An exact Riemann solver and a Godunov scheme for simulating highly transient mixed flows

The current research aims at deriving a one-dimensional numerical model for describing highly transient mixed flows. In particular, this paper focuses on the development and assessment of a unified numerical scheme adapted to describe free-surface flow, pressurized flow and mixed flow (characterized by the simultaneous occurrence of free-surface and pressurized flows). The methodology includes three steps. First, the authors derived a unified mathematical model based on the Preissmann slot model. Second, a first-order explicit finite volume Godunov-type scheme is used to solve the set of equations. Third, the numerical model is assessed by comparison with analytical, experimental and numerical results. The key results of the paper are the development of an original negative Preissmann slot for simulating sub-atmospheric pressurized flow and the derivation of an exact Riemann solver for the Saint-Venant equations coupled with the Preissmann slot.     

The Modeling Dynamics of HIV and CD4$$^{}$$ T-cells During Primary Infection in Fractional Order: Numerical Simulation

This article is devoted to introduce a numerical treatment using Adams–Bashforth–Moulton method of the fractional model of HIV-1 infection of CD4\(^{+}\) T-cells. We study the effect of the changing the average number of viral particles N with different sets of initial conditions on the dynamics of the presented model. The fractional derivative is described in Caputo sense. Special attention is given to present the local stability of the proposed model using fractional Routh–Hurwitz stability criterion. Qualitative results show that the model has two equilibria: the disease-free equilibrium and the endemic equilibrium points. We compare our numerical solutions with those numerical solutions using fourth-order Runge–Kutta method (RK4). The obtained numerical results of the proposed model show the simplicity and the efficiency of the proposed method.     

Numerical predictions of three-dimensional flow in a ventilated room using turbulence models

A three-dimensional recirculation flow in a ventilated room was predicted by the numerical methods in which the turbulence models are applied. The predicted results are compared with the experimental results obtained in a model room in order to estimate the practical utilities of such methods from the viewpoint of engineering. Taking account of the practicability of prediction method which the engineers regard as important, two turbulence models were selected and they were incorporated into the numerical prediction methods respectively. One is the two-equation model, in which transport equations of turbulence energy and its rate of dissipation are adopted. The other is the Deardoff's model, in which the subgrid scale eddy coefficient is utilized. The prediction was made by each numerical method. Consequently, no noticeable difference is recognized between both predicted results. Each result is compared with the experimental results. Generally speaking, each agreement is good with regard to the mean velocity. Thus we can conclude that the numerical method using the two-equation model has more practical utility than that using Deardoff's model, because it can give the solutions in a shorter computer time.     

A boundary preserving numerical algorithm for the Wright-Fisher model with mutation

The Wright-Fisher model is an It? stochastic differential equation that was originally introduced to model genetic drift within finite populations and has recently been used as an approximation to ion channel dynamics within cardiac and neuronal cells. While analytic solutions to this equation remain within the interval [0,1], current numerical methods are unable to preserve such boundaries in the approximation. We present a new numerical method that guarantees approximations to a form of Wright-Fisher model, which includes mutation, remain within [0,1] for all time with probability one. Strong convergence of the method is proved and numerical experiments suggest that this new scheme converges with strong order 1/2. Extending this method to a multidimensional case, numerical tests suggest that the algorithm still converges strongly with order 1/2. Finally, numerical solutions obtained using this new method are compared to those obtained using the Euler-Maruyama method where the Wiener increment is resampled to ensure solutions remain within [0,1].     

Numerical approximation of a quantum drift-diffusion model

This Note is devoted to the discretization and numerical simulation of a new quantum drift-diffusion model that was recently derived. We define an implicit numerical scheme which is equivalent to a convex minimization problem and which preserves the physical properties of the continuous model: charge conservation, positivity of the density and dissipation of an entropy. We illustrate these results by some numerical simulations. To cite this article: S. Gallego, F. Méhats, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Probability density function of leaky integrate-and-fire model with Lévy noise and its numerical approximation

We investigate the numerical analysis of leaky integrate-and-fire model with Lévy noise. We consider a neuronal model in which probability density function of a neuron in some potential at any time is modeled by a transport equation. Lévy noise is included due to jumps by excitatory and inhibitory impulses. Due to these jumps the resulting equation is a transport equation containing two integral in right-hand side (jumps). We design, implement, and analyze numerical methods of finite volume type. Some numerical examples are also included.     

