This paper formulates and analyzes a weak solution to the coupling of time-dependent Navier-Stokes flow with Darcy flow under certain boundary conditions, one of them being the Beaver-Joseph-Saffman law on the interface. Existence and a priori estimates for the weak solution are shown under additional regularity assumptions. We introduce a fully discrete scheme with the unknowns being the Navie...

In this paper a finite difference method for solving 2-dimensional diffusion equation is presented. The method employs Crank-Nicolson scheme to improve finite difference formulation and its convergence and stability. The obtained solution will be a recursive formula in each step of which a system of linear equations should be solved. Given the specific form of obtained matrices, rather than sol...

The temperature distribution due to a rotating or dithering Gaussian laser beam on a finite body is obtained numerically. The authors apply various techniques to solve the nonhomogeneous heat equation in different spatial dimensions. The authors’ approach includes the Crank-Nicolson method, the Fast Fourier Transform (FFT) method and the commercial software COMSOL. It is found that the maximum ...

The objectives of the present study are to investigate thermal radiation of a viscous incompressible unsteady chemically reacting and hydromagnetic fluid flow past an impulsively started vertical plate with heat and mass transfer is analyzed. The fluid is a gray, absorbing-emitting but nonscattering medium and the Rosseland approximation is used to describe the radiative heat flux in the energy...

In the present investigation, the fully implicit and Crank–Nicolson difference schemes for solving option prices are analyzed. It is proved that the error expansions for the difference methods have the correct form for applying Richardson extrapolation to increase the order of accuracy of the approximations. The difference methods are applied to European, American, and down-and-out knock-out ca...

In this paper, we present a new pressure-correction projection scheme for solving the time-dependent Navier-Stokes equations, which is based on the Crank-Nicolson extrapolation method in the time discretization. Error estimates for the velocity and the pressure of semidiscretized scheme are derived by interpreting the projection scheme as second-order time discretization of a perturbed system w...

In this work we generate the numerical solutions of the Burgers’ equation by applying the Crank-Nicolson method directly to the Burgers’ equation, i.e., we do not use Hopf-Cole transformation to reduce Burgers’ equation into the linear heat equation. Absolute error of the present method is compared to the absolute error of the two existing methods for two test problems. The method is also analy...

A semi-Lagrangian method is presented to price continuously observed fixed strike Asian options. At each timestep a set of one dimensional partial integral differential equations (PIDEs) is solved and the solution of each PIDE is updated using semi-Lagrangian timestepping. Crank-Nicolson and second order backward differencing timestepping schemes are studied. Monotonicity and stability results ...

This paper addresses the construction of absorbing boundary conditions for the one-dimensional Schrödinger equation with a general variable repulsive potential or with a cubic nonlinearity. Semi-discrete time schemes, based on Crank-Nicolson approximations, are built for the associated initial boundary value problems. Finally, some numerical simulations give a comparison of the various absorbin...