We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) n-dimensional nonhomogeneous wave equation, n? 1. Their construction is accomplished by both classical Numerov approach and alternative technique based on averaging together with further necessary improvements arising scheme 2. The applicable to other types PDEs including pa...

We establish the error bounds of fourth-order compact finite difference (4cFD) methods for Dirac equation in massless and nonrelativistic regime, which involves a small dimensionless parameter $0 < \varepsilon \le 1$ inversely proportional to speed light. In this solution propagates waves with wavelength $O(\varepsilon)$ time $O(1)$ space, as well wave $O(1/\varepsilon)$ rapid outgoing waves. a...

We present a finite difference method for discretizing a Heaviside function H(u(~x)), where u is a level set function u : Rn 7→ R that is positive on a bounded region Ω ⊂ R. There are two variants of our algorithm, both of which are adapted from finite difference methods that we proposed for discretizing delta functions in [13–15]. We consider our approximate Heaviside functions as they are use...

4 Initial Value Problems for PDEs 18 4.1 Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 FDMs for Parabolic Problems . . . . . . . . . . . . . . . . . . . . 19 4.2.1 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3 Mixed IVP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.3.1 Implicit Scheme . . . . . . . . . . . . . . . . . . ....

Finite difference schemes providing an improved representation of a range of scales (spectral-like resolution) in the evaluation of first, second, and higher order derivatives are presented and compared with well-known schemes. The schemes may be used on non-uniform meshes and a variety of boundary conditions may be imposed. Schemes are also presented for derivatives at mid-cell locations, for ...

Finite-difference approximations are succinctly represented by their stencils, a set of weights that when applied to adjacent samples of a function approximate some differential operator. In image processing the samples are pixels or voxels, and the differential operators must be inverted for smoothing or painting applications. For efficiency in such applications requiring inverses, the finited...

Atmospheric flows are characterized by a large range of length scales as well as strong gradients. The accurate simulation of such flows requires numerical algorithms with high spectral resolution, as well as the ability to provide nonoscillatory solutions across regions of high gradients. These flows exhibit a large range of time scales as well—the slowest waves propagate at the flow velocity ...

A class of generalized high order finite compact difference schemes is proposed for shock/vortex, shock/boundary layer interaction problems. The finite compact difference scheme takes the region between two shocks as a compact stencil. The high order WENO fluxes on shock stencils are used as the internal boundary fluxes for the compact scheme. A lemma of the property of smoothness estimators on...