Abstract. We study the convergence of multigrid schemes for the Helmholtz equation, focusing in particular on the choice of the coarse scale operators. Let Gc denote the number of points per wavelength at the coarse level. If the coarse scale solutions are to approximate the true solutions, then the oscillatory nature of the solutions implies the requirement Gc > 2. However, in examples the requirement is more like Gc 10, in a trade-off involving also the amount of damping present and the number of multigrid iterations. We conjecture that this is caused by the difference in phase speeds between the coarse and fine scale operators. Standard 5-point finite differences in 2-D are our first example. A new coarse scale 9-point operator is constructed to match the fine scale phase speeds. We then compare phase speeds and multigrid performance of standard schemes with a scheme using the new operator. The required Gc is reduced from about 10 to about 3.5, with less damping present so that waves propagate over > 100 wavelengths in the new scheme. Next we consider extensions of the method to more general cases. In 3-D comparable results are obtained with standard 7-point differences and optimized 27-point coarse grid operators, leading to an order of magnitude reduction in the number of unknowns for the coarsest scale linear system. Finally we show how to include PML boundary layers, using a regular grid finite element method. Matching coarse scale operators can easily be constructed for other discretizations. The method is therefore potentially useful for a large class of discretized high-frequency Helmholtz equations.

Black-Scholes equation is a well known partial differential equation in financial mathematics. In this article we discuss about some solution methods for the Black Scholes model with the European options (Call and Put) analytically as well as numerically. We study a weighted average method using different weights for numerical approximations. In fact, we approximate the model using a finite difference scheme in space first followed by a weighted average scheme for the time integration. Then we present the numerical results for the European Call and Put options. Finally, we investigate some linear algebra solvers to compare the superiority of the solvers. GANIT J. Bangladesh Math. Soc. Vol. 33 (2013) 103-115 DOI: http://dx.doi.org/10.3329/ganit.v33i0.17664

In this paper, we use the integration technique together with the finite element method to approximate the numerical solution of an initial value problem of differential equations. The function of two variables is expanded into Taylor's series up to order two. We exploit GaussLegendre quadrature rules evaluating the integrals arising in the formulation of the present method to get the better accuracy.

Bisection and regular false-position methods are widely used to find roots of a transcendental function f(x) in a certain interval [a,b] satisfying f(a). f(b) < 0. The paper develops a new algorithm to find roots of transcendental functions based on false position method. We use two end points of the interval to interpolate f(x) by an equivalent cubic polynomial using clamped cubic spline formula. We consider one of the roots of the interpolated function to define a new interval and to approximate the root of f(x).

This paper deals with a new multigrid method with reduced phase error for solving 2D damped Helmholtz equations. The method is obtained by taking the high‐effective, reduced phase error 5‐point finite difference (FD) scheme as a coarse grid operator and the regular 5‐point FD scheme as a fine grid operator. It is found that the proposed method gives a faster convergent rate than the regular multigrid method. A local mode Fourier analysis confirms the validity of our proposed method. Finally, some numerical results demonstrate the efficiency of the method.