. Incompressible Deflagration Model
A new method for numerical modeling of premixed turbulent combustion is introduced based on the thin flame model. Applications are given, e.g., to flames in turbulent flow in channels and over backward facing ...
