Norikazu Saito

Notes on finite difference schemes to a parabolic-elliptic system modelling chemotaxis

We consider numerical solutions to a nonlinear parabolic-elliptic system in mathematical biology. An important property of solutions to the system is the conservation of L^1 norm. Applying the upwind technique, we make a finite difference scheme satisfying the conservation of a discrete L^1 norm. We also describe a remark on time discretization from the view point of Lyapunovs property which is another important feature of the system. Several numerical examples clarify the nature of our scheme.

Finite volume methods for a Keller---Segel system: discrete energy, error estimates and numerical blow-up analysis

We consider the finite volume approximation for a non-linear parabolic-elliptic system, which describes the aggregation of slime molds resulting from their chemotactic features, called a simplified Keller–Segel system. First, we present a linear finite volume scheme that satisfies both positivity and mass conservations, which are important features of the original system. We derive some inequalities on the discrete free energy. Then, under some assumptions on the regularity of solution, admissible mesh and a priori estimates of the discrete solution, we establish error estimates in

ANALYSIS OF THE FICTITIOUS DOMAIN METHOD WITH AN L 2 -PENALTY FOR ELLIPTIC PROBLEMS

L^p

Remarks on the Rational Approximation of Holomorphic Semigroups with Nonuniform Partitions

Lp norm with a suitable

Hybridized Discontinuous Galerkin Method for Elliptic Interface Problems: Error Estimates Under Low Regularity Assumptions of Solutions

p>2

Some Outflow Boundary Conditions for the Navier-Stokes Equations

p>2 for the two dimensional case. In the last part of this paper, we restrict our attention to the radially symmetric solution of chemotaxis system, and we derive some inequalities concerned with the blow-up phenomenon of numerical solution. Several numerical experiments are presented to verify the theoretical results.