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With the historical review, it can determine that plasma scientists devolved many numerical schemes in literature to solve the kinetic plasma models such as Vlasov- Maxwell system, but the fatal problem is that most of the designed algorithms violate the conservation laws. This issue is significant and cannot disregard because the Vlasov Maxwell system is connected with conservations properties and can evaluate after performing some effective mathematical operations. In this prospective, we formulated the Atangana-Baleanu fractional derivative with Caputo concepts based fractional-order plasma model to study the fractional performance of the plasma particles and further suggested a higher-order conservative numerical algorithm that follows the conservation laws, based on operational matrices theory which is evaluated by using the concepts of Shifted Gegenbauer estimations. The numerical convergence of the designed algorithm is also inspected to validate the competency and compatibility. The comprehensive analysis of the fractional performance of the plasma particles is also part of the study. Moreover, this concept can extend to the variable order and multi-dimensional problems for Vlasov and Boltzmann system.

A highly efficient new three-step derivative-free family of numerical iterative schemes for estimating all roots of polynomial equations is presented. Convergence analysis proved that the proposed simultaneous iterative method possesses 12th-order convergence locally. Numerical examples and computational cost are given to demonstrate the capability of the method presented.

Shallow water equations are widely used in the simulation of those geophysical flows for which the flow horizontal length scale is much greater than the vertical one. Inspired by the example of lava flows, we consider here a modified model with an additional transport equation for a scalar quantity (e.g., temperature), and the derivation of the shallow water equations from depth-averaging the Navier-Stokes equations is presented. The assumption of constant vertical profiles for some of the model variables is relaxed allowing the presence of vertical profiles, and it follows that the non-linearity of the flux terms results in the introduction of appropriate shape coefficients. The space discretization of the resulting system of hyperbolic partial differential equations is obtained with a modified version of the finite volume central-upwind scheme introduced by Kurganov and Petrova in 2007. The time discretization is based on an implicit-explicit Runge-Kutta method which couples properly the hyperbolic part and the stiff source terms, avoiding the use of a very small time step; the use of complex arithmetic increases accuracy in the implicit treatment of stiff terms. The whole scheme is proved to preserve the positivity of flow thickness and the stationary steady-states. Some numerical experiments are performed to validate the proposed method and to show the incidence on the numerical solutions of shape coefficients introduced in the model.

Fractional order differential equations are utilized for modeling many complicated physical and natural phenomena in nonlinear sciences and related fields. In this manuscript, the fractional order Schrodinger-KdV equation in the sense of Atangana-Baleanu derivative is investigated. The Schrodinger-KdV equation demonstrates various types of wave propagation such as Langmuir wave, dust-acoustic wave and electromagnetic waves in plasma physics. Using the fixed-point theorem, the existence and uniqueness to the solution of the studied nonlinear model is established. Using the modified Laplace decomposition method, we establish the exact solution to fractional order Schrodinger-KdV equation. The numerical simulations to the reported result are presented. The comparison between analytical and numerical approximations is also presented. It is shown that the approximate-analytical results are compatible with the analytical results via the L-2 and L-infinity error norms. We compare our result with some existing results in the literature. (C) 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University.

In this work, we develop a sophisticated mathematical model for dengue transmission dynamics based on vaccination, reinfection and carrier class assumptions. We utilize Caputo-Fabrizio fractional framework to represent the transmission phenomena of dengue. For the model analysis, the basic theory and findings of the Caputo-Fabrizio fractional operator are provided. We utilize the next-generation approach to calculate the threshold parameter R0 for our proposed dengue infection model. We have shown that the disease-free steady-state of the hypothesized dengue system is locally asymptotically stable if R0

In this paper, we consider a vector-host epidemic model with saturated incidence and treatment functions. This type of function is biologically more suitable in situations where the number of infected individuals is getting larger and the medical facilities are limited. Further, as most of the biological phenomena possess the fading memory and show crossover behavior. Therefore, in the present study, initially we develop the proposed model using integer-order derivative and then two different fractional operators namely Caputo and Atangana-Baleanu in Caputo sense (ABC) are used to formulate the fractional model. Some of the basic properties including positivity, existence, uniqueness and stability results at the disease free equilibrium of the Caputo model are shown. The existence and uniqueness of the ABC model are explored via fixed point theorem. Furthermore, the fractional models are solved numerically using an efficient iterative scheme. Finally, detailed simulation results are depicted in order to demonstrate the significance of the order of fractional operators and model parameters on the disease dynamics.