Constitutive modeling and FEM for a nonlinear Cosserat continuum

The choice of non-standard material parameters in the presented micropolar continuum model leads to a highly nonlinear problem which calls also for a nonlinear numerical treatment. The model benefits in describing length scale effects as really non-linear effects. The numerical torsion test is one example for the possibilities of our model. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new model for gas flow in pipe networks

We introduce a new model for gas dynamics in pipe networks by asymptotic analysis. The model is derived from the isothermal Euler equations. We present the derivation of the model as well as numerical results illustrating the validity and its properties. We compare the new model with existing models from the mathematical and engineering literature. We further give numerical results on a sample network. Copyright © 2009 John Wiley & Sons, Ltd.     

Finite-difference methods for solving the reaction-diffusion equations of a simple isothermal chemical system

Numerical methods are proposed for the numerical solution of a system of reaction-diffusion equations, which model chemical wave propagation. The reaction terms in this system of partial differential equations contain nonlinear expressions. Nevertheless, it is seen that the numerical solution is obtained by solving a linear algebraic system at each time step, as opposed to solving a nonlinear algebraic system, which is often required when integrating nonlinear partial differential equations. The development of each numerical method is made in the light of experience gained in solving the system of ordinary differential equations, which model the well-stirred analogue of the chemical system. The first-order numerical methods proposed for the solution of this initialvalue problem are characterized to be implicit. However, in each case it is seen that the numerical solution is obtained explicitly. In a series of numerical experiments, in which the ordinary differential equations are solved first of all, it is seen that the proposed methods have superior stability properties to those of the well-known, first-order, Euler method to which they are compared. Incorporating the proposed methods into the numerical solution of the partial differential equations is seen to lead to two economical and reliable methods, one sequential and one parallel, for solving the travelling-wave problem. © 1994 John Wiley & Sons, Inc.     

Approximations of a Ginzburg-Landau model for superconducting hollow spheres based on spherical centroidal Voronoi tessellations

In this paper the numerical approximations of the Ginzburg- Landau model for a superconducting hollow spheres are constructed using a gauge invariant discretization on spherical centroidal Voronoi tessellations. A reduced model equation is used on the surface of the sphere which is valid in the thin spherical shell limit. We present the numerical algorithms and their theoretical convergence as well as interesting numerical results on the vortex configurations. Properties of the spherical centroidal Voronoi tessellations are also utilized to provide a high resolution scheme for computing the supercurrent and the induced magnetic field.

     

Local discontinuous Galerkin method for solving an N‐carrier system

In this article, we present local discontinuous Galerkin (LDG) method for solving a model of energy exchanges in an N ‐carrier system with Neumann boundary conditions. This model extends the concept of the well‐known parabolic two‐step model for microheat transfer to the energy exchanges in a generalized N ‐carrier system with heat sources. The energy norm stability and error estimate of the LDG method is proved for solving N ‐carrier system. Some numerical examples are given. The numerical results when compared with the exact solution and other numerical results indicate that the present method is seen to be a very good alternative to some existing techniques for realistic problems. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Solutions of two-factor models with variable interest rates

The focus of this work is on numerical solutions to two-factor option pricing partial differential equations with variable interest rates. Two interest rate models, the Vasicek model and the Cox–Ingersoll–Ross model (CIR), are considered. Emphasis is placed on the definition and implementation of boundary conditions for different portfolio models, and on appropriate truncation of the computational domain. An exact solution to the Vasicek model and an exact solution for the price of bonds convertible to stock at expiration under a stochastic interest rate are derived. The exact solutions are used to evaluate the accuracy of the numerical simulation schemes. For the numerical simulations the pricing solution is analyzed as the market completeness decreases from the ideal complete level to one with higher volatility of the interest rate and a slower mean-reverting environment. Simulations indicate that the CIR model yields more reasonable results than the Vasicek model in a less complete market.     

A sigma coordinate 3D k– model for turbulent free surface flow over a submerged structure

A fully three-dimensional unsteady flow model is developed to simulate free surface flow over a submerged structure. A new sigma coordinate is used to map the physical domain containing the wavy free surface and uneven bottom to a rectangular prism, and to keep the size of the submerged block unchanged in the sigma coordinate system. The numerical difficulty encountered in the conventional sigma coordinate system in which the block changes dynamically due to the time varying free surface is thus eliminated. A split operator scheme is used in the numerical solution so that different numerical schemes can be purposely chosen to deal with the distinctive mathematical and physical characteristics of the phenomena at different steps. k– model is used in the parameterization of turbulence due to its efficiency and reasonable performance. The model is applied to simulate the propagation of a solitary wave with good results. It is subsequently used to simulate a free surface flow against a submerged cube with one face perpendicular (or 45° inclined) to the flow. The numerical results compare favorably with the experimental measurements. In particular, no excessive turbulent kinetic energy is accumulated at the impingement regions.     

