We present a new smoothness indicator that evaluates the local smoothness of a function inside of a stencil. The corresponding weighted essentially non-oscillatory (WENO) finite difference scheme can provide the fifth convergence order in smooth regions. The proposed WENO scheme provides at least the same or improved behavior over the classical fifth-order WENO scheme [9] and other fifth-order ...

This paper is concerned with the construction and developing of several nonstandard finite difference (NSFD) schemes in matrix form in order to obtain numerical solutions of epidemic models. In particular, we deal with a classical SIR epidemic model and a seasonal model associated with the evolution of the transmission of respiratory syncytial virus RSV in the human population. The first model ...

A two-dimensional mountainous mass flow dynamic procedure solver (Massflow-2D) using the MacCormack-TVD finite difference scheme is proposed. The solver is implemented in Matlab on structured meshes with variable computational domain. To verify the model, a variety of numerical test scenarios, namely, the classical one-dimensional and two-dimensional dam break, the landslide in Hong Kong in 199...

Abstract. The solutions of the nonlinear Schrödinger equation are of great importance for ab initio calculations. It can be shown that such solutions conserve a countable number of quantities, the simplest being the local norm square conservation law. Numerical solutions of high quality, especially for long time intervals, must necessarily obey these conservation laws. In this work we first giv...

The work presented here describes a method of coordinate transformation that enables spectral methods to be applied efficiently to differential problems with steep solutions. The approach makes use of the adaptive finite difference method presented by Huang and Sloan [SIAM J. Sci. Comput., 15 (1994), pp. 776–797]. This method is applied on a coarse grid to obtain a rough approximation of the so...

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 coupled system of two singularly perturbed ordinary differential equations of first order with the prescribed initial values are considered. The leading term of each equation is multiplied by a small positive parameter and the parameters may differ. The solution exhibits overlapping layers. A Shishkin mesh is constructed. A classical finite difference scheme applied on this mesh (which is pie...

The use of finite difference schemes to compute the scattering of acoustic waves by surfaces made up of different materials with sharp surface discontinuities at the joints would, invariably, result in the generations of spurious reflected waves. The spurious waves are caused by the use of a discrete computational method. Spurious scattered waves are produced even if a high-order scheme capable...

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...