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Nowadays, hybrid forms of teaching are increasingly used, combining traditional face-to- face teaching with elements of e-learning. This form of teaching is optimal because it combines all the socio-educational advantages of traditional teaching with new technologies and methods of learning and teaching, such as flipped classroom and ubiquitous learning, where learning can take place anytime and anywhere without spatial or temporal constraints. In traditional face-to-face teaching, data about the learning process is very difficult to access, there is very little of it, and it is rarely used to improve teaching. In e-learning, there are a handful of these data, and the e-learning systems themselves are often developed based on these data, with various machine learning algorithms guiding and optimizing the process of knowledge acquisition. However, in e-learning systems, the algorithms that drive the learning process are very often based on so-called black- box models that are outcome-oriented, without a deeper analysis and understanding of the problem. In both forms of teaching, and especially in the classical classroom model, the adaptation of the learning methodology is mainly based on the experience and intuition of the teacher and is therefore often subjective. One of the important facts of which educators should be aware is that the development of science and technology requires more frequent and increasing interventions in the curriculum, both at the level of content and method, and that these interventions must be directed toward progress, that is, toward more efficient and better acquisition of new knowledge. What is happening in industry, i.e. the fourth industrial revolution, must have a corresponding impact on education and modern teaching should evolve within the concept now known as Education 4.0. It is clear that a subjective approach based on qualitative analysis and self-reflection is not the optimal tool for this and that a new method must be found that is, above all, extremely objective and fast. In this paper, we explore the applications of classical data analysis methods such as correlation and cluster analysis, as well as the possibilities of modern approaches to data processing based on advanced data-driven algorithms. An example of such methods are techniques based on the Koopman operator. We pay special attention to the interpretation of the obtained results and the possibility of their implementation on concrete examples of hybrid educational systems. As an example we take a hybrid model of mathematics education developed for students in the field STEM. The content focuses on the elements of differential and integral calculus and vector analysis with applications in engineering. We analyze the ways to improve the whole teaching process based on the data extracted from the corresponding e- learning system and the interpretation of the results of the mentioned analyzes. We focus on several important areas: Identifying key factors of the teaching process and detecting weaknesses, detecting clusters among students in terms of their performance and interests, and identifying necessary changes in the course content.

For any non-autonomous dynamical system, the family of Koopman operators, as well as related Koopman eigenvalues and eigenfunctions, are parameterized by a time pair. Therefore, a logical approach in the data-driven algorithms for the non-autonomous Koopman mode decomposition is the application of a DMD method on the moving stencils of snapshots in order to capture the time-dependency. In this paper, we investigate the issues that arise in such an approach. These issues do not appear if we use the moving stencil approach as the model fitting method ; they appear as significant errors in the computed non-autonomous Koopman operator eigenvalues. The first issue manifests itself in the hybrid dynamical systems when the moving stencil passes over a non-autonomous switching point. We show that such stencils can be detected through the Krylov subspace projection error and propose an algorithm that computes correct eigenvalues by avoiding such stencils. The second issue appears in the continuous-in- time non-autonomous systems. Even if we apply techniques of finding good observables that solve all issues in the autonomous case, the non-autonomous Koopman eigenvalues will still be computed with a significant error. In the presented theorems, we reveal the nature of this error and propose a second algorithm that is based on the reduction of the stencil size. The application of the two new data-driven algorithms on various non- autonomous systems shows complete recovery from the errors otherwise present in computation of the non- autonomous Koopman operator eigenvalues.

In this paper we consider the nonstationary 3D flow of a compressible viscous and heat-conducting micropolar fluid, which is in the thermodynamical sense perfect and polytropic. The fluid domain is the subset of R 3 bounded with two coaxial cylinders that present solid thermoinsulated walls. We assume that the initial mass density, temperature, as well as the velocity and microrotation vectors are radially dependent only. The corresponding solution is also spatially radially dependent. We derive the mathematical model in the Lagrangian description and by using the Faedo–Galerkin method we introduce a system of approximate equations and construct its solutions. We also analyze two numerical examples.

