Balance Laws Research Articles

A novel, structure-preserving, second-order-in-time relaxation scheme for Schrödinger-Poisson systems

We introduce a new structure preserving, second order in time relaxation-type scheme for approximating solutions of the Schrödinger-Poisson system. More specifically, we use the Crank-Nicolson scheme as a time stepping mechanism, whilst the nonlinearity is handled by means of a relaxation approach in the spirit of [10,11,34] for the nonlinear Schrödinger equation. For the spatial discretisation we use the standard conforming finite element scheme. The resulting scheme is explicit with respect to the nonlinearity, i.e. it requires the solution of a linear system for each time-step, and satisfies discrete versions of the system's mass conservation and energy balance laws for constant meshes. The scheme is seen to be second order in time. We conclude by presenting some numerical experiments, including an example from cosmology and an example with variable time-steps which demonstrate the effectiveness and robustness of the new scheme.

The Identifiability of Mathematical Models in Epidemiology: Tuberculosis, HIV, COVID-19

The paper is devoted to the short review and application of sensitivity-based identifiability approaches for analyzing mathematical models of epidemiology and related processes described by systems of differential equations and agent-based models. It is shown that for structural identifiability of basic SIR models (describe the dynamic of Susceptible, Infected and Removed groups based on nonlinear ordinary differential equations) of epidemic spread and linear compartmental models it is possible to use a priori information about the process. It is demonstrated that a model can be structurally identifiable but be practically non-identifiable due to incomplete data. The paper uses methods for analyzing the sensitivity of parameters to data variation, as well as analyzing the sensitivity of model states to parameter variation, based on linear and differential algebra, Bayesian, and Monte Carlo approaches. It was shown that in the SEIR-HCD model of COVID-19 propagation, described by a system of seven ordinary differential equations and based on the mass balance law, the parameter of humoral immunity acquisition is the least sensitive to changes in the number of diagnosed, critical and mortality cases of COVID-19. The spatial SEIR-HCD model of COVID-19 propagation demonstrated an increase the sensitivity of the partial immunity duration parameter over time, as well as a decrease in the limits of change in the infectivity and infection parameters. In the case of the SEIR-HCD mean-field model of COVID-19 propagation, the sensitivity of the system to the self-isolation index and the lack of sensitivity of the stochastic parameters of the system are shown. In the case of the agent-based COVID-19 propagation model, the change in the infectivity parameter was reduced by more than a factor of 2 compared to the statistics. A differential model of co-infection HIV and tuberculosis spread with multiple drug resistance was developed and its local identifiability was shown.

Uniform asymptotic stability for convection–reaction–diffusion equations in the inviscid limit towards Riemann shocks

The present contribution proves the asymptotic orbital stability of viscous regularizations of stable Riemann shocks of scalar balance laws, uniformly with respect to the viscosity/diffusion parameter \varepsilon . The uniformity is understood in the sense that all constants involved in the stability statements are uniform and that the corresponding multiscale \varepsilon -dependent topology reduces to the classical W^{1, \infty} -topology when restricted to functions supported away from the shock location. Main difficulties include that uniformity precludes any use of parabolic regularization to close regularity estimates, that the global-in-time analysis is also spatially multiscale due to the coexistence of nontrivial slow parts with fast shock-layer parts, that the limiting smooth spectral problem (in fast variables) has no spectral gap and that uniformity requires a very precise and unusual design of the phase shift encoding orbital stability. In particular, our analysis builds a phase that somehow interpolates between the hyperbolic shock location prescribed by the Rankine–Hugoniot conditions and the nonuniform shift arising merely from phasing out the nondecaying 0-mode, as in the classical stability analysis for fronts of reaction–diffusion equations.

