Nessyahu Tadmor Research Articles

A space time conservation element and solution element method for solving two-species chemotaxis model

This work is related to the numerical investigation of two species chemotaxis models in both one and two dimensions. This system in its simpler form is a set of non-linear partial differential equations. First equation represents the dynamics of cell densities and the others are responsible for chemoattractant concentration. The system is known to produce delta-type singularities in finite time. Therefore, a less order accurate numerical schemes are not capable of handling such a complicated spiky solution. A Space Time Conservation Element and Solution Element (CE/SE) method is proposed for solving such systems. This method has distinct attributes which includes treatment of space and time variables in unified fashion. Furthermore, numerical diffusion is reduced on the basis that both conserved quantities and derivatives are anonymous. Due to this feature, diffusion in numerical schemes is inherently reduced. The Nessyahu–Tadmor (NT) central scheme is also implemented for validation and comparison. Several one and two dimensional case studies are carried out. Numerical results obtained through proposed numerical method are analyzed in comparison to Nessyahu–Tadmor (NT) central scheme. Moreover, effects of the different densities and concentration functions are explored to see generic applicability of the proposed method for current system of equations. It is observed that the CE/SE and NT central method has the capability to capture the delta type singularities in the solution profile. However, CE/SE resolves the solution peaks better as compare to NT central scheme.

Space-time CE/SE method for solving repulsive chemotaxis model

This work is concerned with the numerical investigation of nonlinear mathematical systems that describe repulsive chemotaxis phenomena in biology. A space-time conservation element (CE) and solution element (SE) method-based splitting procedure is proposed in one space dimension. In chemotaxis, the orientation of the cells changes. They move to the location which are chemically more favorable or move away from repellents or toxin. Mathematically, these are the systems of nonlinear coupled partial differential equations. The analytical solutions for this set of equations is not possible. A traditional less order accurate solution may fail to describe the underlying phenomena. Therefore, the state of the art numerical procedures is the need for such systems to get physically reliable solution in acceptable computational cost. Unlike tradition numerical procedures, the CE/SE-method has distinct attributes like it treats time and space in a unified fashion. Both conserved quantities and corresponding derivatives are considered to be unknowns due to which inherited numerical diffusion is reduced. Several benchmark numerical test problems are simulated for validation of the scheme. The numerical solutions of case studies are obtained for one space dimension. Moreover, one more central numerical scheme introduced by Nessyahu–Tadmor (NT) is also adapted on staggard grids which is considered to be a black box solver for such models for comparison. It was observed that both schemes perform well for the current mathematical model. However, the CE/SE scheme is more capable of capturing the peaks produced in the solution profile. Further, convergence study is also carried out from both schemes which reveal that the proposed method is fast as compared to NT central scheme.

Accuracy of MUSCL-Type Schemes in Shock Wave Calculations

The central-difference Nessyahu–Tadmor (NT) scheme is considered, which is built using second-order MUSCL reconstruction of fluxes. The accuracy of the NT scheme is studied as applied to calculating shock waves propagating with a variable velocity. It is shown that this scheme has the first order of integral convergence on intervals with one of the boundaries lying in the region of influence of the shock wave. As a result, the local accuracy of the NT scheme is significantly reduced in these areas. Test calculations are presented that demonstrate these properties of the NT scheme.

Discontinuous Galerkin method for solving incompressible two-phase shallow granular flow model

This article deals with the numerical study of two-phase shallow flow model describing the mixture of fluid and solid granular particles. The model under investigation consists of coupled mass and momentum equations for solid granular material and fluid particles through non-conservative momentum exchange terms. The non-conservativity of model equations poses major challenges for any numerical scheme, such as well balancing, positivity preservation, accurate approximation of non-conservative terms, and achievement of steady-state conditions. Thus, in order to approximate the present model an accurate, well-balanced, robust, and efficient numerical scheme is required. For this purpose, in this article, Runge–Kutta discontinuous Galerkin method is applied successfully for the first time to solve the model equations. Several test problems are also carried out to check the performance and accuracy of our proposed numerical method. To compare the results, the same model is solved by staggered central Nessyahu–Tadmor scheme. A good comparison is found between two schemes, but the results obtained by Runge–Kutta discontinuous Galerkin scheme are found superior over the central Nessyahu–Tadmor scheme.