In practice, the dynamics of open systems subject to time-dependent or random forcing is much more present than the dynamics of autonomous systems. Therefore, extension of the Koopman operator theory and applications to such systems is of great importance. At the same time, it brings a new viewpoint to the existing theory of nonautonomous as well as random dynamical systems, particularly, with application of Koopman-based data-driven algorithms. In this chapter, we first review the nonautonomous Koopman operator family based on the two standard nonautonomous dynamical system definitions: skew product and process. Then, we state basic properties of the operator and compare performance of the DMD and Arnoldi-type algorithms in the context of both definitions. In the case of the random dynamical systems (RDS), we introduce the associated stochastic Koopman operator family. We show that when RDS is Markovian, this family satisfies semigroup property and we present some properties for RDS generated by the stochastic differential equations. Finally, we discuss data-driven algorithms in the stochastic framework and illustrate their performance on numerical examples.

In this paper we consider the nonstationary 1D flow of the compressible viscous and heat-conducting micropolar fluid, assuming that it is in the thermodynamically sense perfect and polytropic. The fluid is between a static solid wall and a free boundary connected to a vacuum state. We take the homogeneous boundary conditions for velocity, microrotation and heat flux on the solid border and that the normal stress, heat flux and microrotation are equal to zero on the free boundary. The proof of the global existence of the solution is based on a limit procedure. We define the finite difference approximate equations system and construct the sequence of approximate solutions that converges to the solution of our problem globally in time.

The intention of this work is to concern the numerical solutions to the model of the nonstationary 1d micropolar compressible viscous and heat conducting fluid flow that is in the thermodynamical sense perfect and polytropic. The mathematical model consists of four partial differential equations, transformed from the Eulerian to the Lagrangian description, and which are associated with different boundary conditions. By using the finite difference scheme and the Faedo Galerkin method we make different numerical simulations to the results of our problems. The properties of both numerical schemes are analyzed and numerical results are compared on the chosen test examples. The comparison of the numerical results on problems that have the homogeneous or the non-homogeneous boundary conditions for velocity and microrotation show good agreement of both approaches. However, the advantage of the used finite difference method over the Faedo-Galerkin method lies in the simple implementation of the non-homogeneous boundary conditions and in the possibility of approximation of the free boundary problem on which the Faedo-Galerkin method is not applicable.

In this paper we consider the nonstationary shear flow between two parallel solid and thermoinsulated horizontal plates with the upper one moving irrotationally. The fluid is compressible, micropolar, viscous and heat-conducting, as well as in the thermodynamical sense perfect and polytropic. We assume that, given a Cartesian coordinate system x , y and z , solutions of corresponding problem are x -dependent only. Mathematical model is derived in the Lagrangian description. By using the Faedo–Galerkin method, as well as homogenization of boundary conditions, we derive an approximate system, which we use to obtain a numerical solution to the given problem.

In this paper we propose a generalized Koopman operator framework for discrete and continuous time random dynamical systems. For the particular classes of ran- dom dynamical systems, we provide the results that characterize the spectrum and the eigenfunctions of the stochastic Koopman operator. We discuss the relationship between the spectral properties of the generator of the evolution and the Koopman operator family. The numerical approximations of the spectral objects (eigenval- ues, eigenfunctions) of the stochastic Koopman operator are computed by using the state of the art DMD RRR algorithm. We explore its behavior in the stochastic case on several test examples. Moreover, the DMD RRR algorithm is applied in combination with the Hankel matrix and a convergence theorem for Hankel DMD RRR in the stochastic case is proved.

We consider a nonstationary 3D flow of a compressible viscous heat-conducting micropolar fluid, which is in the thermodynamical sense perfect and polytropic, in the domain bounded with two concentric spheres in R 3 . In this paper we establish the existence of a global solution to the free boundary problem defined with the homogeneous boundary conditions for velocity, microrotation, heat flux on the fixed border and homogeneous boundary conditions for strain, microrotation, and heat flux on the free boundary. We suppose that the initial data are spherically symmetric, with positive initial density and temperature, having the zero density on the free boundary. Because of the spherical symmetry, the starting three-dimensional problem is transformed to the one-dimensional problem in Lagrangian coordinates in the domain that is a segment. The solution to our problem is obtained as a limit of the sequence of approximate solutions derived from suitable semi-discrete finite difference approximate systems. By using the derived bounded estimates of the approximate solutions and the results of the weak and strong compactness, we establish the convergence to the generalized solution of our problem globally in time.

