Convergence of the 2×2 Godunov method for a general resonant nonlinear balance law
SIAM Journal on Applied Mathematics
A well-balanced scheme for the numerical processing of source terms in hyperbolic equations
SIAM Journal on Numerical Analysis
Representation of weak limits and definition of nonconservative products
SIAM Journal on Mathematical Analysis
Hyperbolic models for chemotaxis in 1-D
Nonlinear Analysis: Real World Applications
Localization effects and measure source terms in numerical schemes for balance laws
Mathematics of Computation
Approximation of Hyperbolic Models for Chemosensitive Movement
SIAM Journal on Scientific Computing
Upwinding of the source term at interfaces for Euler equations with high friction
Computers & Mathematics with Applications
Asymptotic High-Order Schemes for $2\times2$ Dissipative Hyperbolic Systems
SIAM Journal on Numerical Analysis
The Finite Volume-Complete Flux Scheme for Advection-Diffusion-Reaction Equations
Journal of Scientific Computing
On the Advantage of Well-Balanced Schemes for Moving-Water Equilibria of the Shallow Water Equations
Journal of Scientific Computing
Operator splittings and numerical methods
LSSC'05 Proceedings of the 5th international conference on Large-Scale Scientific Computing
Asymptotic High Order Mass-Preserving Schemes for a Hyperbolic Model of Chemotaxis
SIAM Journal on Numerical Analysis
Asymptotic High Order Mass-Preserving Schemes for a Hyperbolic Model of Chemotaxis
SIAM Journal on Numerical Analysis
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We focus on the numerical simulation of a one-dimensional model of chemotaxis dynamics (proposed by Greenberg and Alt [Trans. Amer. Math. Soc., 300 (1987), pp. 235-258] in a bounded domain by means of a previously introduced well-balanced (WB) and asymptotic-preserving (AP) scheme [L. Gosse, J. Math. Anal. Appl., (2011)]. We are especially interested in studying the decay onto numerical steady-states for two reasons: (1) conventional upwind schemes have been shown to stabilize onto spurious non-Maxwellian states (with a very big mass flux; see, e.g., [F. Guarguaglini et al., Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), pp. 39-76]) and (2) the initial data lead to a dynamic which is mostly super-characteristic in the sense of [S. Jin and M. Katsoulakis, SIAM J. Appl. Math., 61 (2000), pp. 273-292]; thus the stability results of Gosse do not apply. A reflecting boundary condition which is compatible with the well-balanced character is presented; a mass-preservation property is proved and some results on super-characteristic relaxation are recalled. Numerical experiments with coarse computational grids are presented in detail: they illustrate the bifurcation diagrams of Guarguaglini et al., which relate the total initial mass of cells with the time-asymptotic values of the chemoattractant substance on each side of the domain. It is shown that the WB scheme stabilizes correctly onto zero-mass flow rate (hence Maxwellian) steady-states which agree with the aforementioned bifurcation diagrams. The evolution in time of residues is commented for every considered test case.