Nonlinear hyperbolic equations

Motivated by geological carbon dioxide (CO2) storage, we present a vertical-equilibrium sharp-interface model for the migration of immiscible gravity currents with constant residual trapping in a two-dimensional confined aquifer. The residual acts as a loss term that reduces the current volume continuously. In the limit of a horizontal aquifer, the interface shape is self-similar at early and at late times. The spreading of the current and the decay of its volume are governed by power-laws. At early times the exponent of the scaling law is independent of the residual, but at late times it decreases with increasing loss. Owing to the self-similar nature of the current the volume does not become zero, and the current continues to spread. In the hyperbolic limit, the leading edge of the current is given by a rarefaction and the trailing edge by a shock. In the presence of residual trapping, the current volume is reduced to zero in finite time. Expressions for the up-dip migration distance and the final migration time are obtained. Comparison with numerical results shows that the hyperbolic limit is a good approximation for currents with large mobility ratios even far from the hyperbolic limit. In gently sloping aquifers, the current evolution is divided into an initial near-parabolic stage, with power-law decrease of volume, and a later near-hyperbolic stage, characterized by a rapid decay of the plume volume. Our results suggest that the efficient residual trapping in dipping aquifers may allow CO2 storage in aquifers lacking structural closure, if CO2 is injected far enough from the outcrop of the aquifer.

Decentralized intersection control techniques have received recent attention in the literature as means to overcome scalability issues associated with network-wide intersection control. Chief among these techniques are backpressure (BP) control algorithms, which were originally developed of for large wireless networks. In addition to being lightweight computationally, they come with guarantees of performance at the network level, specifically in terms of network-wide stability. The dynamics in backpressure control are represented using networks of point queues and this also applies to all of the applications to traffic control. As such, BP in traffic fail to capture the spatial distribution of vehicles along the intersection links and, consequently, spill-back dynamics. This paper derives a position weighted backpressure (PWBP) control policy for network traffic applying continuum modeling principles of traffic dynamics and thus capture the spatial distribution of vehicles along network roads and spill-back dynamics. PWBP inherits the computational advantages of traditional BP. To prove stability of PWBP, (i) a Lyapunov functional that captures the spatial distribution of vehicles is developed; (ii) the capacity region of the network is formally defined in the context of macroscopic network traffic; and (iii) it is proved, when exogenous arrival rates are within the capacity region, that PWBP control is network stabilizing. We conduct comparisons against a real-world adap-tive control implementation for an isolated intersection. Comparisons are also performed against other BP approaches in addition to optimized fixed timing control at the network level. These experiments demonstrate the superiority of PWBP over the other control policies in terms of capacity region, network-wide delay, congestion propagation speed, recov-erability from heavy congestion (outside of the capacity region), and response to incidents.

We present a computational analysis of a 2×2 hyperbolic system of balance laws whose solutions exhibit complex nonlinear behavior. Traveling-wave solutions of the system are shown to undergo a series of bifurcations as a parameter in the model is varied. Linear and nonlinear stability properties of the traveling waves are computed numerically using accurate shock-fitting methods. The model may be considered as a minimal hyperbolic system with chaotic solutions and can also serve as a stringent numerical test problem for systems of hyperbolic balance laws.

Melt extraction from the Earth's mantle through high-porosity channels is required to explain the composition of the oceanic crust. Feedbacks from reactive melt transport are thought to localize melt into a network of high-porosity channels. Recent studies invoke lithological heterogeneities in the Earth's mantle to seed the localization of partial melts. Therefore, it is necessary to understand the reaction fronts that form as melt flows across the lithological interface between the heterogeneity and the ambient mantle. Here we present a chromatographic analysis of reactive melt transport across lithological boundaries, using the theory of hyperbolic conservation laws. This is an extension of linear trace element chromatography to the coupling of major elements and energy transport. Our analysis allows the prediction of the nonlinear feedbacks that arise in reactive melt transport due to changes in porosity. This study considers the special case of a partially molten porous medium with binary solid solution. As melt traverses a lithological contact, binary solid solution leads to the formation of a reacted zone between an advancing reaction front and the initial contact. The analysis also shows that the behavior of a fertile heterogeneity depends on its absolute concentration, in addition to compositional differences between itself and the refractory background. We present a regime diagram that predicts if melt emanating from a fertile heterogeneity localizes into high-porosity channels or develops a zero porosity shell. The theoretical framework presented here provides a useful tool for understanding nonlinear feedbacks in reactive melt transport, because it can be extended to more complex and realistic phase behaviors.

