Constraint-consistent Runge-Kutta methods for one-dimensional incompressible multiphase flow

New time integration methods are proposed for simulating incompressible multiphase flow in pipelines described by the one-dimensional two-fluid model. The methodology is based on 'half-explicit' Runge-Kutta methods, being explicit for the mass and momentum equations and implicit for the volume constraint. These half-explicit methods are constraint-consistent, i.e., they satisfy the hidden constraints of the two-fluid model, namely the volumetric flow (incompressibility) constraint and the Poisson equation for the pressure. A novel analysis shows that these hidden constraints are present in the continuous, semi-discrete, and fully discrete equations. Next to constraint-consistency, the new methods are conservative: the original mass and momentum equations are solved, and the proper shock conditions are satisfied; efficient: the implicit constraint is rewritten into a pressure Poisson equation, and the time step for the explicit part is restricted by a CFL condition based on the convective wave speeds; and accurate: achieving high order temporal accuracy for all solution components (masses, velocities, and pressure). High-order accuracy is obtained by constructing a new third order Runge-Kutta method that satisfies the additional order conditions arising from the presence of the constraint in combination with time-dependent boundary conditions. Two test cases (Kelvin-Helmholtz instabilities in a pipeline and liquid sloshing in a cylindrical tank) show that for time-independent boundary conditions the half-explicit formulation with a classic fourth-order Runge-Kutta method accurately integrates the two-fluid model equations in time while preserving all constraints. A third test case (ramp-up of gas production in a multiphase pipeline) shows that our new third order method is preferred for cases featuring time-dependent boundary conditions

Engineering technology for networks

Space Network (SN) modeling and evaluation are presented. The following tasks are included: Network Modeling (developing measures and metrics for SN, modeling of the Network Control Center (NCC), using knowledge acquired from the NCC to model the SNC, and modeling the SN); and Space Network Resource scheduling

Mathemagics

Dr. Arthur Benjamin is the Smallwood Family Professor of Mathematics at Harvey Mudd College in Claremont, California. He is also a professional magician, and in his entertaining and fast-paced performance, Dr. Benjamin will demonstrate how to mentally add and multiply numbers faster than a calculator, how to figure out the day of the week of any date in history, and other amazing feats of mind.https://egrove.olemiss.edu/math_dalrymple/1001/thumbnail.jp

Self-Avoiding Walks and Fibonacci Numbers

By combinatorial arguments, we prove that the number of self-avoiding walks on the strip {0, 1} × Z is 8Fn − 4 when n is odd and is 8Fn − n when n is even. Also, when backwards moves are prohibited, we derive simple expressions for the number of length n self-avoiding walks on {0, 1} × Z, Z × Z, the triangular lattice, and the cubic lattice

The Bisection Method: Which Root?

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Book Review: Across the Board: The Mathematics of Chessboard Problems by John J. Watkins

I think I became a mathematician because I loved to play games as a child. I learned about probability and expectation by playing games like backgammon, bridge, and Risk. But I experienced the greater thrill of careful deductive reasoning through games like Mastermind and chess. In fact, for many years I took the game of chess quite seriously and played in many tournaments. But I gave up the game when I started college and turned my attention to more serious pursuits, like learning real mathematics

An Amazing Mathematical Card Trick

A magician gives a member of the audience 20 cards to shuffle. After the cards are thoroughly mixed, the magician goes through the deck two cards at a time, sometimes putting the two cards face to face, sometimes back to back, and sometimes in the same direction. Before dealing each pair of cards into a pile, he asks random members of the audience if the pair should be flipped over or not. He goes through the pile again four cards at a time and before each group of four is dealt to a pile, the audience gets to decide whether each group should be flipped over or not. Then the cards are dealt into four rows of five cards. The audience can decide, for each row, whether it should be dealt from left to right or from right to left, producing an arrangement like the one shown

Mathematical Constance (A Poem Dedicated to Constance Reid)

Mathematical Constance (A Poem Dedicated to Constance Reid) I think that I shall never see A constant lovelier than e, Whose digits are too great too state, They\u27re 2.71828… And e has such amazing features It\u27s loved by all (but mostly teachers). With all of e\u27s great properties Most integrals are done with … ease. Theorems are proved by fools like me But only Euler could make an e. I suppose, though, if I had to try To choose another constant, I Might offer i or phi or pi. But none of those would satisfy. Of all the constants I know well, There\u27s only one that rings the Bell. Not pi, not i, nor even e. In fact, my Constance is a she. It\u27s Constance Reid, I would not fool ya\u27 With Books like Hilbert, Courant, and Julia. Of all the constants you will need, There’s only one that you should Reid

Combinatorics and Campus Security

One day I received electronic mail from our director of campus security [Gilbraith 1993]: I have a puzzle for you that has practical applications for me. I need to know how many different combinations there are for our combination locks. A lock has 5 buttons. In setting the combination you can use only 1button or as many as 5. Buttons may be pressed simultaneously and / or successively, but the same button cannot be used more than once in the same combination. I had a student (obviously not a math major) email me that there are only 120 possibilities, but even I know this is only if you press all five buttons one at a time. It doesn\u27t take into account 1-23-4-5, for instance. My question to you is how many combinations exist, and is it enough to keep our buildings adequately protected? To clarify, combinations like 1-25-4 (which is the same as 1-52-4 but different from 4-25-1) and 1-2-5-43 are legal, whereas 13-35 is illegal because the number 3 is used twice. I gave this problem to the students in my discrete mathematics class as a bonus exercise. Most arrived at the (correct) answer of 1081 or 1082 by breaking the problem into oodles of cases, but this would not have been a convenient method if the locks contained 10 buttons instead of 5. I use this problem as an excuse to demonstrate the power of generating functions by solving the n-button problem. Most students are amazed that the problem is essentially solved by the function ex /(2 - ex), which leads to a surprisingly accurate approximation of n!/(ln 2)n+1

Sensible Rules for Remembering Duals -- The S-O-B Method

We present a natural motivation and simple mnemonic for creating the dual LP of any linear programing problem