Inverse problems for a mathematical model of redox sorption

We consider a mathematical model of sorption that allows for external diffusion kinetics and a redox reaction. Two inverse problems are considered for this model, uniqueness is proved, and numerical solution methods are proposed. The efficiency of the numerical methods is investigated by computer experiments. __________ Translated from Prikladnaya Matematika i Informatika, No. 23, pp. 15–23, 2006.     

Oscillation analysis of θ‐methods for the Nicholson's blowflies model

This paper is concerned with the oscillation of numerical solution for the Nicholson's blowflies model. Using two kinds of θ‐methods, namely, the linear θ‐method and the one‐leg θ‐method, several conditions under which the numerical solution oscillates are derived. Moreover, it is shown that every non‐oscillatory numerical solution tends to equilibrium point of the original continuous‐time model. Finally, numerical experiments are provided to illustrate the analytical results. Copyright © 2015 John Wiley & Sons, Ltd.     

A staggered-grid numerical algorithm for the extended Boussinesq equations

A second-order accurate numerical scheme is developed to solve Nwogu’s extended Boussinesq equations. A staggered-grid system is introduced with the first-order spatial derivatives being discretized by the fourth-order accurate finite-difference scheme. For the time derivatives, the fourth-order accurate Adams predictor–corrector method is used. The numerical method is validated against available analytical solutions, other numerical results of Navier–Stokes equations, and experimental data for both 1D and 2D nonlinear wave transformation problems. It is shown that the new algorithm has very good conservative characteristics for mass calculation. As a result, the model can provide accurate and stable results for long-term simulation. The model has proven to be a useful modeling tool for a wide range of water wave problems.     

An improved Serre model: Efficient simulation and comparative evaluation

The so-called Serre or Green and Naghdi equations are a well-known set of fully nonlinear and weakly dispersive equations that describe the propagation of long surface waves in shallow water. In order to extend its range of application to intermediate water depths, some modifications have been proposed in the literature. In this work, we analyze a new Serre model with improved linear dispersion characteristics. This new Serre system, herein denoted by Serre_{α, β}, presents additional terms of dispersive origin, thus extending its applicability to more general depth to wavelength ratios.A careful development of the Serre_{α, β} model allows a straightforward and efficient numerical implementation. This model is suitable for numerical integration by a splitting strategy which requires the solution of a hyperbolic problem and a dispersive problem. The hyperbolic part is discretized using a high-order finite volume method. For the dispersive part standard finite differences are used. A set of numerical experiments are conducted to validate the Serre_{α, β} model and to test the robustness of our numerical scheme. Theoretical solutions and benchmark experimental data are used. Moreover, comparisons against the classical Serre equations and against another well established Serre model with improved dispersion characteristics are also made.     

Topological structure preserving numerical simulations of dynamical models

This paper brings together two methods producing numerical solutions with a statement of their quality — the nonstandard finite difference method and the method of validated computing. It deals with the construction and the analysis of reliable numerical discretizations of dynamical systems by employing these two techniques. An epidemiological model is used as a model example for their combined application.     

Numerical simulation of oblique and multidirectional wave propagation and breaking on steep slope based on FEM model of Boussinesq equations

Creating a representative numerical simulation of the propagation and breaking of waves along slopes is an important problem in engineering design. Most studies on wave breaking have focused on the propagation of normal incident waves on gentle slopes. In practice, however, waves on steep slopes are obliquely incident or multidirectional irregular waves. In this paper, the eddy viscosity term is introduced to the momentum equation of the improved Boussinesq equations to model wave dissipation caused by breaking and friction, and a numerical model based on an unstructured finite element method (FEM) is established based on the governing equations. It is applied to simulate wave propagation on a steep slope of 1:5. Parallel physical experiments are conducted for comparative analysis that considered a large number of cases, including those featuring of normal and oblique incident regular and irregular waves, and multidirectional waves. The heights of the incident wave increase for different periods to represent different kinds of waves breaking. Based on examination, the effectiveness and accuracy of the numerical model is verified through a comprehensive comparison between the numerical and the experimental results, including in terms of variation in wave height, wave spectrum, and nonlinear parameters. Satisfactory agreement between the numerical and experimental values shows that the proposed model is effective in representing the breaking of oblique incident regular waves, irregular waves, and multidirectional incident irregular waves. However, the initial threshold of the breaking parameter η_t^(I) takes different values for oblique and multidirectional waves. This needs to be paid attention when the breaking of waves is simulated using the Boussinesq equations.