Global Well-Posedness of Shock Front Solutions to One-Dimensional Piston Problem for Combustion Euler Flows

This paper is devoted to the well-posedness theory of piston problem for compressible combustion Euler flows with physical ignition condition. A significant combustion phenomenon called detonation will occur provided that the reactant is compressed and ignited by a leading shock. Mathematically, the problem can be formulated as an initial-boundary value problem for hyperbolic balance laws with a large shock front as free boundary. In the present paper, we establish the global well-posedness of entropy solutions via wave front tracking scheme within the framework of space. The main difficulties here stem from the discontinuous source term without uniform dissipation structure and from the characteristic boundary associated with degenerate characteristic field. In dealing with the obstacles caused by ignition temperature, we develop a modified Glimm-type functional to control the oscillation growth of combustion waves, even if the exothermic source fails to uniformly decay. As to the characteristic boundary, the degeneracy of contact discontinuity is fully employed to get elegant stability estimates near the piston boundary. Meanwhile, we devise a weighted Lyapunov functional to balance the nonlinear effects arising from large shock, characteristic boundary, and exothermic reaction and then obtain the -stability of combustion wave solutions. Our results reveal that one-dimensional Zeldovich–von Neumann–Döring detonation waves supported by a forward piston are indeed nonlinearly stable under small perturbation in sense. This is the first work on well-posedness of inviscid reacting Euler fluids dominated by ignition temperature.

Implementation and verification of a user‐defined element (UEL) for coupled thermal‐hydraulic‐mechanical‐chemical (THMC) processes in saturated geological media

AbstractEfficient and accurate modeling of the coupled thermal‐hydraulic‐mechanical‐chemical (THMC) processes in various rock formations is indispensable for designing energy geo‐structures such as underground repositories for high‐level nuclear wastes. This work focuses on developing and verifying an implicit finite element solver for generic coupled THMC problems in geological settings. Starting from the mass, momentum, and energy balance laws, a specialized set of governing equations and a thermoporoelastic constitutive model is derived. This system is then solved by an implicit finite element (FE) scheme. Specifically, the residuals and the Jacobians are scripted in a user‐defined element (UEL) subroutine which is then combined with the general‐purpose FE software Abaqus Standard to solve initial‐boundary value problems. Considering the complexity of the system, the UEL development follows a stepwise manner by first solving the coupled hydraulic‐mechanical (HM) and thermal‐hydraulic‐mechanical (THM) equations before moving on to the full THMC problem. Each implementation step consists of at least one verification test by comparing computed results with closed‐form analytical solutions to ensure that the various coupling effects are correctly realized. To demonstrate the robustness of the algorithm and to validate the UEL, a three‐dimensional case study is performed with reference to the in‐situ heating test of ATLAS at Belgium in 1980s. A hypothetical radionuclide leakage event is then simulated by activating the chemical‐concentration degree of freedom and prescribing a constant high concentration at the heater's surface. The model predicts a limited contaminated regime after six years considering both diffusion and advection effects on species transport.

On the formulation of energy conservation in the eeKdV equation

The Korteweg-de Vries (KdV) equation is a well-known model equation for unidirectional shallow-water (long) surface waves. The equation includes dispersion and weak non-linearity. The derivation of the equation originates in assuming that the velocity potential takes the form of an asymptotic expansion, and applying this in the classical surface wave problem. While a typical assumption on the relative size of non-dimensional key parameters introduced in the derivation will give the KdV equation as a final result, one can change the assumption on the relative size of parameters, and end up with an equation including terms in higher orders in desired parameters. The present article presents the derivation of an extended form referred to as the eeKdV equation.Information regarding various properties of the flow can be found by studying the derivation of the eeKdV equation itself, and some of the relations found can be used for studying the energy balance of a system modeled by the equation. In line with previous work for the KdV equation, we present here corresponding formulations of energy balance laws for an inertial reference frame, in context of the eeKdV equation. We also present a partial verification of the KdV and eeKdV energy flux expressions by looking at a far-field, uniform flow situation, as well as performing a numerical study to confirm the assumed behaviour of the error in the eeKdV energy equation in the context of a undular bore flow setup. Further, conserved integrals for the eeKdV equation are presented and numerically checked.

