A well-balanced scheme for the numerical processing of source terms in hyperbolic equations
SIAM Journal on Numerical Analysis
Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations
SIAM Journal on Scientific Computing
Hyperbolic models for chemotaxis in 1-D
Nonlinear Analysis: Real World Applications
A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows
SIAM Journal on Scientific Computing
Approximation of Hyperbolic Models for Chemosensitive Movement
SIAM Journal on Scientific Computing
Implicit---Explicit Runge---Kutta Schemes and Applications to Hyperbolic Systems with Relaxation
Journal of Scientific Computing
Upwinding of the source term at interfaces for Euler equations with high friction
Computers & Mathematics with Applications
3D simulations of early blood vessel formation
Journal of Computational Physics
Asymptotic High-Order Schemes for $2\times2$ Dissipative Hyperbolic Systems
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
Maxwellian Decay for Well-balanced Approximations of a Super-characteristic Chemotaxis Model
SIAM Journal on Scientific Computing
Maxwellian Decay for Well-balanced Approximations of a Super-characteristic Chemotaxis Model
SIAM Journal on Scientific Computing
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We introduce a new class of finite difference schemes for approximating the solutions to an initial-boundary value problem on a bounded interval for a one-dimensional dissipative hyperbolic system with an external source term, which arises as a simple model of chemotaxis. Since the solutions to this problem may converge to nonconstant asymptotic states for large times, standard schemes usually fail to yield a good approximation. Therefore, we propose a new class of schemes, which use an asymptotic higher order correction, second and third order in our examples, to balance the effects of the source term and the influence of the asymptotic solutions. Special care is needed to deal with boundary conditions to avoid harmful loss of mass. Convergence results are proved for these new schemes, and several numerical tests are presented and discussed to verify the effectiveness of their behavior.