International audience; Monte-Carlo (MC) schemes for photonics have been intensively studied all along the last decades, see [1] for a complete review. The recent ISMC scheme described in [2] presents many advantages (no teleportation error, converging behaviour with respect to the spatial and time discretisations). But it is rather different from the IMC one (it is based on a different linearisation and needs a slightly different code architecture). On another hand, legacy codes are often based on IMC implementations. For this reason, it remains important to be able to cancel the teleportation error within IMC codes. Cancelling the teleportation error within the IMC framework is also important for fair comparisons between both the IMC and the ISMC linearisations. This paper aims at suggesting some simple corrections to apply to an IMC implementation to completely cancel the teleportation error.

We propose a quantization-based numerical scheme for a family of decoupled FBSDEs. We simplify the scheme for the control in Pag\`es and Sagna (2018) so that our approach is fully based on recursive marginal quantization and does not involve any Monte Carlo simulation for the computation of conditional expectations. We analyse in detail the numerical error of our scheme and we show through some examples the performance of the whole procedure, which proves to be very effective in view of financial applications., Comment: 22 pages

This article describes the corona virus spread in a population under certain assumptions with the help of a fractional order mathematical model. The fractional order derivative is the well-known fractal fractional operator. We have given the existence results and numerical simulations with the help of the given data in the literature. Our results show similar behavior as the classical order ones. This characteristic shows the applicability and usefulness of the derivative and our numerical scheme.

Human immunodeficiency virus (HIV) is a life life-threatening and serious infection caused by a virus that attacks CD4+ T-cells, which fight against infections and make a person susceptible to other diseases. It is a global public health problem with no cure, therefore, it is highly important to study and understand the intricate phenomena of HIV. In this article, we focus on the numerical study of the path-tracking damped oscillatory behavior of a model for the HIV infection of CD4+ T-cells. We formulate fractional dynamics of HIV with a source term for the supply of new CD4+ T-cells depending on the viral load via the Caputo–Fabrizio derivative. In the formulation of fractional HIV dynamics, we replaced the constant source term for the supply of new CD4+ T-cells from the thymus with a variable source term depending on the concentration of the viral load, and introduced a term that describes the incidence of the HIV infection of CD4+ T-cells. We present a novel numerical scheme for fractional view analysis of the proposed model to highlight the solution pathway of HIV. We inspect the periodic and chaotic behavior of HIV for the given values of input factors using numerical simulations.

Continuous hybrid methods are now recognized as efficient numerical methods for problems whose solutions have finite domains or cannot be solved analytically. In this work, the continuous hybrid numerical method for the solution of general second order initial value problems of ordinary differential equations is considered. The method of collocation of the differential system arising from the approximate solution to the problem is adopted using the power series as a basis function. The method is zero stable, consistent, convergent. It is suitable for both non-stiff and mildly-stiff problems and results were found to compete favorably with the existing methods in terms of accuracy.

In this paper, we develop a new approximation for the generalized fractional derivative, which is characterized by a scale function and a weight function. The new approximation is then used in the numerical treatment of a class of generalized fractional sub-diffusion equations. The theoretical aspects of solvability, stability and convergence are established rigorously in maximum norm by discrete energy methodology. Due to the new approximation, the theoretical temporal convergence order of the numerical scheme improves those of earlier work. To confirm, four examples are presented to illustrate the accuracy of the proposed scheme and to compare with other methods in the literature.

In this paper, we build wellposed intrusive generalised Polynomial Chaos (gPC) based reduced models for uncertain photonics. We solve the reduced models with a Monte-Carlo (MC) scheme. Care is taken to highlight under which condition a reduced model (gPC based or not) is wellposed. The analysis is carried out thanks to an analogy between the construction of reduced models for uncertainty quantification and the construction of reduced models for kinetic equations. In order to enforce the aforementioned wellposedness conditions, several strategies, inspired from the hyperbolicity-preserving ones [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] are reviewed, adapted and analysed. The resolution of the reduced models is performed thanks to an astute combination of the Implicit Semi-analog MC (ISMC, see [9] ) scheme for photonics and of MC-gPC (see [10] ) for uncertainty propagation. This work demonstrates that MC-gPC can be efficiently applied to a stiff nonlinear set of partial derivative equations if the MC resolution allows a fast convergence with respect to both the time and spatial discretisations (the latter properties being allowed by ISMC). Several benchmarks are investigated in the last section, they allow illustrating important aspects of the new ISMC-gPC solver for uncertain photonics.

