This article proposes a family of non-standard methods coupled with compact finite differences to numerically integrate the non-linear Burgers’ equation. Firstly, is derived deal system ordinary differential equations (ODEs) arising from semi-discretization initial-boundary value partial (PDEs). Further, method this considered as special case and fourth-order difference resulting in combined nu...

We study necessary conditions for stability of a Numerov-type compact higher-order finite-difference scheme the 1D homogeneous wave equation in case non-uniform spatial meshes. first show that uniform time cannot be valid any norm provided complex eigenvalues appear associated mesh eigenvalue problem. Moreover, we prove then solution grows exponentially making strongly non-dissipative and there...

We present a simple and efficient compact fourth-order Poisson solver in polar coordinates. This solver relies on the truncated Fourier series expansion, where the differential equations of the Fourier coefficients are solved by the compact fourthorder finite difference scheme. By shifting a grid a half mesh away from the origin and incorporating the symmetry constraint of Fourier coefficients,...

This paper presents a simple and computationally-efficient algorithm for solving steady two-dimensional subsonic and transonic compressible flow over an airfoil. This work uses an interactive viscous-inviscid solution by incorporating the viscous effects in a thin shear-layer. Boundary-layer approximation reduces the Navier-Stokes equations to a parabolic set of coupled, non-linear partial diff...

We study compact finite difference methods for the Schrödinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave function ψ and external potential V(x). The CrankNicolson compact finite difference method and the semi-implicit compact finite difference method are both of order O(h4+τ2) in discrete l2 ,H1 and l norms with ...

This paper discusses several aspects related to the consistency and stability of the discrete adjoints of upwind numerical schemes. First and third order upwind discretizations of the one-dimensional advection equation are considered in both the finite difference and finite volume formulations. We show that the discrete adjoints may lose consistency and stability near the points where upwinding...

We prove the convergence of a semi-implicit monotone finite difference scheme approximating an initial-boundary value problem for a spatially one-dimensional quasilinear strongly degenerate parabolic equation, which is supplied with two different inhomogeneous flux-type boundary conditions. This problem arises in the modeling of the sedimentation-consolidation process. We formulate the definiti...