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 ...

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....

An efficient and accurate numerical scheme is proposed, analyzed and implemented for the Kawahara and modified Kawahara equations which model many physical phenomena such as gravity-capillary waves and magneto-sound propagation in plasmas. The scheme consists of dual-Petrov-Galerkin method in space and Crank-Nicholson-leap-frog in time such that at each time step only a sparse banded linear sys...

Two discrete-time orthogonal spline collocation schemes are formulated and analyzed for solving the linear time-dependent Schrr odinger equation in two space variables. These are Crank-Nicolson and alternating direction implicit (ADI) schemes employing C 1 piecewise polynomial spaces of arbitrary degree 3 in each space variable. The stability of the schemes is examined and optimal order a prior...

The Trapezoidal Rule with second order Backward Difference Formula (TR-BDF2) time stepping method was applied to the Black-Scholes PDE for option pricing. It is proved that TR-BDF2 time stepping method is unconditionally stable, and compared to the usual Crank-Nicolson time stepping method, the TR-BDF2 shows fewer oscillations when computing the derivatives of the solution, which are important ...

This study proposes one-dimensional advection–diffusion equation (ADE) with finite differences method (FDM) using implicit spreadsheet simulation (ADEISS). By changing only the values of temporal and spatial weighted parameters with ADEISS implementation, solutions are implicitly obtained for the BTCS, Upwind and Crank–Nicolson schemes. The ADEISS uses iterative spreadsheet solution technique. ...