The Hermite targeted essentially non-oscillatory (HTENO) scheme offers an excellent small-scale resolution and better shock-capturing capability in solving compressible flow problems. However, the scheme has a high computational cost due to global reference smoothness indicators and artificial limiters. In this paper, we improve the computational efficiency by simplifying the smoothness indicator of the HTENO scheme and adopting a hybrid reconstruction method, where the HTENO scheme is only employed near the discontinuous region and uses linear reconstruction for the smooth parts. The numerical results show that the proposed scheme achieves the desired order of accuracy and inherits HTENO’s numerical resolutions while improving the computational efficiency.

International audience; We present and analyse a numerical framework for the approximation of nonlinear degenerate elliptic equations of the Stefan or porous medium types. This framework is based on piecewise constant approximations for the functions, which we show are essentially necessary to obtain convergence and error estimates. Convergence is established without regularity assumption on the solution. A detailed analysis is then performed to understand the design properties that enable a scheme, despite these piecewise constant approximations and the degeneracy of the model, to satisfy high-order error estimates if the solution is piecewise smooth. Numerical tests, based on continuous and discontinuous approximation methods, are provided on a variety of 1D and 2D problems, showing the influence on the convergence rate of the nature of the degeneracy and of the design choices.

In this paper, we consider numerical solutions for Riesz space fractional partial differential equations with a second order time derivative. We propose a Galerkin finite element scheme for both the temporal and spatial discretizations. For the proposed numerical scheme, we derive sharp stability estimates as well as optimal a priori error estimates. Extensive numerical experiments are conducted to verify the promising features of the newly proposed method.

The current work is of interest to introduce a detailed analysis of the novel fractional COVID-19 model. Non-local fractional operators are one of the most efficient tools in order to understand the dynamics of the disease spread. For this purpose, we intend as an attempt at investigating the fractional COVID-19 model through Caputo operator with order χ ∈ ( 0 , 1 ) . Employing the fixed point theorem, it is shown that the solutions of the proposed fractional model are determined to satisfy the existence and uniqueness conditions under the Caputo derivative. On the other hand, its iterative solutions are indicated by making use of the Laplace transform of the Caputo fractional operator. Also, we establish the stability criteria for the fractional COVID-19 model via the fixed point theorem. The invariant region in which all solutions of the fractional model under investigation are positive is determined as the non-negative hyperoctant R + 7 . Moreover, we perform the parameter estimation of the COVID-19 model by utilizing the non-linear least squares curve fitting method. The sensitivity analysis of the basic reproduction number R 0 c is carried out to determine the effects of the proposed fractional model’s parameters on the spread of the disease. Numerical simulations show that all results are in good agreement with real data and all theoretical calculations about the disease.

International audience; In this paper, we present a new implicit Monte-Carlo scheme for photonics. The new solver combines the benefits of both the IMC solver of Fleck & Cummings and the SMC solver of Ahrens & Larsen. It is implicit hence allows taking affordable time steps (as IMC) and has no teleportation error (as SMC). The paper also provides some original analysis of existing schemes (IMC, tilted IMC, SMC), especially with respect to the teleportation error in the equilibrium diffusion regime. In particular, we demonstrate that any small spatial inaccuracies during the sampling of source particles for IMC lead to a competing behaviour between the spatial and time discretisation parameters. The new scheme we suggest is implicit, conservative, has no teleportation error (and as a consequence does not need tilting), does not rely on source sampling for the emission of source particles, captures the equilibrium diffusion limit (provided a small enough time step), can be used with arbitrary equations of state and does not suffer the above competing behaviour. All those properties are either demonstrated or numerically highlighted in the paper.