Investigation of Fluid-structure Interaction by Explicit Central Finite Difference Methods

Fluid-structure interaction (FSI) occurs when the dynamic water hammer forces; cause vibrations in the pipe wall. FSI in pipe systems due to Poisson and junction coupling has been the center of attention in recent years. It causes fluctuations in pressure heads and vibrations in the pipe wall. The governing equations of this phenomenon include a system of first order hyperbolic partial differential equations (PDEs) in terms of hydraulic and structural quantities. In the present paper, a two-step variant of the Lax-Friedrichs (LxF) method, and a method based on the Nessyahu-Tadmor (NT) are used to simulate FSI in a reservoir-pipe-valve system. The computational results are compared with those of the Method of Characteristics (MOC), Godunov's scheme and also with the exact solution of linear hyperbolic four-equation system to verify the proposed numerical solution. The results reveal that the proposed LxF and NT schemes can predict discontinuous in fluid pressure with an acceptable order of accuracy. The independency of time and space steps allows for setting different spatial grid size with a unique time step, thus increasing the accuracy with respect to the conventional MOC. In these schemes, no Riemann problems were solved and hence field-by-field decompositions were avoided which led to reduced run times compared with Godunov scheme.

A splitting scheme based on the space–time CE/SE method for solving multi-dimensional hydrodynamical models of semiconductor devices

Numerical solutions of the hydrodynamical model of semiconductor devices are presented in one and two-space dimension. The model describes the charge transport in semiconductor devices. Mathematically, the models can be written as a convection–diffusion type system with a right hand side describing the relaxation effects and interaction with a self consistent electric field. The proposed numerical scheme is a splitting scheme based on the conservation element and solution element (CE/SE) method for hyperbolic step, and a semi-implicit scheme for the relaxation step. The numerical results of the suggested scheme are compared with the splitting scheme based on Nessyahu–Tadmor (NT) central scheme for convection step and the same semi-implicit scheme for the relaxation step. The effects of various parameters such as low field mobility, device length, lattice temperature and voltages for one-space dimensional hydrodynamic model are explored to further validate the generic applicability of the CE/SE method for the current model equations. A two dimensional simulation is also performed by CE/SE method for a MESFET device, producing results in good agreement with those obtained by NT-central scheme.

An Application of 3-D Kinematical Conservation Laws: Propagation of a 3-D Wavefront

Three-dimensional (3-D) kinematical conservation laws (KCL) are equations of evolution of a propagating surface $\Omega_t$ in three space dimensions. We start with a brief review of the 3-D KCL system and mention some of its properties relevant to this paper. The 3-D KCL, a system of six conservation laws, is an underdetermined system to which we add an energy transport equation for a small amplitude 3-D nonlinear wavefront propagating in a polytropic gas in a uniform state and at rest. We call the enlarged system of 3-D KCL with the energy transport equation equations of weakly nonlinear ray theory (WNLRT). We highlight some interesting properties of the eigenstructure of the equations of WNLRT, but the main aim of this paper is to test the numerical efficacy of this system of seven conservation laws. We take several initial shapes for a nonlinear wavefront with a suitable amplitude distribution on it and let it evolve according to the 3-D WNLRT. The 3-D WNLRT is a weakly hyperbolic $7\times7$ system that is highly nonlinear. Here we use the staggered Lax–Friedrichs and Nessyahu–Tadmor central schemes and have obtained some very interesting shapes of the wavefronts. We find the 3-D KCL to be suitable for solving many complex problems for which there presently seems to be no other method capable of giving such physically realistic features.

FORCE schemes on unstructured meshes I: Conservative hyperbolic systems

This paper is about the construction of numerical fluxes of the centred type for one-step schemes in conservative form for solving general systems of conservation laws in multiple space dimensions on structured and unstructured meshes. The work is a multi-dimensional extension of the one-dimensional FORCE flux and is closely related to the work of Nessyahu–Tadmor and Arminjon. The resulting basic flux is first-order accurate and monotone; it is then extended to arbitrary order of accuracy in space and time on unstructured meshes in the framework of finite volume and discontinuous Galerkin methods. The performance of the schemes is assessed on a suite of test problems for the multi-dimensional Euler and Magnetohydrodynamics equations on unstructured meshes.

Three-Dimensional Adaptive Central Schemes on Unstructured Staggered Grids

We present an explicit second-order finite volume generalization of the one-dimensional (1D) Nessyahu–Tadmor schemes for hyperbolic equations on adaptive unstructured tetrahedral grids. The nonoscillatory central difference scheme of Nessyahu and Tadmor, in which the resolution of the Riemann problem at the cell interfaces is bypassed thanks to the use of the staggered Lax–Friedrichs scheme, is extended here to a two-steps scheme. In order to reduce artificial viscosity, we start with an adaptively refined primal grid in three dimensions (3D), where the theoretical a posteriori result of the first-order scheme is used to derive appropriate refinement indicators. We apply those methods to solve Euler's equations. Numerical experimental tests on classical problems are obtained by our method and by the computational fluid dynamics software Fluent. These tests include results for the 3D Euler system (shock tube problem) and flow around an NACA0012 airfoil.

