Philsu Kim

Permanence and stability of an Ivlev-type predator-prey system with impulsive control strategies

In this paper, we study a predator-prey system with an Ivlev-type functional response and impulsive control strategies containing a biological control (periodic impulsive immigration of the predator) and a chemical control (periodic pesticide spraying) with the same period, but not simultaneously. We find conditions for the local stability of the prey-free periodic solution by applying the Floquet theory of an impulsive differential equation and small amplitude perturbation techniques to the system. In addition, it is shown that the system is permanent under some conditions by using comparison results of impulsive differential inequalities. Moreover, we add a forcing term into the prey populations intrinsic growth rate and find the conditions for the stability and for the permanence of this system.

An Error Corrected Euler Method for Solving Stiff Problems Based on Chebyshev Collocation

In this paper, we present error corrected Euler methods for solving stiff initial value problems, which not only avoid unnecessary iteration process that may be required in most implicit methods but also have such a good stability as all implicit methods possess. The proposed methods use a Chebyshev collocation technique as well as an asymptotical linear ordinary differential equation of first-order derived from the difference between the exact solution and the Eulers polygon. These methods with or without the Jacobian are analyzed in terms of convergence and stability. In particular, it is proved that the proposed methods have a convergence order up to 4 regardless of the usage of the Jacobian. Numerical tests are given to support the theoretical analysis as evidences.

Convergence on error correction methods for solving initial value problems

Higher-order semi-explicit one-step error correction methods(ECM) for solving initial value problems are developed. ECM provides the excellent convergence O(h^2^p^+^2) one wants to get without any iteration processes required by most implicit type methods. This is possible if one constructs a local approximation having a residual error O(h^p) on each time step. As a practical example, we construct a local quadratic approximation. Further, it is shown that special choices of parameters for the local quadratic polynomial lead to the known explicit second-order methods which can be improved into a semi-explicit type ECM of the order of accuracy 6. The stability function is also derived and numerical evidences are presented to support theoretical results with several stiff and non-stiff problems. It should be remarked that the ECM approach developed here does not yield explicit methods, but semi-implicit methods of the Rosenbrock type. Both ECM and Rosenbrocks methods require to solve a few linear systems at each integration step, but the ECM approach involves 2p+2 evaluations of the Jacobian matrix per integration step whereas the Rosenbrock method demands one evaluation only. However, it is much easier to get high order methods by using the ECM approach.

Development of semi-Lagrangian gyrokinetic code for full-f turbulence simulation in general tokamak geometry

In this work, we report the development of a new gyrokinetic code for full-f simulation of electrostatic turbulence in general tokamak geometry. Backward semi-Lagrangian scheme is employed for noise-free simulation of the gyrokinetic Vlasov equation with finite Larmor radius effects. Grid systems and numerical interpolations are implemented to deal with arbitrary equilibrium information given in a GEQDSK format file. In particular, we introduce an adaptive interpolation technique for fluctuating quantities, which have elongated structures along equilibrium magnetic fields. This field-aligned interpolation can reduce the required number of grid points to represent the fluctuating quantities. Also, it is shown that the new interpolation allows us to choose bigger time sizes with better simulation accuracy. Several benchmark simulation results are presented for comparison with previously known cases. It is demonstrated that the new code can reproduce the well known results of zonal flow and linear ITG instabilities in concentric circular equilibrium. It is also shown that the code can capture the effects of plasma shaping on the zonal flow and ITG instabilities, and the stabilization effects of the shaping result in significant up-shifts of the threshold of ion temperature gradient for ITG in nonlinear simulation.

Simple ECEM Algorithms Using Function Values Only

In this paper, we improve the error corrected Euler method(ECEM) intro- duced in (11) by evaluating function values only at local nodes in each time interval. As a result, one can avoid computations of Jacobian matrices on each time interval so that the algorithms become simpler to implement in solving various class of time dependent differ- ential equations numerically. The proposed ECEM formula resembles to the Runge-Kutta method in its representations but both methods have different characteristic properties.

A Chebyshev Collocation Method for Stiff Initial Value Problems and Its Stability

The Chebyshev collocation method in [21] to solve stiff initial-value problems is generalized by using arbitrary degrees of interpolation polynomials and arbitrary collocation points. The convergence of this generalized Chebyshev collocation method is shown to be independent of the chosen collocation points. It is observed how the stability region does depend on collocation points. In particular, A-stability is shown by taking the mid points of nodes as collocation points.

Alon-Babai-Suzuki's Conjecture Related to Binary Codes in Nonmodular Version

Let and be sets of nonnegative integers. Let be a family of subsets of with for each and for any . Every subset of can be represented by a binary code a such that if and if . Alon et al. made a conjecture in 1991 in modular version. We prove Alon-Babai-Sukukis Conjecture in nonmodular version. For any and with , .

AN ITERATION FREE BACKWARD SEMI-LAGRANGIAN SCHEME FOR GUIDING CENTER PROBLEMS ∗

In this paper, we develop an iteration free backward semi-Lagrangian method for nonlinear guiding center models. We apply the fourth-order central difference scheme for the Poisson equation and employ the local cubic interpolation for the spatial discretization. A key problem in the time discretization is to find the characteristic curve arriving at each grid point which is the solution of a system of highly nonlinear ODEs with a self-consistency imposed by the Poisson equation. The proposed method is based on the error correction method recently developed by the authors. For the error correction method, we introduce a modified Eulers polygon and solve the induced asymptotically linear differential equation with the midpoint quadrature rule to get the error correction term. We prove that the proposed iteration free method has convergence order at least

An iteration free backward semi-Lagrangian scheme for solving incompressible Navier-Stokes equations

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An error embedded method based on generalized Chebyshev polynomials

in space and