The importance of numerical schemes in hydrological models has been increasingly recognized in the hydrological community. However, the relationship between model performance and the properties of numerical schemes remains unclear. In this study, we employed two types of numerical schemes (i.e., explicit Runge-Kutta schemes with different orders of accuracy and partially implicit Euler schemes with different implicit factors) in the hydrological model (HYMOD) to simulate the flow hydrograph of the Leaf River basin from 1948 to 1988. Results computed by different numerical schemes were compared and the relationships between model performance and two scheme properties (i.e., the order of accuracy and the implicit factor) were discussed. Results showed that the more explicit schemes generally lead to the overestimation of flow hydrographs, whereas the more implicit schemes lead to underestimation. In addition, the numerical error tended to decrease with increasing orders of accuracy. As a result, the optimal parameter sets found by low-order schemes significantly deviated from those found by the analytical solution. The findings of this study can provide useful implications for designing suitable numerical schemes for hydrological models.

In this article, the generalized Rosenau–KdV equation is split into two subequations such that one is linear and the other is nonlinear. The resulting subequations with the prescribed initial and boundary conditions are numerically solved by the first order Lie–Trotter and the second-order Strang time-splitting techniques combined with the quintic B-spline collocation by the help of the fourth order Runge–Kutta (RK-4) method. To show the accuracy and reliability of the proposed techniques, two test problems having exact solutions are considered. The computed error norms L2 and L? with the conservative properties of the discrete mass Q(t) and energy E(t) are compared with those available in the literature. The convergence orders of both techniques have also been calculated. Moreover, the stability analyses of the numerical schemes are investigated. © 2019 Wiley Periodicals, Inc.

Fourth order curvature driven interface evolution equations frequently appear in the natural sciences. Often axisymmetric geometries are of interest, and in this situation numerical computations are much more efficient. We will introduce and analyze several new finite element schemes for fourth order geometric evolution equations in an axisymmetric setting, and for selected schemes we will show existence, uniqueness and stability results. The presented schemes have very good mesh and stability properties, as will be demonstrated by several numerical examples., Revised version. 37 pages, 27 figures. This article is closely related to arXiv:1805.04322

In this paper, we are interested in the resolution of the time-dependent problem of particle transport in a media whose composition is uncertain. The most common resolution strategy consists of running, at prescribed points in uncertain space (at experimental designs points), a simulation device as a black-box. This ensures performing the uncertainty propagation resolution (i.e. resolution of SPDE). This kind of strategy is commonly called non-intrusive. The non-intrusive resolution can be carried with Monte-Carlo, quasi Monte-Carlo, generalized Polynomial Chaos etc. The latter is of interest in this paper for its fast convergence rate. First, we go over and illustrate the main drawbacks of the non-intrusive (gPC or otherwise) uncertainty propagation resolution for the linear Boltzmann equation in a simplified configuration. We then build a new gPC based MC scheme solving intrusively the uncertain counterpart of the problem. The paper ends with some numerical examples.

International audience; Erythropoiesis, the red blood cell production process, involves interactions between cell populations with different differentiation states, mainly immature progenitor cells and mature erythrocytes, and growth factors such as erythropoietin and glucocorticoids, known to respectively inhibit cell apoptosis, stimulate proliferation and differentiation, and stimulate self-renewal. The feedback regulation of this process allows a very fast and efficient recovery in the case of a severe anemia. We consider an age-structured model of red blood cell production accounting for these feedback regulations and the dynamics of growth factors. We theoretically show the existence of a unique positive steady state for the model and we propose a numerical method to obtain an approximation to its solution. Experiments are reported to show numerically, on one hand, the optimal convergence order of the numerical scheme and, on the other hand, a fine approximation to real experimental data, with a suitable selection of the parameters involved.

In this paper, we shall construct a new numerical scheme for fourth-order fractional diffusion wave model. The solvability, stability and convergence of proposed method are established in l2 norm and it is shown that the numerical scheme improves the earlier work done. Simulation is carried out to verity the accuracy and efficiency of the numerical scheme. Published version

International audience; Parallel transport on Riemannian manifolds allows one to connect tangent spaces at different points in an isometric way and is therefore of importance in many contexts, such as for statistics on manifolds. The existing methods to compute parallel transport require either the computation of Riemannian logarithms, such as the Schild's ladder, or the Christoffel symbols. The Logarithm is rarely given in closed form, and therefore costly to compute whereas the number of Christoffel symbols explodes with the dimension of the manifold, making both these methods intractable. From an identity between parallel transport and Jacobi fields, we propose a numerical scheme to approximate the parallel transport along a geodesic. We find and prove an optimal convergence rate for the scheme, which is equivalent to Schild's ladder's. We investigate potential variations of the scheme and give experimental results on the Euclidean two-sphere and on the manifold of symmetric positive-definite matrices.