In this paper we propose a family of well-balanced semi-implicit numerical schemes for hyperbolic conservation and balance laws. The basic idea of the proposed schemes lies in the combination of the finite volume WENO discretization with Roe’s solver and the strong stability preserving (SSP) time integration methods, which ensure the stability properties of the considered schemes [S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89–112]. While standard WENO schemes typically use explicit time integration methods, in this paper we are combining WENO spatial discretization with optimal SSP singly diagonally implicit (SDIRK) methods developed in [L. Ferracina, M.N. Spijker, Strong stability of singly diagonally implicit Runge–Kutta methods, Appl. Numer. Math. 58 (2008) 1675–1686]. In this way the implicit WENO numerical schemes are obtained. In order to reduce the computational effort, the implicit part of the numerical scheme is linearized in time by taking into account the complete WENO reconstruction procedure. With the proposed linearization the new semi-implicit finite volume WENO schemes are designed. A detailed numerical investigation of the proposed numerical schemes is presented in the paper. More precisely, schemes are tested on one-dimensional linear scalar equation and on non-linear conservation law systems. Furthermore, well-balanced semi-implicit WENO schemes for balance laws with geometrical source terms are defined. Such schemes are then applied to the open channel flow equations. We prove that the defined numerical schemes maintain steady state solution of still water. The application of the new schemes to different open channel flow examples is shown.

Strain-life fatigue parameters of unalloyed steels, low-alloy steels, high-alloy steels, aluminium alloys and titanium alloys and their strain-life behaviour across low-cycle, medium-cycle and high-cycle fatigue range were analyzed. Substantial differences were found to exist both among fatigue parameters and among (delta epsilon)/2 - 2Nf relationships of materials belonging to these main metallic materials groups. This turns out to be in contrast with the fact that most of the existing methods for the estimation of strain-life fatigue parameters from monotonic properties show rather limited regard as to which group material in question belongs to. From results of the analysis it can be concluded that in estimation of strain-life fatigue parameters differential approach might be desirable, i.e. that separate expressions should be developed for every material group. It is expected that such an approach and resulting expressions could result in more accurate estimations of fatigue parameters and hence, more accurate estimations of time to crack initiation.

A new approach to numerical modeling of water hammer is proposed. An unsteady pipe flow model incorporating Brunone’s unsteady friction model is used, but in contrast to the standard treatment of the unsteady friction term as a source term, the writers propose a nonconservative formulation of source term. Second-order flux limited and high order weighted essentially nonoscillating numerical schemes were applied to the proposed formulation, and results are in better agreement with measurements when compared with results obtained with standard form.

The importance of spatial resolution for two-dimensional shallow-water model accuracy has been investigated by testing the effect of mesh refinement on two test cases based on laboratory dam-break experiments. A balanced first-order accurate upwind Q-Scheme and a second-order accurate upwind Hancock Monotone Upstream-centered Scheme for Conservation Laws scheme were both first validated on an analytical test, and then applied to the experimental dam-break test cases on four meshes of different density. Simulation results were evaluated through comparison of experimental and computed water level values at several available gauge points. Model sensitivity analysis showed that (1) mesh density was not critical for results accuracy ; (2) excessive mesh refinement somewhat deteriorated the results ; and (3) optimal spatial resolution was relatively low. Response is shown to be highly complex and no simple relation between spatial resolution and model accuracy has been found.

In this paper, we present an extension of balanced upwind explicit finite volume schemes designed in Vukovic and Sopta (SIAM J Sci Comput 24(5):1630–1649, 2003) to semi-implicit ones. Particularly, those schemes are applied to open-channel flows with general geometries and we verify the exact conservation property (C-property). We present the algorithm, the proof of exact C-property and results for several test cases. Also, we test and compare balanced semi-implicit and explicit schemes on standard test cases and on cases involving friction, non-uniform bed slopes and strong channel width variations.

In this paper we consider the numerical approximations of the nonstationary 1D flow of the compressible micropolar fluid, assuming that it is in the thermodynamically sense perfect and polytropic. The flow equations are considered in Eulerian formulation. It is proved that the inviscid micropolar flow equations are hyperbolic and the corresponding eigensystem is determined. The numerical approximation is based on the upwind Roe solver applied to the inviscid part of the flux, while the viscous part of the flux is approximated by using central differences. The numerical results for the inviscid flow show that the numerical scheme approximate the solutions very accurately. The numerical test for the viscous and heat conducting flow are also performed.

