In this paper, we describe a comparison of two spatial discretization schemes for the advection equation, namely the first finite difference method and the method of lines. The stability of the methods has been studied by Von Neumann method and with the matrix analysis. The methods are applied to a number of test problems to compare the accuracy and computational efficiency. We show that both d...

We approximate the price of the American put for jump diffusions by a sequence of functions, which are computed iteratively. This sequence converges to the price function uniformly and exponentially fast. Each element of the approximating sequence solves an optimal stopping problem for geometric Brownian motion, and can be numerically constructed using the classical finite difference methods. W...

A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct. The components of the solution exhibit overlapping layers. Shishkin piecewise-uniform meshes are introduced, which ...

In this paper, we derive a highly accurate numerical method for the solution of one-dimensional wave equation with Neumann boundary conditions. This hyperbolic problem is solved by using semidiscrete approximations. The space direction is discretized by wavelet-Galerkin method and the time variable is discretized by using various classical finite difference schemes. The numerical results show t...

This paper proposes an approach to speed up seismic forward modeling when the Green’s function of a given model is structured in the sense that the Green’s function has a sparse representation in some known transform domain. The first step of our approach runs a forward finite-difference (FD) simulation for a duration longer than each conventional run, but with all sources activated simultaneou...

A generalized framework is presented that extends the classical theory of finite-difference summation-by-parts (SBP) operators to include a wide range of operators, where the main extensions are i) non-repeating interior point operators, ii) nonuniform nodal distribution in the computational domain, iii) operators that do not include one or both boundary nodes. Necessary and sufficient conditio...

We propose and analyze several finite difference schemes for the Hunter–Saxton equation (HS) ut + uux = 1 2 ∫ x 0 (ux) 2 dx, x > 0, t > 0. This equation has been suggested as a simple model for nematic liquid crystals. We prove that the numerical approximations converge to the unique dissipative solution of (HS), as identified by Zhang and Zheng. A main aspect of the analysis, in addition to th...

The stability theory for hyperbolic initial boundary value problems relies most of the time on the Laplace transform with respect to the time variable. For technical reasons, this usually restricts the validity of stability estimates to the case of zero initial data. In this article, we consider the class of non-glancing finite difference approximations to the hyperbolic operator. We show that ...

We present a multilevel high order ADI method for separable generalized Helmholtz equations. The discretization method we use is a onedimensional fourth order compact finite difference applied to each directional component of the Laplace operator, resulting in a discrete system efficiently solvable by ADI methods. We apply this high order difference scheme to all levels of grids, and then start...

We propose a fast and stable numerical method to evaluate two-dimensional partial differential equation (PDE) for pricing arithmetic average Asian options. The numerical method is deduced by combining an alternating-direction technique and the central difference scheme on a piecewise uniform mesh. The numerical scheme is stable in the maximum norm, which is true for arbitrary volatility and arb...