In this paper, we introduce a numerical scheme for fractional differential equations with feedback control. Due to the possibility of dealing with a feedback control as a functional delay, we construct a numerical method based on Euler method accompanied with piecewise constant interpolation. The method is based on the idea of separating the current state and the prehistory function. The convergence of the method is stated and proved. Numerical experiments are given to clarify the good agreement between numerical and theoretical results.

This paper continues optimization of numerical solution algorithm of iterative scheme grid for the droplet task, which was presented in the first article of this series. Assumptions were made by optimal assessable number of members which was already defined in numerical experiment in case of compound heat spread by conduction and radiation and an unsteady temperature field was described by infinite integral equation sum. For the convenience of numerical analysis, droplet thermal parameters PT were described by universal Fourier criteria Fo and by dimensionless radial coordinate η function PT(Fo,η). This function is given in form of infinite integral equation sum with each thermal parameter having a distinct initial member and individually defined subsidiary function. This function is given in form of infinite integral equation sum with each thermal parameter having a distinct initial member and individually defined subsidiary function. The droplet time and radial coordinate grading change influence for calculated function graphs PT(Fo,η) was evaluated by water droplets heat transfer and phase transformation numerical experiment. Summarizing by conduction and radiation heated water droplets thermal parameter variation patterns a methodology of forming an optimal grid for droplet task' task iterative solving, is provided.

Made available in DSpace on 2020-12-10T19:50:16Z (GMT). No. of bitstreams: 0 Previous issue date: 2020-02-01 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) CEFT (Centro de Estudos de Fenomenos de Transporte) FEDER funds through COMPETE2020 - Programa Operational Competitividade e Intemacionalizacao (POCI) FCT - Fundacao para a Ciencia e Tecnologia Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG) The capillary thinning dynamics can be considered one of the fingerprints of extensionally-dominated viscoelastic flows. Notably, the rheological behavior of complex fluids in extensional flow has been investigated in different rheometric devices, as for instance in the Capillary Breakup Extensional Rheometer (CaBER), the Dripping-ontoSubstrate (DoS) rheometry, or the Rayleigh Ohnesorge Jetting Extensional Rheometer (ROJER). In recent years, the Computational Rheology community has made a considerable effort to better understand the interplay of viscoelasticity and capillarity effects in such transient rheometric experiments. In this work, we present a numerical study on the dynamics of extensional flows of dilute polymeric solutions at small scales. Our numerical investigation focus primarily on the potential of using small scale two-phase extensional viscoelastic flows as suitable platforms for performing rheometry of weakly viscoelastic polymer solutions. In particular, we have adopted the setup used in the experiments of Sousa et al. [Rheol. Acta 56 (2017) 11-20]. In such set up, the filament stretching is conducted using oil as an outer phase, avoiding sample evaporation, or allowing visualization of the filament interior. In order to handle with the moving interface problem, we have employed a two-phase viscoelastic fluid flow solver, based on a finite differences scheme. In this methodology, the interface between the fluids is approximated by the volume-of-fluid interface reconstruction algorithm, and a second-order operator-split method is used to solve the advection equation. We observed a negligible influence of the use of different types of low viscosity oils as outer fluid on the measurement of the sample relaxation time. We also found that the use of an external low viscosity immiscible oil did not prevent the formation of beads-on-a-string structures. Univ Fed Uberlandia, Fac Matemat, Bloco 1F,Av Joao Naves de Avila, BR-38408100 Uberlandia, MG, Brazil Univ Estadual Paulista, Fac Ciencias & Tecnol, Dept Matemat & Comp, BR-19060900 Presidente Prudente, SP, Brazil Univ Porto, Ctr Estudos Fenomenos Transporte, Fac Engn, Rua Dr Roberto Frias S-N, P-4200465 Porto, Portugal Univ Porto, Dept Engn Quim, CEFT, Fac Engn, Rua Dr Roberto Frias S-N, P-4200465 Porto, Portugal Univ Estadual Paulista, Fac Ciencias & Tecnol, Dept Matemat & Comp, BR-19060900 Presidente Prudente, SP, Brazil FAPESP: 2013/07375-0 CNPq: 307459/2016-0 FEDER funds through COMPETE2020 - Programa Operational Competitividade e Intemacionalizacao (POCI): PTDC/EMS-ENE/3362/2014 - POCI-01-0145-FEDER016665 FCT - Fundacao para a Ciencia e Tecnologia: UID/EMS/00532/2013 FAPEMIG: APQ-03075-18

The main challenge in the analysis of numerical schemes for the Zakharov system originates from the presence of derivatives in the nonlinearity. In this article, a new trigonometric time-integration scheme for the Zakharov system is constructed and convergence is proved. The time-step restriction is independent of a spatial discretization. Numerical experiments confirm the findings.

