. In this paper, we develop a quadratic spline collocation method for integrating the nonlinear partial differential equations (PDEs) of a plug flow reactor model. The method is proposed in order to be used for the operation of control design and/or numerical simulations. We first present the Crank-Nicolson method to temporally discretize the state variable. Then, we develop and analyze the pro...

The time dependent transport equation is solved with stabilized continuous, piecewise linear finite elements and the Crank-Nicolson scheme [1]. Finite elements with large aspect ratio are advocated in order to account for boundary layers. The error due to space discretization has already been studied in [2]. Here, the error due to the use of the Crank-Nicolson scheme is taken into account. Anis...

Abstract. We consider a partial differential equation of Schrödinger type, known as the ‘parabolic’ approximation to the Helmholtz equation in the theory of sound propagation in an underwater, rangeand depth-dependent environment with a variable bottom. We solve an associated initialand boundary-value problem by a finite difference scheme of Crank-Nicolson type on a variable mesh. We prove that...

We consider a model initialand boundary-value problem for a third-order p.d.e., a wide-angle ‘parabolic’ equation frequently used in underwater acoustics, with depthand rangedependent coefficients in the presence of horizontal interfaces and dissipation. After commenting on the existence–uniqueness theory of solution of the equation, we discretize the problem by a secondorder finite difference ...

Blood flow is modeled as non-Newtonian micropolar fluid. The non-linear governing equations of continuum and momentum in the cylindrical coordinate are being discretized using a finite difference approach and have been solved iteratively ,through Crank-Nicolson method. The blood velocity distribution, volumetric flow rate and Resistance to blood flow at the stenosis throat are computed for vari...

We consider discrete schemes for a nonlinear model of non-Fickian diffusion in viscoelastic polymers. The model is motivated by, but not the same as, that proposed by Cohen et al. in SIAM J. Appl. Math., 55, pp. 348–368, 1995. The spatial discretisation is effected with both the symmetric and non-symmetric interior penalty discontinuous Galerkin finite element method, and the time discretisatio...

We analyze a single step method for solving second-order parabolic initial–boundary value problems. The method uses a step-doubling extrapolation scheme in time based on backward Euler and a Galerkin approximation in space. The technique is shown to be a second-order correct approximation in time. Since step-doubling can be used as a mechanism for step-size control, the analysis is done for var...

We derive residual-based a posteriori error estimates of optimal order for fully discrete approximations for linear parabolic problems. The time discretization uses the Crank– Nicolson method, and the space discretization uses finite element spaces that are allowed to change in time. The main tool in our analysis is the comparison with an appropriate reconstruction of the discrete solution, whi...

We deal with an initial-boundary value problem for the generalized time-dependent Schrödinger equation with variable coefficients in an unbounded n–dimensional parallelepiped (n ≥ 1). To solve it, the Crank-Nicolson in time and the polylinear finite element in space method with the discrete transparent boundary conditions is considered. We present its stability properties and derive new error e...

In this paper, we suggest a jump diffusion model in markets during financial crisis. Using risk-neutral pricing, we derive a partial differential equation (P.D.E.) for the prices of European options. We find a closed form solution of the P.D.E. in the particular case where the stock price is too large. Then, we use such a solution as a boundary condition in the numerical treatment of the P.D.E....