We introduce new Godunov-type semidiscrete central schemes for hyperbolic systems of conservation laws and Hamilton--Jacobi equations. The are based on the use more precise information about local speeds propagation can be viewed as a generalization from [A. Kurganov E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241--282; A. D. Levy, SIAM Sci. Comput., 22 1461--1488; G. Petrova, A third-order genuinely multidimensional scheme related problems, Numer. Math., to appear] 720--742]. main...

A family of Godunov-type central-upwind schemes for the Saint-Venant system shallow water equations has been first introduced in [A.Kurganov and D. Levy, M2AN Math.Model.Numer.Anal., 36, 397-425, 2002].Depending on reconstruction step, second-order versions there could be made either well-balanced or positivity preserving, but fail to satisfy both properties simultaneously.Here, we introduce an improved scheme which, unlike its forerunners, is capable preserve stationary steady states (lake...

Abstract We report here on our numerical study of the two‐dimensional Riemann problem for compressible Euler equations. Compared with relatively simple 1‐D configurations, 2‐D case consists a plethora geometric wave patterns that pose computational challenge high‐resolution methods. The main feature in present computations these waves is use Riemann‐solvers‐free central schemes presented by Kurganov et al. This family avoids intricate and time‐consuming computation eigensystem hence offers...

We present one- and two-dimensional central-upwind schemes for approximating solutions of the Saint-Venant system with source terms due to bottom topography. The has steady-state in which nonzero flux gradients are exactly balanced by terms. It is a challenging problem preserve this delicate balance numerical schemes. Small perturbations these states also very difficult compute. Our approach based on extending semi-discrete central systems hyperbolic conservation laws laws. Special attention...

We present a new third-order, semidiscrete, central method for approximating solutions to multidimensional systems of hyperbolic conservation laws, convection-diffusion equations, and related problems. Our is high-order extension the recently proposed second-order, semidiscrete in [A. Kurgonov E. Tadmor, J. Comput Phys., 160 (2000) pp. 241--282]. The derived independently specific piecewise polynomial reconstruction which based on previously computed cell-averages. demonstrate our results by...

Abstract Aquatic bacteria like Bacillus subtilis are heavier than water yet they able to swim up an oxygen gradient and concentrate in a layer below the surface, which will undergo Rayleigh–Taylor-type instabilities for sufficiently high concentrations. In literature, simplified chemotaxis–fluid system has been proposed as model bio-convection modestly diluted cell suspensions. It couples convective chemotaxis oxygen-consuming oxytactic with incompressible Navier–Stokes equations subject...

Shallow-water equations are widely used to model water flow in rivers, lakes, reservoirs, coastal areas, and other situations which the depth is much smaller than horizontal length scale of motion. The classical shallow-water equations, Saint-Venant system, were originally proposed about 150 years ago still a variety applications. For many practical purposes, it extremely important have an accurate, efficient robust numerical solver for system related models. As their solutions typically...

We discover that the choice of a piecewise polynomial reconstruction is crucial in computing solutions nonconvex hyperbolic (systems of) conservation laws. Using semidiscrete central-upwind schemes, we illustrate obtained numerical approximations may fail to converge unique entropy solution or convergence be so slow achieving proper resolution would require use (almost) impractically fine meshes. For example, scalar case, all computed seem are for some pairs. However, most applications, one...

We derive a second-order semidiscrete central-upwind scheme for one- and two-dimensional systems of two-layer shallow water equations. prove that the presented is well-balanced in sense stationary steady-state solutions are exactly preserved by positivity preserving; is, depth each fluid layer guaranteed to be nonnegative. also propose new technique treatment nonconservative products describing momentum exchange between layers. The performance proposed method illustrated on number numerical...

We propose a PDE chemotaxis model, which can be viewed as regularization of the Patlak-Keller-Segel (PKS)system. Our modification is based on fundamental physical property chemotactic flux function---itsboundedness. This means that cell velocity proportional to magnitude chemoattractant gradientonly when latter small, while gradient tends infinity velocitysaturates. Unlike original PKS system, solutions modified model do not blow up in either finiteor infinite time any number spatial...

We introduce a new second-order central-upwind scheme for the Saint-Venant system of shallow water equations on triangular grids. prove that both preserves “lake at rest” steady states and guarantees positivity computed fluid depth. Moreover, it can be applied to models with discontinuous bottom topography irregular channel widths. demonstrate these features scheme, as well its high resolution robustness in number numerical examples.

Summary Shallow water models are widely used to describe and study free‐surface flow. While in some practical applications the bottom friction does not have much influence on solutions, there still many applications, where is important. In particular, terms will play a significant role when depth of very small. this paper, we shallow equations with develop semi‐discrete second‐order central‐upwind scheme that capable exactly preserving physically relevant steady states maintaining positivity...

The Darcy–Boussinesq equations at infinite Darcy–Prandtl number are used to study convection and heat transport in a basic model of porous-medium over broad range Rayleigh back down onset.

We develop a family of new interior penalty discontinuous Galerkin methods for the Keller–Segel chemotaxis model. This model is described by system two nonlinear PDEs: convection-diffusion equation cell density coupled with reaction-diffusion chemoattractant concentration. It has been recently shown that convective part this mixed hyperbolic–elliptic-type, which may cause severe instabilities when studied solved straightforward numerical methods. Therefore, first step in derivation our made...

We develop path-conservative central-upwind schemes for nonconservative one-dimensional hyperbolic systems of nonlinear partial differential equations. Such arise in a variety applications and the most challenging part their numerical discretization is robust treatment product terms. Godunov-type were developed as an efficient, highly accurate ``black-box’’ solver conservation balance laws. They successfully applied to large number including several ones. To overcome difficulties related...