We shall develop a high order numerical scheme for a fourth-order fractional sub-diffusion problem. Theoretical results will be established in maximum norm and it is shown that the convergence order is higher than some earlier work done. Numerical experiments will be carried out to demonstrate the efficiency of the proposed scheme as well as to compare with other methods. Published version

A finite-dimensional Port-Hamiltonian formulation for the presented. A numerical scheme based on this formulation is shallow water equations. The scheme is verified against conservation of mass and energy to the discrete level. dynamics of smooth open channel flows is developed for both the linear and nonlinear exact solutions and has the advantage of (c) 2012 Elsevier B.V. All rights reserved.

16 pages; International audience; We derive in this paper a numerical scheme in order to calculate solutions of 1D transport equations. This 2nd-order scheme is based on the method of characteristics and consists of two steps: The first step is about the approximation of the foot of the characteristic curve whereas the second one deals with the computation of the solution at this point. The main idea in our scheme is to combine two 2nd-order interpolation schemes so as to preserve the maximum principle. The resulting method is unconditionaly stable and designed for classical solutions but turns out to remain valid when shocks occur.

In this article, we propose a simple method for detecting an inclusion Ω2 embedded in a host electrostatic medium Ω1 from a single Cauchy pair of voltage and current flux measurements on the exterior boundary of Ω1. A nonlinear constrained minimization regularized method of fundamental solutions is developed for the numerical solution of this inverse problem. The stability of the numerical scheme is investigated by inverting measurements contaminated by random noise. © 2010 Taylor & Francis. 18 4 567 583 Cited By :20

We consider integrodifferential equations of the abstract form $\mathbf{H} (\partial_{t})\Phi=\mathbf{G}(\nabla)\Phi+f$, where $\mathbf{H}(\partial_{t})$ is a diagonal convolution operator and $\mathbf{G}(\nabla)$ is a linear anti-selfadjoint differential operator. On the basis of an original approach devoted to integral causal operators, we propose and study a time-local augmented formulation under the form of a Cauchy problem $\partial_{t}\Psi= \mathcal{A}\Psi+\mathcal{B}f$ such that $\Phi=\mathcal{C}\Psi$. We show that under a suitable hypothesis on the symbol $\mathbf{H}(p)$, this new formulation is dissipative in the sense of a natural energy functional. We then establish the stability of numerical schemes built from this time-local formulation, thanks to the dissipation of appropriate discrete energies. Finally, the efficiency of these schemes is highlighted by concrete numerical results relating to a model recently proposed for 1D acoustic waves in porous media.

The gradient scheme framework is based on a small number of properties and encompasses a large number of numerical methods for diffusion models. We recall these properties and develop some new generic tools associated with the gradient scheme framework. These tools enable us to prove that classical schemes are indeed gradient schemes, and allow us to perform a complete and generic study of the well-known (but rarely well-studied) mass lumping process. They also allow an easy check of the mathematical properties of new schemes, by developing a generic process for eliminating unknowns via barycentric condensation, and by designing a concept of discrete functional analysis toolbox for schemes based on polytopal meshes., To appear in Mathematical Modelling and Numerical Analysis (M2AN), special edition on Polyhedral meshes

We study the weak approximation of the second-order backward SDEs (2BSDEs), when the continuous driving martingales are approximated by discrete time martingales. We establish a convergence result for a class of 2BSDEs, using both robustness properties of BSDEs, as proved in Briand, Delyon and M\'{e}min [Stochastic Process. Appl. 97 (2002) 229-253], and tightness of solutions to discrete time BSDEs. In particular, when the approximating martingales are given by some particular controlled Markov chains, we obtain several concrete numerical schemes for 2BSDEs, which we illustrate on specific examples., Comment: Published at http://dx.doi.org/10.1214/14-AAP1055 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In this paper, we propose a model of transmission of arboviruses, which take into account a future vaccination strategy in human population. A qualitative analysis based on stability and bifurcation theory reveals that the phenomenon of backward bifurcation may occur; the stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the associated reproduction number is less than unity. We show that the backward bifurcation phenomenon is caused by the arbovirus induced mortality in humans. Using theВ direct Lyapunov method, we show the global stability of the trivial equilibrium. Through global sensitivity analysis, wedetermine the relative importance of model parameters for disease transmission. Simulation results using a qualitatively stable numerical scheme, are provide to illustrate the impact of vaccination strategy in human community.

