The long-time stability properties of a few multistep numerical schemes for the two-dimensional incompressible Navier–Stokes equations (formulated in vorticity-stream function) are investigated in this article. These semi-implicit numerical schemes use a combination of explicit Adams–Bashforth extrapolation for the nonlinear convection term and implicit Adams–Moulton interpolation for the visco...

By applying the Adams-Bashforth-Moulton method (ABM), this paper explores complexity and synchronization of a fractional-order laser dynamical model. The dynamics under variance derivative order q parameters system have examined using multiscale algorithm bifurcation diagram. Numerical simulation outcomes demonstrate that generates chaos with decreasing q. Moreover, designs coupled network syst...

In this paper, I explore the solution of the contaminant transport problem over a random conductivity field. First, I generate a random, correlated conductivity field by convolving independent, uniformly distributed random variables with the identity function. I use finite differences to discretize the diffusion equation to solve for the steady-state pressure gradient using Backward Euler. This...

We outline a new class of robust and efficient methods for solving the Navier–Stokes equations with a Boussinesq model for buoyancy driven flow. We describe a general solution strategy that has two basic building blocks: an implicit time integrator using a stabilized trapezoid rule with an explicit Adams–Bashforth method for error control, and a robust Krylov subspace solver for the spatially d...

This paper constructs multirate linear multistep time discretizations based on Adams-Bashforth methods. These methods are aimed at solving conservation laws and allow different timesteps to be used in different parts of the spatial domain. The proposed family of discretizations is second order accurate in time and has conservation and linear and nonlinear stability properties under local CFL co...

Even the simplest advection-diffusion problems can exhibit multiple time scales. This means that robust variable step time integrators are a prerequisite if such problems are to be efficiently solved computationally. The performance of the second order trapezoid rule using an explicit Adams–Bashforth method for error control is assessed in this work. This combination is particularly well suited...

In this chapter, some real-life model problems that can be formulated as ordinary differential equations (ODEs) are introduced and numerically studied. These models are the variable-order fractional Hodgkin–Huxley model of neuronal excitation (VOFHHM) and other models with the variable-order fractional (VOF) time delay, such as the 4year life cycle of a population of lemmings model, the enzyme ...

In this paper, a fractional-order model for the spread of computer viruses with saturated treatment rate is presented. This model consists of three components: Susceptible, Infectious, and Recovered. The SIR epidemic model with saturated treatment rate is considered. The stability of the disease free and endemic equilibrium points are studied. The global stability of the disease free equilibriu...

Assuming that the population remains small, one can truncate at given values of n1 and n2 to yield a finite set of differential equations. For an initial condition, we take the population to be zero, that is, ρ1,1,1,1 = 1 with all other elements of the density matrix set to zero. Eq. A.6 can then be solved numerically (we used the Adams-Moulton-Bashforth procedure) until a steady state is reach...