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 . . . . . . . . . . . . . . . . . . ....

This paper reviews and extends the theory and application of mimetic finite difference methods for the solution of diffusion problems in strongly heterogeneous anisotropic materials. These difference operators satisfy the fundamental identities, conservation laws and theorems of vector and tensor calculus on nonorthogonal, nonsmooth, structured and unstructured computational grids. We provide e...

We introduce the use of mimetic methods to the imaging community, for the solution of the initial-value problems ubiquitous in the machine vision and image processing and analysis fields. PDE-based image processing and analysis techniques comprise a host of applications such as noise removal and restoration, deblurring and enhancement, segmentation, edge detection, inpainting, registration, mot...

We develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes. To approximate the velocity (vector variable), the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n-gonal mesh face in three dimensions. To approximate the pressure (scalar variable), the method uses ...

This paper will address the problem of time marching function approximated solutions inherent in emerging meshfree Computational Fluid Dynamics (CFD) solution techniques. The numerical solutions of partial differential equations (PDEs) of CFD has been dominated by either finite difference methods (FDM), finite element methods (FEM), and finite volume methods (FVM). These methods can be derived ...

is obtained where A (u) is the Jacobian matrix of the components of / with respect to the components of u. Equation (1.2) is said to be hyperbolic if the eigenvalues of the matrix pi + 6A are real for all real numbers m, 0. Several authors have proposed finite-difference schemes for the numerical integration of (1.1) (or (1.2)). In [6], Lax and Wendroff introduced an explicit scheme which is st...

In this review paper, the finite difference methods (FDMs) for the fractional differential equations are displayed. The considered equations mainly include the fractional kinetic equations of diffusion or dispersion with time, space and time-space derivatives. In some way, these numerical methods have similar form as the case for classical equations, some of which can be seen as the generalizat...

Advanced finite-difference methods for seismic modeling Yang Liu1,2 and Mrinal K Sen2 State Key Laboratory of Petroleum Resource and Prospecting (China University of Petroleum, Beijing), Beijing, 102249, China The Institute for Geophysics, John A. and Katherine G. Jackson School of Geosciences, The University of Texas at Austin, 10100 Burnet Road, R2200 Austin, TX 78758, USA E-mail: wliuyang@vi...

In this work we present a general approach for developing high-order compact differencing schemes by utilizing the governing differential equation to help approximate truncation error terms. As an illustrative application we consider the stream-function vorticity form of the Navier Stokes equations, and provide driven cavity results. Some extensions to treat non-constant metric coefficients res...