Principle of virtual power and drilling degrees of freedom for dynamic modeling of the behavior of liquid crystal elastomer films

In this work, we aim to model the reorientation process of mesogens in nematic liquid crystal elastomers within the context of dynamics. We consider a continuum model with separate mappings for the deformation of the monolithic material and the orientation of the nematic director, where the latter describes the inclination of the mesogens. We achieve the inextensibility of the nematic director through the introduction of drilling degrees of freedom. We combine this approach with the application of the principle of virtual power and a mixed finite element formulation, in order to formulate distinct momentum and angular momentum balance laws for the two separate mappings. Furthermore, we include in our continuum model a volume load and a surface load associated only with the orientation mapping. We show in the presented three numerical examples that our formulation enables the fulfillment of all momentum and angular momentum balance laws.

Well Posedness and Characterization of Solutions to Non Conservative Products in Non Homogeneous Fluid Dynamics Equations

AbstractConsider a balance law where the flux depends explicitly on the space variable. At jump discontinuities, modeling considerations may impose the defect in the conservation of some quantities, thus leading to non conservative products. Below, we deduce the evolution in the smooth case from the jump conditions at discontinuities. Moreover, the resulting framework enjoys well posedness and solutions are uniquely characterized. These results apply, for instance, to the flow of water in a canal with varying width and depth, as well as to the inviscid Euler equations in pipes with varying geometry.

A computational framework based on explicit local chemical equilibrium for coupled chemo-hydro-mechanical effects on fluid-infiltrating porous media

Conventionally, a finite element framework for coupled hydro-mechanical-chemical processes in porous media adopts simplified chemical reactions, which consider superficial chemical effects of complicated multiphysics mechanisms. This paper presents a novel computational framework for geochemistry of reactive multi-component systems to explicitly capture the mineral dissolution/precipitation of saturated porous media. The resultant of this framework can be applied in oil recovery engineering and used to simulate the production scenarios. The model is also fundamental for studying on creep behaviors from the view of the coupling effects between hydro-geochemistry. In particular, a complete chemistry system of ▪ is considered, including intra-aqueous chemical reactions and kinetic rate-limiting chemical reactions. Besides, an algorithm is proposed such that the complete chemistry system is scalable for various chemical reactions and is easily incorporated into existing finite element frameworks for hydro-mechanical processes of porous media. This complete chemistry system is implemented to establish a local chemical equilibrium. The geochemical reactive transport equations are then sequentially coupled with the balance laws such that coupling among fluid advection-diffusion, solid deformation, chemical concentration advection-diffusion, intra-aqueous chemical reaction, and kinetic rate-limiting chemical reaction can be simulated. Phenomenological relations for porosity and permeability changes with mineral saturation in solutions are employed. We first verify the proposed framework to capture the advection-diffusion of different chemical species under steady-state and transient conditions. Then a consolidation problem under the chemical reactions is presented to further demonstrate the capabilities of the proposed computational framework.

A new two-dimensional blood flow model with arbitrary cross sections

A new two-dimensional model for blood flows in arteries with arbitrary cross sections is derived. The domain consists of a narrow, large vessel that extends along an axial direction, with cross sections described by radial and angular coordinates. The model consists of a system of balance laws for conservation of mass and balance of momentum in the axial and angular directions. The equations are derived by applying asymptotic analysis to the incompressible Navier-Stokes equations in a moving domain with an elastic membrane, and integrating in the radial direction in each cross section. The resulting model is a system of hyperbolic balance laws with source terms. The main properties of the system are discussed and a positivity-preserving well-balanced central-upwind scheme is presented. The merits of the scheme will be tested in a variety of scenarios. In particular, simulations using an idealized aorta model are shown. We analyze the time evolution of the blood flow under different initial conditions such as perturbations to steady states, which parametrizes a bulging in a vessel’s wall. We consider different situations given by distinct variations in the vessel’s elasticity.

Numerical approximation of living-man steady state solutions for blood flow in arteries using a well-balanced discontinuous Galerkin scheme

This study developed a well-balanced discontinuous Galerkin method for the one-dimensional blood flow model capable of preserving zero and non-zero velocity steady state solutions. The strategy used was to incorporate the source term into the flux function to rewrite the law of balance in a conservative form, in terms of global equilibrium variables that remained constant in space and time under steady states. Subsequently, the conservative variables were recalculated using the equilibrium variables. Recently, this concept was applied to shallow water and the Euler equations of gas dynamics. Numerical tests verified the well-balanced property of the designed scheme and its ability to capture small steady-state perturbations on relatively coarse meshes accurately.