Most of the standard papers about the WENO schemes consider their implementation to uniform meshes only. In that case the WENO reconstruction is preformed efficiently by using the algebraic expressions for evaluating the reconstruction values and the smoothness indicators from cell averages. The coefficients appearing in these expressions are constant, dependent just on the scheme order, not on the mesh size or the reconstruction function values, and can be found, for example, in Jiang, Shu, Efficient implementation of WENO schemes, J. Comp. Physics, 1996. In problems where the geometrical properties must be taken into account or the solution has localized fine scale structure that must be resolved, it is computationally efficient to do local grid refinement, and therefore also desirable to have numerical schemes, which can be applied to nonuniform meshes. Finite volume WENO schemes extend naturally to nonuniform meshes although the reconstruction becomes quite complicated, depending on the complexity of the grid structure. In this paper we propose an efficient implementation of finite volume WENO schemes to nonuniform meshes. In order to save the computational cost in the nonuniform case, we suggest the way for precomputing the coefficients and linear weights for different orders of WENO schemes. Furthermore, for the smoothness indicators that are defined in an integral form we present the corresponding algebraic expressions in which the coefficients obtained as a linear combination of divided differences arise. In order to validate the new implementation, resulting schemes are applied in different test examples.

The goal of this work is to extend finite volume WENO and central WENO schemes to the hyperbolic balance laws with geometrical source term and spatially variable flux function. In particular, we apply proposed schemes to the shallow water and the open-channel flow equations where the source term depends on the channel geometry. For obtaining stable numerical schemes that are free of spurious oscillations, it becomes crucial to use the decomposed source term evaluation, which maintains the balancing between the flux gradient and the source term. In addition, the open-channel flow equations contain spatially variable flux function. The appropriate definitions of the terms that arise in the source term decomposition, in combination with the Roe approximate Riemann solver that includes the spatial derivative of the flux function, lead to the finite volume WENO scheme that satisfies the exact conservation property - the property of preserving the quiescent flow exactly. When the central WENO schemes are applied, additional reformulations are introduced for the transition from the staggered values to the nonstaggered ones and vice versa by using the WENO reconstruction procedure. The proposed central WENO schemes also preserve the quiescent flow, but only in prismatic channels. In various test problems the obtained balanced schemes show improvements in comparison with the standard versions of the proposed type schemes, as well as with some other first- and second-order numerical schemes.

In this paper we extend the non-staggered version of the central NT (Nessyahu–Tadmor) scheme to the balance laws with geometrical source term. This extension is based on the source term evaluation that includes balancing between the flux gradient and the source term with an additional reformulation that depends on the source term discretization. The main property of the scheme obtained by the proposed reformulation is preservation of the particular set of the steady-state solutions. We verify the improved scheme on two types of balance laws with geometrical source term: the shallow water equations and the non-homogeneous Burger's equation. The presented results show good behaviour of the considered scheme when compared with the analytical or numerical results obtained by using other numerical schemes. Furthermore, comparison with the numerical results obtained by the classical central NT scheme where the source term is simply pointwise evaluated shows that the proposed reformulations are essential. Copyright © 2004 John Wiley & Sons, Ltd.

Essentially nonoscillatory and weighted essentially nonoscillatory schemes are high order resolution schemes constructed for the hyperbolic conservation laws. In this paper we extend these schemes to the one-dimensional bed-load sediment transport equations. The difficulties that arise in the numerical modelling come from the fact that a nonconservative product is present in the system. Our specific numerical approximations for the nonconservative product are based on two ideas. First is to include the influence of that term in the system upwinding and the second is to define the numerical approximation in such a way that the obtained scheme solves the system for the quiescent flow case exactly. As a consequence, the resulting schemes give excellent results, as it can be seen from the numerical tests we present. On the opposite, the numerical results obtained by applying the pointwise evaluation of nonconservative product on the same tests present unacceptably large numerical errors.

The irruption of gas and oil into the Gulf of Mexico during the Deepwater Horizon event fed a deep sea bacterial bloom that consumed hydrocarbons in the affected waters, formed a regional oxygen anomaly, and altered the microbiology of the region. In this work, we develop a coupled physical–metabolic model to assess the impact of mixing processes on these deep ocean bacterial communities and their capacity for hydrocarbon and oxygen use. We find that observed biodegradation patterns are well-described by exponential growth of bacteria from seed populations present at low abundance and that current oscillation and mixing processes played a critical role in distributing hydrocarbons and associated bacterial blooms within the northeast Gulf of Mexico. Mixing processes also accelerated hydrocarbon degradation through an autoinoculation effect, where water masses, in which the hydrocarbon irruption had caused blooms, later returned to the spill site with hydrocarbon-degrading bacteria persisting at elevated abundance. Interestingly, although the initial irruption of hydrocarbons fed successive blooms of different bacterial types, subsequent irruptions promoted consistency in the structure of the bacterial community. These results highlight an impact of mixing and circulation processes on biodegradation activity of bacteria during the Deepwater Horizon event and suggest an important role for mixing processes in the microbial ecology of deep ocean environments.