International audience; We propose a variational framework for the resolution of a non-hydrostatic Saint-Venant type model with bottom topography. This model is a shallow water type approximation of the incompressible Euler system with free surface and slightly differs from the Green-Nagdhi model, see [13] for more details about the model derivation. The numerical approximation relies on a prediction-correction type scheme initially introduced by Chorin-Temam [17] to treat the incompressibility in the Navier-Stokes equations. The hyperbolic part of the system is approximated using a kinetic finite volume solver and the correction step implies to solve a mixed problem where the velocity and the pressure are defined in compatible finite element spaces. The resolution of the incompressibility constraint leads to an elliptic problem involving the non-hydrostatic part of the pressure. This step uses a variational formulation of a shallow water version of the incompressibility condition.Several numerical experiments are performed to confirm the relevance of our approach.

8th Vienna International Conference on Mathematical Modelling, VIENNE, AUTRICHE, 18-/02/2015 - 20/02/2015; This paper is concerned with the macroscopic modeling and simulation of traffic flowon junctions. More precisely, we deal with a generic class of second order models, known in the literature as the GSOM family. While classical approaches focus on the Eulerian point-of-view, here we recast the model using its Lagrangian coordinates and we treat the junction as a specific discontinuity in Lagrangian framework. We propose a complete numerical methodology based on a finite difference scheme for solving such a model and we provide a numerical example.

We introduce a discrete time reflected scheme to solve doubly reflected Backward Stochastic Differential Equations with jumps (in short DRBSDEs), driven by a Brownian motion and an independent compensated Poisson process. As in Dumitrescu-Labart (2014), we approximate the Brownian motion and the Poisson process by two random walks, but contrary to this paper, we discretize directly the DRBSDE, without using a penalization step. This gives us a fully implementable scheme, which only depends on one parameter of approximation: the number of time steps $n$ (contrary to the scheme proposed in Dumitrescu-Labart (2014), which also depends on the penalization parameter). We prove the convergence of the scheme, and give some numerical examples., arXiv admin note: text overlap with arXiv:1406.3612

We consider numerical approximations of stochastic Langevin equations by implicit methods. We show a weak backward error analysis result in the sense that the generator associated with the numerical solution coincides with the solution of a modified Kolmogorov equation up to high order terms with respect to the stepsize. This implies that every measure of the numerical scheme is close to a modified invariant measure obtained by asymptotic expansion. Moreover, we prove that, up to negligible terms, the dynamic associated with the implicit scheme considered is exponentially mixing., Comment: arXiv admin note: substantial text overlap with arXiv:1105.0489 by other authors; and substantial text overlap with arXiv:1310.2404

A one-dimensional model problem for computation of the age field in ice sheets, which is of great importance for dating deep ice cores, is considered. the corresponding partial differential equation (PDE) is of purely advective (hyperbolic) type, which is notoriously difficult to solve numerically. By integrating the PDE over a space–time element in the sense of a finite-volume approach, a general difference equation is constructed from which a hierarchy of solution schemes can be derived. Iteration rules are given explicitly for central differences, first-, second- and third-order (QUICK) upstreaming as well as modified TVD Lax–Friedrichs schemes (TVDLFs). the performance of these schemes in terms of convergence and accuracy is discussed. Second-order upstreaming, themodifiedTVDLF scheme with Minmod slope limiter and, with limitations of the accuracy directly at the base, first-order upstreaming prove to be the most suitable for numerical age computations in ice-sheet models.

