The Solution of the Riemann Problem for a Hyperbolic System of Conservation Laws Modeling Polymer Flooding

On the Construction and Comparison of Difference Schemes

Existence and uniqueness of classical solutions for the multidimensional expanding Hele{Shaw problem are proved.

We consider the general system of n first order linear ordinary differential equations y'(t)=A(t)y(t)+g(t), a<t< b, subject to "boundary" conditions, or rather linear constraints, of the form Σ^(N)_(ν=1) B_(ν)y(τ_ν)=β Here y(t), g(t) and II are n-vectors and A(t), Bx,..., BN are n × n matrices. The N distinct points {τ_ν} lie in [a,b] and we only require N ≧ 1. Thus as special cases initial value problems, N=1, are included as well as the general 2-point boundary value problem, N=2, with τ_1=a, τ_2=b. (More general linear constraints are also studied, see (5.1) and (5.17).)

We formulate and study a one–dimensional single–species diffusive–delay population model. The time delay is the time taken from birth to maturity. Without diffusion, the delay differential model extends the well–known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes. This delayed model is known to have simple global dynamics similar to that of the logistic equation. Through the use of a sub/supersolution pair method, we show that the diffusive delay model continues to generate simple global dynamics. This has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics. We also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population, connecting the zero solution of that equation with the positive steady state. Our main finding here is that our fronts appear to be all monotone, regardless of the size of the delay. This is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.

Classes of machines using very limited amounts of nondeterminism are studied. The P=? NP question is related to questions about classes lying within P. Complete sets for these classes are given.

It is shown that the pathwidth of a cograph equals its treewidth, and a linear time algorithm to determine the pathwidth of a cograph and build a corresponding path-decomposition is given.MSC codes05C0505C8568R10Keywordsgraph algorithmscographstreewidthpathwidth

A sequential approach for numerical simulation of two-phase multicomponent flow with reactive transport in porous media

We describe an algorithm for the numerical computation of the rank-one convex envelope of a function $f:\MM^{m\times n}\rightarrow\RR$. We prove its convergence and an error estimate in $L^\infty$.

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

Numerial Methods for Volterra Integral Equations with Singular Kernels

Buoyant incompressible multiphase flow and transport in porous media typically is simulated by coupling an elliptic flow equation with a hyperbolic transport description. The tight coupling between flow and transport either requires a fully implicit solution algorithm or very small time steps, if solved sequentially. In any case, however, a large linear system has to be solved each time step.Here a new solution approach is devised, which relies on solving a coupled hyperbolic system of conservation laws with an explicit finite volume method. Consequently, only local operations have to be performed, which is a great advantage in case of massive parallel simulations and if GPUs are employed.The devised method is based on the isothermal Euler equations with momentum source terms accounting for resistance due to the porous medium and buoyancy. In this system the pressure is proportional to the density and in case of very small Mach numbers and relative density variation, the computed velocity field is divergence free and converges towards the mean total volumetric flow in the porous domain. To account for saturation transport, the system was augmented by an additional hyperbolic equation. In order to obtain the numerical fluxes, a characteristic based approximative Riemann solver was developed and 2nd order accuracy in space and time is achieved by piecewise linear reconstruction.Two-phase flow test cases with buoyant plumes and a lock exchange problem demonstrate that the devised hyperbolic approach recovers the correct incompressible solutions. Numerical experiments also demonstrate that the method is accurate and efficient.Thus the devised method is very promising, and conceptually it also can be applied for flow and transport in heterogeneous media; but it remains to be investigated how well it performs for such cases.

Next article Contraction Mappings in the Theory Underlying Dynamic ProgrammingEric V. DenardoEric V. Denardohttps://doi.org/10.1137/1009030PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Richard Bellman, Dynamic programming, Princeton Univeristy Press, Princeton, N. J., 1957xxv+342 MR0090477 Google Scholar[2] David Blackwell, Discrete dynamic programming, Ann. Math. Statist., 33 (1962), 719–726 MR0149965 0133.12906 CrossrefISIGoogle Scholar[3] David Blackwell, Discounted dynamic programming, Ann. Math. Statist., 36 (1965), 226–235 MR0173536 0133.42805 CrossrefGoogle Scholar[4] A. Charnes and , R. G. Schroeder, On some tactical antisubmarine games, Systems Research Memorandum No. 131, The Technological Institute, Northwestern University, Evanston, Illinois, 1965 Google Scholar[5] E. V. Denardo, Masters Thesis, Sequential decision processes, Doctoral thesis, Northwestern University, Evanston, Illinois, 1965 Google Scholar[6] F. 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Denardo and B. L. July 2006 | SIAM Journal on Applied Mathematics, Vol. No. 3AbstractPDF of Stationary Optimal Policies for Some Markov July 2006 | SIAM Review, Vol. 9, No. 3AbstractPDF Programming by Linear August 2006 | SIAM Journal on Applied Mathematics, Vol. 14, No. 9, August July 2006 for and Applied & for and Applied Mathematics

Over the past few decades, due to the special features (i.e., easily produced, large-surface-area-to-volume ratio, and engineered particles with designed surface properties), nanoparticles have not only attracted great attentions from the oil and gas industry but also had various applications from drilling and completion, reservoir characterization, to enhanced oil recovery (EOR). As sensors or EOR agents, thus, fate and behavior of nanoparticles in porous media are essential and need to be investigated thoroughly. Nevertheless, most of the published review papers focus on particle transport in saturated porous media, and all of them are about steady-state flow conditions. So far, no attempts have been extended to systematically review current knowledge about nanoparticle transport in porous media with single-phase and two-phase flow systems under both steady-state and unsteady-state conditions. Accordingly, this review will discuss nanoparticle transport phenomena in porous media with its focus on the filtration mechanisms, the underlying interaction forces, and factors dominating nanoparticle transport behavior in porous media. Finally, mathematical models used to describe nanoparticle transport in porous media for both single-phase flow and two-phase flow under steady-state and transient flow conditions will be summarized, respectively.

A triangle-based positive semi-discrete Lagrangian–Eulerian scheme via the weak asymptotic method for scalar equations and systems of hyperbolic conservation laws

The Numerical Solution of Integral Equations on the Half-Line