In this paper we present numerical simulations for the dam break flood wave propagation from Tribalj accumulation to Crikvenica (Croatia). The mathematical models we used were the one and the two-dimensional shallow water equations. They were solved with the well-balanced finite volume numerical schemes which additionally include special numerical treatment of the wetting/drying front boundary. These schemes were tested on CADAM test problems. The aim of this study was to assess potential damage in the cities of Tribalj and Crikvenica. Results of these simulations were used as the basis for urban planning and micro-zoning of the flood-risk areas. Several different dam break scenarios were considered, ranging from sudden dam disappearance to partial and dynamic breach formation. The research showed some advantages of numerical simulation over physical models.

In this paper we focus our attention on obtaining well-balanced schemes for balance laws by using Marquina’s flux in combination with the finite difference and finite volume WENO schemes. We consider also the Rusanov flux splitting and the HLL approximate Riemann solver. In particular, for the presented numerical schemes we develop corresponding discretizations of the source term, based on the idea of balancing with the flux gradient. When applied to the open-channel flow and to the shallow water equations, we obtain the finite difference WENO scheme with Marquina’s flux splitting, which satisfies the approximate conservation property, and also the balanced finite volume WENO scheme with Marquina’s solver satisfying the exact conservation property. Finally, we also present an improvement of the balanced finite difference WENO scheme with the Rusanov (locally Lax–Friedrichs) flux splitting, we previously developed in [Vukovic S, Sopta L. ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations. J Comput Phys 2002;179:593–621].

Extended finite difference WENO schemes for hyperbolic balance laws with spatially varying flux and geometrical source term were developed by the authors. In these schemes high order ENO and WENO reconstruction for flux and source term characteristicwise components are used, therefore schemes give high resolution results. In this paper these high order properties are analyzed hrough several convergence tests: on two shallow water tests, one linear acoustics test, and one test for the Burgers equation with a source term. Experimentally established orders of accuracy vs. the theoretically expected orders are compared for the new schemes, as well as for the original WENO schemes combined with pointwise source term evaluation.

In this paper we present main points in the process of application of numerical schemes for hyperbolic balance laws to water wave propagation and flooding. The appropriate mathematical models are the one-dimensional open channel flow simulation results and two-dimensional shallow water equations. Therefore good simulation results can only be obtained with well-balanced numerical schemes such as the ones developed by Bermudez and Vazquez, Hubbard and Garcia-Navarro, LeVeque, etc. as well as the ones developed by the authors of this papers. We also propose a modification of the well-balanced Q-scheme for the two-dimensional shallow water equations that solves the wetting and drying problem. Finally, we present numerical results for three simulation tasks: the CADAM dam break experiment, the water wave propagation in the Toce river, and the catastrophic dam break on the Malpasset river.

The numerical method we consider is based on the nonstaggered central scheme proposed by Jiang, Levy, Lin, Osher, and Tadmor (SIAM J. Numer. Anal. 35, 2147(1998)) that was obtained by conversion of the standard central NT scheme to the nonstaggered mesh. The generalization we propose is connected with the numerical evaluation of the geometrical source term. The presented scheme is applied to the nonhomogeneous shallow water system. Including an appropriate numerical treatment for the source term evaluation we obtain the scheme that preserves quiescent steady-state for the shallow water equations exactly. We consider two different approaches that depend on the discretization of the riverbed bottom. The obtained schemes are well balanced and present accurate and robust results in both steady and unsteady flow simulations.

The mathematical model describing the physical phenomenon of one-dimensional bed-load sediment transport in channels and rivers consists of three equations. Two of them represent conservation laws for one-dimensional shallow water equations, and third is the conservation law governing bed-load sediment transport. Here we considerone possibletype of the sediment flux proposed by Hudson and Sweby [7]. We compare numerical results for test problems using different numerical schemes: Q-scheme, Hubbard’s scheme, ENO Roe and ENO locally Lax-Friedrichs scheme. The obtained results illustrate good properties of ENO schemes with the source term decomposition, developed by authors. We also prove that these schemes have the exact C-property when applied to the sediment transport equations.