When functionally graded materials meet higher order simplified strain gradient elasticity theory

We investigate the constitutive equations when the functionally graded materials (FGMs) are considered under the strain gradient elasticity theory (SGET). The SGET considered includes both the Helmholtz and bi-Helmholtz types. The constitutive equations become more complicated and contain more terms due to the interaction between the material gradation and the nonlocal effect of the strain gradient. We also derive the governing partial differential equations (PDEs) by the balance law of linear momentum, and the PDEs also appear to be highly complicated. Under the assumption that material gradation is of the exponential form, a simpler set of governing PDEs can be reached. Solutions to those PDEs under the formulation of different modes of crack problems are discussed.

Global existence of classical solutions for the gas flows near vacuum through ducts expanding with space and time

In this study, we examine the global existence of classical solutions for the gas flows near vacuum through ducts expanding with space and time. It is described by the initial-boundary value problem of compressible Euler equations together with a sufficiently small variable parameter, which can be viewed as a hyperbolic system of balance laws whose source is a non-dissipative term when Riemann invariants are applied. We prove a couple of global existence theorems of classical solutions under the suitable conditions of expanding ducts and the initial-boundary data whose C0 norms can be large. The analysis depends primarily on the local existence theorem and on uniform a priori estimates, which are obtained by giving the maximum principle and introducing new generalized Lax transformations. Furthermore, the limit behavior of expanding ducts at infinity and the long-time behavior of global classical solutions along all characteristic curves and vertical lines are also determined. Lastly, we explore the feasibility of the initial value problem for such expanding ducts.

Tumor Evolution Models of Phase-Field Type with Nonlocal Effects and Angiogenesis

In this survey article, a variety of systems modeling tumor growth are discussed. In accordance with the hallmarks of cancer, the described models incorporate the primary characteristics of cancer evolution. Specifically, we focus on diffusive interface models and follow the phase-field approach that describes the tumor as a collection of cells. Such systems are based on a multiphase approach that employs constitutive laws and balance laws for individual constituents. In mathematical oncology, numerous biological phenomena are involved, including temporal and spatial nonlocal effects, complex nonlinearities, stochasticity, and mixed-dimensional couplings. Using the models, for instance, we can express angiogenesis and cell-to-matrix adhesion effects. Finally, we offer some methods for numerically approximating the models and show simulations of the tumor's evolution in response to various biological effects.

Sticky particle Cucker–Smale dynamics and the entropic selection principle for the 1D Euler-alignment system

We develop a global wellposedness theory for weak solutions to the 1D Euler-alignment system with measure-valued density, bounded velocity, and locally integrable communication protocol. A satisfactory understanding of the low-regularity theory is an issue of pressing interest, as smooth solutions may lose regularity in finite time. However, no such theory currently exists except for a very special class of alignment interactions. We show that the dynamics of the 1D Euler-alignment system can be effectively described by a nonlocal scalar balance law, the entropy conditions of which serves as an entropic selection principle that determines a unique weak solution of the Euler-alignment system. Moreover, the distinguished weak solution of the system can be approximated by the sticky particle Cucker–Smale dynamics. Our approach is inspired by the work of Brenier and Grenier on the pressureless Euler equations.

Historical Laws in the System of Modern Knowledge about the Past of Human Society

This article examines the question of the existence of specific historical (general sociological) laws that operate throughout all stages of human history and differ from other laws of social existence. The authors have attempted to formulate the most obvious basic historical laws based on the available scientific developments, methodological approaches and ideas of researchers from both the distant and very recent past. The authors define the following list of historical laws: the law of cyclicity; the law of civilizational, event-based and personal uniqueness of the historical process; the law of balance and struggle of personal, group, corporate, state and civilizational interests; the law of cause-and-effect relationships; the law of unrealized alternatives; the law of unintended consequences. They reflect in the most concentrated form the current level of knowledge about the world historical process. They relate to the entire array of the past of human society. They are synthetic, complex, multidimensional and all encompassing. They manifest themselves in all periods of human history without exception. Therefore, they open up the possibility of a more conscious perception of the present and the formation of general scientific ideas about the deep mechanisms of the future development of society.

