The main aim of wavelet-based numerical methods for solving partial differential equations is to develop adaptive schemes, in order to achieve accuracy and computational efficiency. The wavelet optimized finite difference method (WOFD) uses wavelets to generate appropriate grids to apply finite difference method. Its standard implementation carries out static-re-griddings after a fixed number o...

A novel hybrid Finite Difference Time Domain–Wavelet-Galerkin (FDTD–WG) technique is presented for the accurate representation of electromagnetic field solution in regions of fast field transitions. Its fundamental concept lies on the combination of the robust FDTD method with the Wavelet–Galerkin formulation, which in its turn can efficiently treat highly varying phenomena. The computational d...

Abstract We consider numerical boundary conditions for high order finite difference schemes for solving convection-diffusion equations on arbitrary geometry. The two main difficulties for numerical boundary conditions in such situations are: (1) the wide stencil of the high order finite difference operator requires special treatment for a few ghost points near the boundary; (2) the physical bou...

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...

Modern computational power makes possible realistic simulations of elastic wavefields at frequencies of interest. The most general numerical methods for modelling are grid-based techniques that track the wavefield on a dense 3D grid of points, e.g. the finite-difference, finite-element and pseudospectral methods. The study of wavefield propagation with 3D finite-difference methods has contribut...

In this paper, an unconditionally stable three-dimensional (3-D) finite-difference time-method (FDTD) is presented where the time step used is no longer restricted by stability but by accuracy. The principle of the alternating direction implicit (ADI) technique that has been used in formulating an unconditionally stable two-dimensional FDTD is applied. Unlike the conventional ADI algorithms, ho...

Computational physics has become an essential research and interpretation tool in many fields. Particularly, in reservoir geophysics, ultrasonic and seismic modeling in porous media is used to study the properties of rocks and characterize the seismic response of geological formations. Here, we give a brief overview of the most common numerical methods used to solve the partial differential equ...

In this paper, numerical methods are proposed for some interface problems in polar or Cartesian coordinates. The new methods are based on a formulation that transforms the interface problem with a non-smooth or discontinuous solution to a problem with a smooth solution. The new formulation leads to a simple second order finite difference scheme for the partial differential equation and a new in...

Purely numerical methods do not always provide an accurate way to find all the global solutions to nonlinear ODE on infinite intervals. For example, finite-difference methods fail to capture the asymptotic behavior of solutions, which might be critical for ensuring global existence. We first show, by way of a detailed example, how asymptotic information alone provides significant insight into t...

Bounds are obtained for the derivatives of the solution of a turning point problem. These results suggest a modification of the El-Mistikawy Werle finite difference scheme at the turning point. A uniform error estimate is obtained for the resulting method, and illustrative numerical results are given.