Samir Kumar Bhowmik

A Multigrid Method for the Helmholtz Equation with Optimized Coarse Grid Corrections

We study the convergence of multigrid schemes for the Helmholtz equation, focusing in particular on the choice of the coarse scale operators. Let

Stable numerical schemes for a partly convolutional partial integro-differential equation

G_{\rm c}

Fast and efficient numerical methods for an extended Black-Scholes model

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

Numerical Study of Rosenau-KdV Equation Using Finite Element Method Based on Collocation Approach

G_{\rm c} > 2

Mathematical analysis of bioethanol production through continuous reactor with a settling unit

. However, in examples the requirement is more like

Solving two dimensional second order elliptic equations in exterior domains using the inverted finite elements method

G_{\rm c} \gtrsim 10

Stability and convergence analysis of a one step approximation of a linear partial integro-differential equation

, 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 two dimensions 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

Preconditioners based on windowed Fourier frames applied to elliptic partial differential equations

G_{\rm c}

Numerical approximation of a convolution model of · θ -neuron networks

is reduced from about 10 to about 3.5, with less dam...

Numerical convergence of a one step approximation of an integro-differential equation

A model partial integro-differential operator (PIDO) that contains both local and nonlocal diffusion operators is considered in this article. This type of operators come in modeling various scientific and financial engineering problems. In most cases, people use finite difference schemes to generate solutions of such model problems. We compare and analyze stability and accuracy of two such finite difference schemes. We first present a discrete analogue of the PIDO and then approximate the semi-discrete time dependent problem using two different one step methods and show the stability conditions and the accuracy of the schemes. We use the Fourier transforms throughout our analysis.