A Well-Balanced Active Flux Method for the Shallow Water Equations with Wetting and Drying

Active Flux is a third order accurate numerical method which evolves cell averages and point values at cell interfaces independently. It naturally uses a continuous reconstruction, but is stable when applied to hyperbolic problems. In this work, the Active Flux method is extended for the first time to a nonlinear hyperbolic system of balance laws, namely, to the shallow water equations with bottom topography. We demonstrate how to achieve an Active Flux method that is well-balanced, positivity preserving, and allows for dry states in one spatial dimension. Because of the continuous reconstruction all these properties are achieved using new approaches. To maintain third order accuracy, we also propose a novel high-order approximate evolution operator for the update of the point values. A variety of test problems demonstrates the good performance of the method even in presence of shocks.

제4차 산업혁명시대 인공지능의 의사결정과 인격권 보장

The key word in the era of the Fourth Industrial Revolution is definitely AI(Artificial Intelligence). Artificial intelligence's decision based on big data increases the efficiency of work processing by quickly reviewing a large amount of data and reducing time and cost. As a result, artificial intelligence technology has begun to be used in all areas of society, and it is expected to be an anti-corruption policy tool against hiring-related corruption and irregularities that raise doubts about trust and process. However, if there is no proper control over artificial intelligence, artificial intelligence can rather strengthen prejudice and infringe on job seekers' personal information. This is because the data based on artificial intelligence's decisions have been accumulated for a long time, so discrimination and prejudice against race, gender, or certain groups in the past can be revealed or reflected in algorithm design. Moreover, since artificial intelligence's decision-making process is difficult to present procedures and grounds that can be generally understood through self-learning, there is a situation in which humans have to accept machine decisions as they are in a difficult state to refute or correct. Therefore, while maximizing the utility of high-tech technology, it is necessary to find a way to use artificial intelligence without damaging human dignity and value.
 As a way to guarantee the personal rights of artificial intelligence's decision-making, the exclusive decision of artificial intelligence should be limited by referring to the European Personal Information Protection Regulations and the final decision authority should be placed on humans. In addition, it is necessary to set guidelines based on careful analysis by audit agencies, the government, ethics committees, and psychological experts for legal and ethical problems that arise in the hiring process using artificial intelligence. It is more urgent to prepare effective specific procedures to prevent and monitor the risks of technology. As the hiring process is the first step in getting a job, which is a means of self-realization, evaluation according to equality of opportunities and job competency is important. We hope that practical and balanced clear laws will be created through intense discussions by legal, philosophical and technical experts. The most important proposition is that artificial intelligence is a tool that helps humans make decisions, and its ethics always depend on the person who makes the tool.

Switched hyperbolic balance laws and differential algebraic equations

Motivated by several applications, we investigate the well-posedness of a switched system composed by a system of linear hyperbolic balance laws and by a system of linear algebraic differential equations. This setting includes networks and looped systems of hyperbolic balance laws. The obtained results are globally in time, provided that the inputs have finite (but not necessarily small) total variation.

On the curvature dependence of gradient damage models: Control and opportunities

Curvature dependence of gradient damage models is an often undesired side effect. It originates from a change of the orientation of the non-local gradient term. The resulting curvature dependence then implicitly affects the loading behaviour, the predicted fracture energy and also the damage distribution. Two extended approaches are presented here for curvature control and applied to a micromorphic gradient damage model and to a phase-field model for fracture. Both allow for a better control of the crack path and consider the Hessian of the non-local damage field by means of an additional micromorphic field, since curvature is connected to second derivatives. The first approach is based on an additional curvature-dependent potential within the Helmholtz energy and automatically fulfils the second law of thermodynamics. The second approach modifies the micro forces of the balance law that is responsible for the curvature dependence directly. Unlike the first approach, it allows to eliminate the curvature dependence completely. However, thermodynamic consistency has to be checked explicitly.