Keywords: noble gases shale gas two-phase flow migration–fractionation geochemical tracers geological carbon storage Environmental monitoring of shale gas production and geological carbon dioxide (CO 2) storage requires identification of subsurface gas sources. Noble gases provide a powerful tool to distinguish different sources if the modifications of the gas composition during transport can be accounted for. Despite the recognition of compositional changes due to gas migration in the subsurface, the interpretation of geochemical data relies largely on zero-dimensional mixing and fractionation models. Here we present two-phase flow column experiments that demonstrate these changes. Water containing a dissolved noble gas is displaced by gas comprised of CO 2 and argon. We observe a characteristic pattern of initial co-enrichment of noble gases from both phases in banks at the gas front, followed by a depletion of the dissolved noble gas. The enrichment of the co-injected noble gas is due to the dissolution of the more soluble major gas component, while the enrichment of the dissolved noble gas is due to stripping from the groundwater. These processes amount to chromatographic separations that occur during two-phase flow and can be predicted by the theory of gas injection. This theory provides a mechanistic basis for noble gas fractionation during gas migration and improves our ability to identify subsurface gas sources after post-genetic modification. Finally, we show that compositional changes due to two-phase flow can qualitatively explain the spatial compositional trends observed within the Bravo Dome natural CO 2 reservoir and some regional compositional trends observed in drinking water wells overlying the Marcellus and Barnett shale regions. In both cases, only the migration of a gas with constant source composition is required, rather than multi-stage mixing and fractionation models previously proposed.

In this paper, we present an analytical study, in the one space dimensional case, of the fluid dynamics system proposed in [4] to model the formation of biofilms. After showing the hyperbolicity of the system, we show that, in a open neighborhood of the physical parameters, the system is totally dissipative near its unique non vanishing equilibrium point. Using this property, we are able to prove existence and uniqueness of global smooth solutions to the Cauchy problem on the whole line for small perturbations of this equilibrium point and the solutions are shown to converge exponentially in time at the equilibrium state.

In this article, we study in details the fluid dynamics system proposed in Clarelli et al (2013) to model the formation of cyanobacteria biofilms. After analyzing the linear stability of the unique non trivial equilibrium of the system, we introduce in the model the influence of light and temperature, which are two important factors for the development of cyanobacteria biofilm. Since the values of the coefficients we use for our simulations are estimated through information found in the literature, some sensitivity and robustness analyses on these parameters are performed. All these elements enable us to control and to validate the model we have already derived and to present some numerical simulations in the 2D and the 3D cases.

We propose a model of a density-dependent compressible-incompressible fluid, which is intended as a simplified version of models based on mixture theory, as for instance those arising in the study of biofilms, tumor growth, and vasculogenesis. Though our compressible-incompressible model seems to be very close to the density-dependent incompressible Euler equations, it presents some differences from an analytical point of view. In this paper, we establish a result of local existence and uniqueness of solutions in Sobolev spaces to this model, using two different singular perturbation approximations.

In this paper we study a semilinear hyperbolic-parabolic system modeling biological phenomena evolving on a network composed by oriented arcs. We prove the existence of global (in time) smooth solutions to this problem. The result is obtained by using energy estimates with suitable transmission conditions at nodes.

Combining a priori estimates with penalization techniques and an implicit function argument based on Campanato's near operators theory, we obtain the existence of periodic solutions for a fourth order integro-differential equation modelling actuators in MEMS devices. New insights are also derived for the stationary problem improving previous existence results by removing smallness assumptions on the domain.

We establish existence and regularity results for a time dependent fourth order integro-differential equation with a possibly singular nonlinearity which has applications in designing MicroElectroMechanicalSystems. The key ingredient in our approach, besides basic theory of hyperbolic equations in Hilbert spaces, exploit Near Operators Theory introduced by Campanato.

Gathering together some existing results, we show that the solutions to the one-dimensional Burgers equation converge for long times towards the stationary solutions to the steady Burgers equation, whose Fourier spectrum is not integrable. This is one of the main features of wave turbulence.

In this paper, the performance of single-tone Radio over Fiber (RoF) system has been analyzed by employing different duobinary modulation formats. This single-tone RoF system has been modeled and analyzed using OptiSystem (14.0) software. To evaluate the transmission performance of RoF system, various performance metrics such as Q-factor, BER, and Eye Height are considered. Simulation results indicate that duobinary Hyperbolic-Secant pulse generator format with Single Drive Mach-Zehnder modulator provides better Q-factor and minimum BER as compared to existing modulation format in RoF system.

We develop a polygonal mesh simplification algorithm based on a novel analysis of the mesh geometry. Particularly, we propose first a characterization of vertices as hyperbolic or non-hyperbolic depend-ing upon their discrete local geometry. Subsequently, the simplification process computes a volume cost for each non-hyperbolic vertex, in anal-ogy with spherical volume, to capture the loss of fidelity if that vertex is decimated. Vertices of least volume cost are then successively deleted and the resulting holes re-triangulated using a method based on a novel heuristic. Preliminary experiments indicate a performance comparable to that of the best known mesh simplification algorithms.

This paper presents the generalized differential quadrature (GDQ) simulation for analysis of a nanofluid
over a nonlinearly stretching sheet. The obtained governing equations of flow and heat transfer are
discretized by GDQ method and then are solved by Newton-Raphson method. The effects of stretching
parameter, Brownian motion number (Nb), Thermophoresis number (Nt) and Lewis number (Le), on the
concentration distribution and temperature distribution are evaluated. The obtained results exhibit that the
heat transfer rate can be controlled by choosing different nanoparticles and stretching parameter.

In a paper [Appl. Math. Comput., 188 (2) (2007) 1587--1591], authors have suggested and analyzed a method for solving nonlinear equations. In the present work, we modified this method by using the finite difference scheme, which has a quintic convergence. We have compared this modified Halley method with some other iterative of fifth-orders convergence methods, which shows that this new method having convergence of fourth order, is efficient.