We give a probabilistic interpretation of the Monte Carlo scheme proposed by Fahim, Touzi and Warin [Ann. Appl. Probab. 21 (2011) 1322-1364] for fully nonlinear parabolic PDEs, and hence generalize it to the path-dependent (or non-Markovian) case for a general stochastic control problem. A general convergence result is obtained by a weak convergence method in the spirit of Kushner and Dupuis [Numerical Methods for Stochastic Control Problems in Continuous Time (1992) Springer]. We also get a rate of convergence using the invariance principle technique as in Dolinsky [Electron. J. Probab. 17 (2012) 1-5], which is better than that obtained by viscosity solution method. Finally, by approximating the conditional expectations arising in the numerical scheme with simulation-regression method, we obtain an implementable scheme., Comment: Published in at http://dx.doi.org/10.1214/13-AAP963 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We study steady boundary value problems of nonlinear kinetic theory. Using a continuation argument based on the variation of the Knudsen number we derive a method for the construction of steady solutions of discrete velocity models in a slab. This method is readily transformed into a numerical code. In a preliminary numerical test case the numerical scheme turns out to yield solutions even for Knudsen numbers small enough to calculate with high precision the asymptotic flow field adjacent to a kinetic boundary layer. Thus, we are able to numerically simulate in a simplified situation the transition from a (mesoscopic) kinetic boundary layer to the (macroscopic) far field.

Recently in [3], a new method has been presented providing a family of artificial boundary conditions for parabolic equations which are local in time and stable. This method applies to variable coefficients and curved boundaries but unfortunately there are no error estimates. As an alternative, we present in this paper numerical simulations showing the efficiency of the method. The construction of the artificial boundary conditions depends on the number and values of points at which z 1 2 is interpolated by rational fractions in the complex plane. Particular attention is paid to the choice of those points.

International audience; In this paper, we propose a discretization for the compressible Stokes problem with an equation of state of the form p = ϕ(ρ) (where p stands for the pressure, ρ for the density and ϕ is a superlinear nondecreasing function from R to R). This scheme is based on Crouzeix-Raviart approximation spaces. The discretization of the momentum balance is obtained by the usual finite element technique. The discrete mass balance is obtained by a finite volume scheme, with an upwinding of the density, and two additional terms. We prove the existence of a discrete solution and the convergence of this approximate solution to a solution of the continuous problem.

International audience; We consider numerical approximations of stochastic differential equations by the Euler method. In the case where the SDE is elliptic or hypoelliptic, we show a weak backward error analysis result in the sense that the generator associated with the numerical solution coincides with the solution of a modified Kolmogorov equation up to high order terms with respect to the stepsize. This implies that every invariant measure of the numerical scheme is close to a modified invariant measure obtained by asymptotic expansion. Moreover, we prove that, up to negligible terms, the dynamic associated with the Euler scheme is exponentially mixing.

Markets liquidity is an issue of very high concern in financial risk management. In a perfect liquid market the option pricing model becomes the well-known linear Black-Scholes problem. Nonlinear models appear when transaction costs or illiquid market effects are taken into account. This paper deals with the numerical analysis of nonlinear Black-Scholes equations modeling illiquid markets when price impact in the underlying asset market affects the replication of a European contingent claim. Numerical analysis of a nonlinear model is necessary because disregarded computations may waste a good mathematical model. In this paper we propose a finite-difference numerical scheme that guarantees positivity of the solution as well as stability and consistency. © 2011 IMACS. Published by Elsevier B.V. All rights reserved., This paper has been supported by the Spanish Department of Science and Education grant TRA2007-68006-C02-02 and the Generalitat Valenciana grant GVPRE/20081092.

to appear on SIAM J. Appl. Math.; International audience; An extension of the Colombo phase transition model is proposed. The congestion phase is described by a two-dimensional zone defined around an equilibrium flux known as the classical fundamental diagram. General criteria to build such a set-valued fundamental diagram are enumerated, and instantiated on several equilibrium fluxes with different concavity properties. The solution of the Riemann problem in the presence of phase transitions is obtained through the construction of a Riemann solver, which enables the definition of the solution of the Cauchy problem using wavefront tracking. The free-flow phase is described using a Newell-Daganzo fundamental diagram, which allows for a more tractable definition of phase transition compared to the original Colombo phase transition model. The accuracy of the numerical solution obtained by a modified Godunov scheme is assessed on benchmark scenarios for the different flux functions constructed.

A method of approximate solution of the Riemann problem on the basis of characteristic relations that can be used in numerical schemes of the Godunov method for integration of equations defining a one-velocity flow of a multicomponent mixture in the adiabatic approximation is presented. © 2010 Springer Science+Business Media, Inc.