New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection—Diffusion Equations

High resolution schemes for hyperbolic conservation laws

Let X(t) be the number of tumor cells at time t, and Pr{X(t) = n} = pn(t) is the density of X. A “birth”, i.e., an increase of one of the total population of cancer cells, can occur either by mutation of a normal cell caused by the action of the carcinogen, consisting of randomly (Poisson) distributed hits, or by reproduction of existing cancer cells. A death of a tumor cell occurs as an additive combination of non-immunological and immunological elements. Once a tumor is initiated by carcinogenic action, it undergoes a birth and death process with infinitesimal birth rate linear and infinitesimal death rate composed of a linear and a nonlinear term, the former due to non-immunological deaths, the latter to immunological feedback. The death rate per tumor cell due to immunological response is proportional to the total number of antigen-producing (tumor) cells; thus, the total death rate is quadratic. Although this assumes a very simple mechanism for the action of immunological feedback, it is nevertheless a first step.

The Mathematical Model

Water chemistry has been shown experimentally to affect the stability of water films and the sorption of organic oil components on mineral surfaces. When oil is displaced by water, water chemistry has been shown to impact oil recovery. At least two mechanisms could account for these effects, the water chemistry could change the charge on the rock surface and affect the rock wettability, and/or changes in the water chemistry could dissolve rock minerals and affect the rock wettability. The explanations need not be the same for oil displacement of water as for water imbibition and displacement of oil. This article investigates how water chemistry affects surface charge and rock dissolution in a pure calcium carbonate rock similar to the Stevns Klint chalk by constructing and applying a chemical model that couples bulk aqueous and surface chemistry and also addresses mineral precipitation and dissolution. We perform calculations for seawater and formation water for temperatures between 70 and 130°C. The model we construct accurately predicts the surface potential of calcite and the adsorption of sulfate ions from the pore water. The surface potential changes are not able to explain the observed changes in oil recovery caused by changes in pore water chemistry or temperature. On the other hand, chemical dissolution of calcite has the experimentally observed chemical and temperature dependence and could account for the experimental recovery systematics. Based on this preliminary analysis, we conclude that although surface potential may explain some aspects of the existing spontaneous imbibitions data set, mineral dissolution appears to be the controlling factor.

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The Impact of Pore Water Chemistry on Carbonate Surface Charge and Oil Wettability

Dean of Undergraduates John Hutchinson, the General Council and President David Leebron assembled the Working Group on University Responses to Federal Initiatives on Sexual Assault in June, according to Associate Vice Provost Matthew Taylor. The group formed in order to address federal measures as well as to continue changes already in progress. The federal government began releasing guidelines on how universities and colleges should address sexual assault beginning in April 2014. The White House task force report, Not Alone, features certain recommendations regarding training, prevention and adjudication of sexual assault. “The working group is looking at Rice’s policies, processes and communications about sexual violence and sexual harassment, in light of what the government is either recommending or requiring to make sure that we have a clear sense of the areas in which,” Taylor said. “We either exceed the recommendations or there are things that we need to do to meet the recommendations.” The working group consists of both undergraduate and graduate representatives, as well as leadership from the Student Wellbeing Office, Student Judicial Programs, the Rice University Police Department, the Rice Counseling Center, Athletics and Graduate and Post-Doctoral Studies, among others. There are approximately an equal number of males and females in the group. “I wouldn’t say [we have had] findings yet,” Taylor said. “We have tried to educate ourselves on the government’s recommendations.” According to Taylor, the working group focused on immediate shortterm issues for the summer. Many of the federal recommendations were already being met by Rice, and several aspects were already undergoing changes before the formation of the working group. Before the working group was formed, Associate Dean of Undergraduates Donald Ostdiek had already decided to hire a Title IX Resource Navigator, who will help students navigate the investigation if they choose to come forward and report a sexual assault. The main efforts of the working group are directed towards meeting federal guidelines that state that universities should designate employees to whom students can report sexual assault in confidence. Taylor said although Rice already has designated individuals, the policy’s language did not clearly outline whether those individuals are Title IX officers, responsible employees or non-disclosing employees. He said the new language encourages individuals to report cases. “The Title IX officers, in most cases, don’t have to report a student’s name, but we make it equally fair that, in some cases, the university would have to take action based on the law,” Taylor said. “Some of the people that we know students are most likely to go to for advice after a sexual assault are college masters, resident [associates] and college coordinators. We make it clear that they are responsible employees [who are] required to notify one of the Title IX coordinators if they are alerted by a student to a case of sexual assault or sexual violence.” Director of the Center for Student Wellbeing Kate Noonan and Wellbeing Advisor Saralyn Hernandez also contributed to the working group. Hernandez and Noonan said the Wellbeing Advisors are committed to providing trauma-informed support to students who have experienced sexual misconduct. “For survivors who prefer offcampus resources, Rice contracts with the Houston Area Women’s Center, [which] can provide support to Rice students outside the hedges,” Noonan and Hernandez said. Currently, new students undergo training during Orientation Week through Project Sexual Assault-Free Environment, presented by the Houston Area Women’s Resource Center and the Student Wellbeing Office. Although the working group did not have any influence on this year’s presentation of Project SAFE, the presentation already met federal recommendations by focusing on bystander prevention. The presentation discussed non-disclosing and responsible employees, although it did not talk about consequences for potential perpetrators. Colin Losey, a Martel College sophomore who transferred from the University of Chicago, said although he thought the presentation itself was satisfactory, he did not feel that all of the options for victims of sexual assault were covered, especially the option to report to police outside the university. “I’m a transfer student from a university that was recently hit with a major Title IX investigation, and a lot of what has come out from that is that the university’s adjudication process was used in a way to keep things internal,” Losey said. “Given what has happened at other college campuses across the country, students should be clear that this is criminal behavior, and they can press criminal charges.You don’t need to contact Rice first, and Rice need not be involved.” However, Taylor said offering Project SAFE only during O-Week is unsatisfactory and that there must Honor Council removes undergraduates from graduate cases, violates constitution