One-sided stability and convergence of the Nessyahu–Tadmor scheme

Non-oscillatory schemes are widely used in numerical approximations of nonlinear conservation laws. The Nessyahu–Tadmor (NT) scheme is an example of a second order scheme that is both robust and simple. In this paper, we prove a new stability property of the NT scheme based on the standard minmod reconstruction in the case of a scalar strictly convex conservation law. This property is similar to the One-sided Lipschitz condition for first order schemes. Using this new stability, we derive the convergence of the NT scheme to the exact entropy solution without imposing any nonhomogeneous limitations on the method. We also derive an error estimate for monotone initial data.

Order of convergence of second order schemes based on the minmod limiter

Many second order accurate nonoscillatory schemes are based on the minmod limiter, e.g., the Nessyahu–Tadmor scheme. It is well known that the L p L_p -error of monotone finite difference methods for the linear advection equation is of order 1 / 2 1/2 for initial data in W 1 ( L p ) W^1(L_p) , 1 ≤ p ≤ ∞ 1\leq p\leq \infty . For second or higher order nonoscillatory schemes very little is known because they are nonlinear even for the simple advection equation. In this paper, in the case of a linear advection equation with monotone initial data, it is shown that the order of the L 2 L_2 -error for a class of second order schemes based on the minmod limiter is of order at least 5 / 8 5/8 in contrast to the 1 / 2 1/2 order for any formally first order scheme.

Central finite volume schemes with constrained transport divergence treatment for three-dimensional ideal MHD

We present three-dimensional central finite volume methods for solving systems of hyperbolic equations. Based on the Lax–Friedrichs and Nessyahu–Tadmor one-dimensional central finite difference schemes, the numerical methods we propose involve an original and a staggered grid in order to avoid the resolution of the Riemann problems at the cell interfaces. The cells of the original grid are Cartesian (cubes with faces parallel to the axes) while those of the staggered grid are either Cartesian or diamond-shaped. We apply these methods and solve some ideal magnetohydrodynamics problems. To satisfy the solenoidal property of the magnetic field in the numerical solution, we present an adaptation of Evans and Hawley’s constrained transport method for central schemes which we call “CTCS”. The CTCS method is easy to implement, it deals directly with the computed solution and does not require any additional staggering for the magnetic field components; furthermore, it preserves the second-order accuracy of the base scheme. Even without the application of the CTCS procedure, our numerical base schemes do not break down, and may even in some cases deliver reasonable results. The diamond dual cell scheme has a slight advantage for shocks and contact discontinuities. Our numerical results are in good agreement with corresponding results appearing in the recent literature.

An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws

We discuss the occurrence of oscillations when using central schemes of the Lax-Friedrichs type (LFt), Rusanov's method and the staggered and non-staggered second order Nessyahu-Tadmor (NT) schemes. Although these schemes are monotone or TVD, respectively, oscillations may be in- troduced at local data extrema. The dependence of oscillatory properties on the numerical viscosity coefficient is investigated rigorously for the LFt schemes, illuminating also the properties of Rusanov's method. It turns out, that schemes with a large viscosity coefficient are prone to oscillations at data extrema. For all LFt schemes except for the classical Lax-Friedrichs method, occurring oscillations are damped in the course of a computation. This damping effect also holds for Rusanov's method. Concerning the NT schemes, the non-staggered version may yield oscillatory results, while it can be shown rigorously that the staggered NT scheme does not produce oscillations when using the classical minmod-limiter under a restriction on the time step size. Note that this restriction is not the same as the condition ensuring the TVD property. Numerical investigations of one-dimensional scalar prob- lems and of the system of shallow water equations in two dimensions with respect to the phenomenon complete the paper.

Numerical integration of the plasma fluid equations with a modification of the second-order Nessyahu–Tadmor central scheme and soliton modeling

Here we outline a modification of the second order central difference scheme based on staggered spatial grids due to Nessyahu and Tadmor [H. Nessyahu, E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys. 87 (1990) 408] to a non-staggered scheme for one-dimensional hyperbolic systems which can additionally include source terms. With this modification we integrate the one-dimensional electrostatic plasma fluid-Poisson equations to illustrate ion-acoustic soliton formation and propagation. This application is interesting because, to our knowledge, it is the first time that a high-resolution scheme has been employed on the plasma fluid equations, where in particular, we test its ability to handle a coupled fluid-Poisson system and also, we examine its performance on very long time integrations involving thousands of time steps. As a check on the accuracy of the modified scheme we perform tests on a shock capturing problem in a Broadwell gas, and in both cases, the results obtained are compared with those from previously reported schemes.

