| A MULTI-LENGTH SCALE THEORY OF THE ANOMALOUS MIXING LENGTH GROWTH FOR TRACER FLOW IN HETEROGENEOUS POROUS MEDIA |
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We develop a multi-length scale (multi-fractal) theory for the effect of rock heterogeneity on the growth of the mixing layer of the flow of a passive tracer through porous media. The multi-fractal exponent of the size of the mixing layer is determined analytically from the statistical properties of a random velocity (permeability) field. The anomalous diffusion of the mixing layer can occur both on finite and on asymptotic length scales.
Preprint #SUNYSB-AMS-91-02, Appeared in J. Stat. Phys., vol. 66, pp. 485--499, 1992.
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| INERTIAL RANGE SCALING OF LAMINAR SHEAR FLOW AS A MODEL OF TURBULENT TRANSPORT |
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Asymptotic scaling behavior, characteristic of the inertial range, is obtained for a fractal stochastic system proposed as a model for turbulent transport.
Preprint #SUNYSB-AMS-91-03, Appeared in Comm. Math. Phys., vol. 146, pp. 217--229, 1992.
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| TIME DEPENDENT ANOMALOUS DIFFUSION FOR FLOW IN MULTI-FRACTAL POROUS MEDIA |
F. Furtado, F. Pereira, B.Lindquis , James Glimm and Qiang Zhang |
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Preprint #SUNYSB-AMS-91-06, Appeared in Proceedings of the Workshop on Numerical Methods for the Simulation
of Multi-Phase and Complex Flow, Amsterdam. May, 1990.
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| A GLOBAL FORMALISM FOR NONLINEAR WAVES IN CONSERVATION LAWS |
Eli Isaacson, Dan Marchesin, C. Frederico Palmeira, and Bradley J. Plohr |
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We introduce a unifying framework for treating all of the fundamental waves occurring in general systems of $n$ conservation laws. Fundamental waves are represented as pairs of states satisfying the Rankine-Hugoniot conditions; after trivial solutions have been eliminated by means of a blow-up procedure, these pairs form an $(n+1)$-dimensional manifold $\W$, the fundamental wave manifold. There is a distinguished $n$-dimensional submanifold of $\W$ containing a single one-dimensional foliation that represents the rarefaction curves for all families. Similarly, there is a foliation of $\W$ itself that represents shock curves. We identify other $n$-dimensional submanifolds of $\W$ that are naturally interpreted as boundaries of regions of admissible shock waves. These submanifolds also have one-dimensional foliations, which represent curves of composite waves.
This geometric framework promises to simplify greatly the study of the stability and bifurcation properties of global solutions of Riemann problems for mixed hyperbolic-elliptic systems. In particular, bifurcations of wave curves can be understood as resulting from loss of transversality between foliations and admissibility boundaries.
Preprint #SUNYSB-AMS-91-08,Comm. Math. Phys., vol. 146, pp. 505-552, 1992.
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| SCATTERING BEHAVIOR OF TRANSITIONAL SHOCK WAVES |
Kevin Zumbrun and Dan Marchesin Bradley J. Plohr, |
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We study the stability and asymptotic behavior of transitional shock waves as solutions of a parabolic system of conservation laws. In contrast to classical shock waves, transitional shock waves are sensitive to the precise form of the parabolic term, not only in their internal structure but also in terms of the end states that they connect.
In our numerical investigation, these waves exhibit robust stability. Moreover, their response to perturbation differs from that of classical waves; in particular, the asymptotic state of a perturbed transitional wave depends on the location of the perturbation relative to the shock wave. We develop a linear scattering model that predicts behavior agreeing quantitatively with our numerical results.
Preprint #SUNYSB-AMS-91-09, Appeared in Mat. Contemp., vol. 3, pp. 191-209, 1993.
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| A GLOBAL APPROACH TO SHOCK WAVE ADMISSIBILITY |
Suncica Canic and Bradley J. Plohr |
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We present a new approach to characterizing admissible shock wave solutions for systems of conservation laws. A shock wave is admissible if it has a viscous profile, \ie its end states are joined by an orbit for an associated dynamical system. The family of all such dynamical systems is parameterized by the fundamental wave manifold $\W$, which provides a global setting for studying shock waves. The regions of $\W$ comprising admissible shock waves are bounded by loci of structurally unstable dynamical systems.
We give explicit formulae for many of the admissibility boundaries for systems of two conservation laws with quadratic fluxes. These boundaries include the loci associated with saddle-node, Hopf, and certain heteroclinic bifurcations. Furthermore, we explore other heteroclinic loci and the homoclinic locus using numerical methods. One surprising observation is that the heteroclinic loci appear to form ruled surfaces within $\W$.
Preprint #SUNYSB-AMS-91-10, Appeared in Anais do 18^o Colóquio Brasileiro Matemática,
ed. M. J. Pacífico, pp. 199-216,
Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil, 1993.
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| A QUASI-NEWTON METHOD FOR SOLVING NONLINEAR ALGEBRAIC EQUATIONS |
S. Kim and R.P.Tewarson, |
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An algorithm for solving systems of nonlinear algebraic equations is described. The Jacobian matrix is modified by using a convex combination of Broyden and a weighted update. A q-superlinear convergence theorem and computational evidence exhibiting significant relative efficiency of the proposed method are given.
Preprint #SUNYSB-AMS-91-15, Appeared in Computers and Mathematics with Applications, vol. 24, pp. 93--97, 1992.
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| A QUASI-GAUSS-NEWTON METHOD FOR SOLVING NONLINEAR ALGEBRAIC EQUATIONS |
H. Wang and R.P.Tewarson |
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One of the most popular algorithms for solving systems of nolinear algebraic equations is the sequencing $QR$ factorization implementation of the quasi-Newton method. We propose a significantly better algorithm and give computational results.
Preprint #SUNYSB-AMS-91-16, Appeared in Computers and Mathematics with Applications, vol. 25, pp. 53--63, 1993.
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