A new generation.

On the Advection-Upstream-Splitting-Method Hybrid Scheme: A Simple Transient-Flow Model for Managed-Pressure-Drilling and Underbalanced-Drilling Applications

In this paper, we consider a compressible two-fluid model with constant viscosity coefficients and unequal pressure functions \({P^+\neq P^-}\). As mentioned in the seminal work by Bresch, Desjardins, et al. (Arch Rational Mech Anal 196:599–629, 2010) for the compressible two-fluid model, where \({P^+=P^-}\) (common pressure) is used and capillarity effects are accounted for in terms of a third-order derivative of density, the case of constant viscosity coefficients cannot be handled in their settings. Besides, their analysis relies on a special choice for the density-dependent viscosity [refer also to another reference (Commun Math Phys 309:737–755, 2012) by Bresch, Huang and Li for a study of the same model in one dimension but without capillarity effects]. In this work, we obtain the global solution and its optimal decay rate (in time) with constant viscosity coefficients and some smallness assumptions. In particular, capillary pressure is taken into account in the sense that \({\Delta P=P^+ - P^-=f\neq 0}\) where the difference function \({f}\) is assumed to be a strictly decreasing function near the equilibrium relative to the fluid corresponding to \({P^-}\). This assumption plays an key role in the analysis and appears to have an essential stabilization effect on the model in question.

Global Well-Posedness and Decay Rates of Strong Solutions to a Non-Conservative Compressible Two-Fluid Model

In this paper we present a methodology for constructing accurate and efficient hybrid central-upwind (HCU) type schemes for the numerical resolution of a two-fluid model commonly used by the nuclear and petroleum industry. Particularly, we propose a method which does not make use of any information about the eigenstructure of the Jacobian matrix of the model. The two-fluid model possesses a highly nonlinear pressure law. From the mass conservation equations we develop an evolution equation which describes how pressure evolves in time. By applying a quasi-staggered Lax-Friedrichs type discretization for this pressure equation together with a Modified Lax-Friedrich type discretization of the convective terms, we obtain a central type scheme which allows to cope with the nonlinearity (nonlinear pressure waves) of the two-fluid model in a robust manner. Then, in order to obtain an accurate resolution of mass fronts, we employ a modification of the convective mass fluxes by hybridizing the central type mass flux components with upwind type components. This hybridization is based on a splitting of the mass fluxes into components corresponding to the pressure and volume fraction variables, recovering an accurate resolution of a contact discontinuity. In the numerical simulations, the resulting HCU scheme gives results comparable to an approximate Riemann solver while being superior in efficiency. Furthermore, the HCU scheme yields better robustness than other popular Riemann-free upwind schemes.

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Steinar Evje

Papers

On the Advection-Upstream-Splitting-Method Hybrid Scheme: A Simple Transient-Flow Model for Managed-Pressure-Drilling and Underbalanced-Drilling Applications

Global Well-Posedness and Decay Rates of Strong Solutions to a Non-Conservative Compressible Two-Fluid Model

Hybrid central-upwind schemes for numerical resolution of two-phase flows

A model for interpretation of brine-dependent spontaneous imbibition experiments

A model for reactive flow in fractured porous media