Central finite volume methods with constrained transport divergence treatment for ideal MHD

Two and three-dimensional finite volume extensions of the Lax–Friedrichs (LF) and Nessyahu–Tadmor one-dimensional difference schemes were previously presented and successfully applied to several problems for nonlinear hyperbolic systems, and in particular to typical test cases for both inviscid and viscous compressible flows. These “central” schemes by-pass the resolution, at the cell interfaces, of the Riemann problems, thanks to the use of the staggered LF scheme which serves as the base scheme on which high order finite volume methods can be constructed using van Leer’s MUSCL-type limited reconstruction principle. For this purpose, two dual grids are used at alternate time steps. These methods are extended here to several problems in one- and multi-dimensional ideal compressible magnetohydrodynamics using a modified version of the first author’s central methods with oblique (diamond shaped) dual cells. In two-dimensions the system has eight equations and solving the corresponding Riemann problem is an elaborate and time-consuming process. Central methods lead to significant computing time reductions, and the numerical experiments presented here suggest the accuracy is quite satisfactory. In order to satisfy the physical constraint ∇ · B = 0, we have constructed a strategy (“CTCS”) inspired from the Constrained Transport method of Evans and Hawley. The validity of our base scheme and our CTCS approach is clearly confirmed by the results.

Improved non‐staggered central NT schemes for balance laws with geometrical source terms

AbstractIn 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.

Nessyahu–Tadmor-type central finite volume methods without predictor for 3D Cartesian and unstructured tetrahedral grids

A new modified version of the Nessyahu–Tadmor (NT) 1-dimensional finite volume central scheme is presented, as well as corresponding new versions for 2D-structured and 3D-unstructured grids inspired from the NT-scheme. The modification avoids the intermediate predictor time step between t n and t n+1 . Although this does not really bring about substantial accuracy/computer time improvements in the 1D case, in the 2- and 3-dimensional cases, the modified scheme does lead to important reductions in computer times. 3D comparative simulations for the shock tube problem and for a supersonic inviscid flow through a channel with a 4% circular bump are presented.

Computation of travelling wave solutions of scalar conservation laws with a stiff source term

In this paper we propose a nonoscillatory numerical technique to compute the travelling wave solution of scalar conservation laws with a stiff source term. This procedure is based on the dynamical behavior described by the associated stationary ODE and it reduces/avoids numerical errors usually encountered with these problems, i.e., spurious oscillations and incorrect wave propagation speed. We combine this treatment with either the first order Lax–Friedrichs scheme or the second order Nessyahu–Tadmor scheme. We have tested several model problems by LeVeque and Yee for which the stiffness coefficient can be increased. We have also tested a problem with a nonlinear flux and a discontinuous source term.

New two- and three-dimensional non-oscillatory central finite volume methods on staggered Cartesian grids

We present a 3D finite volume generalization of the 1-dimensional Lax–Friedrichs and Nessyahu–Tadmor schemes for hyperbolic equations on Cartesian grids. The non-oscillatory central difference scheme of Nessyahu and Tadmor, in which the resolution of the Riemann problem at the cell interfaces is by-passed thanks to the use of the staggered Lax–Friedrichs scheme, is extended here to a two-step, 3-dimensional non-oscillatory centered scheme in finite volume formulation. Piecewise linear cell interpolants using several van Leer-type limiting techniques to estimate the gradient (van Leer, van Albada, SuperBee, MinMod) lead to a non-oscillatory spatial resolution of order superior to 1. The fact that the expected second-order resolution is not fully attained in 3D is investigated first by considering an alternate dual grid (in 2D), and by using the original van Albada limiter in 3D. Numerical results for a linear advection problem with continuous and discontinuous initial conditions in 2D and 3D show the accuracy and stability of the method. A comparison is made between the 2D Arminjon–Stanescu–Viallon and Jiang–Tadmor formulations and the new one. A new simple projection method is used for the gradients in the new 2D scheme. We also include results for the 3D Euler system (channel with a forward facing step).

NUMERICAL SOLUTION FOR HYDRODYNAMICAL MODELS OF SEMICONDUCTORS

Numerical solutions of recent hydrodynamical models of semiconductors are computed in one-space dimension. Such models describe charge transport in semiconductor devices. Two models are taken into consideration. The first one has been developed by Blotekjaer, Baccarani et al., and the second one by Anile et al. In both cases the system of equations can be written as a convection-diffusion type system, with a right-hand side describing relaxation effects and interaction with a self-consistent electric field. The numerical scheme is a splitting scheme based on the Nessyahu–Tadmor scheme for the hyperbolic step, and a semi-implicit scheme for the relaxation step. The numerical results are compared to detailed Monte-Carlo simulation.