The paper deals with the Darboux problem for the equation Dxyz(x, y) = f(x, y, z(x,y)) where z(x,y) is a function defined by z(x,y)(t, s) = z(x + t, y + s), (t, s) ∈ [−a0, 0]× [−b0, 0]. We construct a general class of difference methods for this problem. We prove the existence and uniqueness of solutions to implicit functional difference equations by means of a comparison method; moreover we gi...

We study mixed-moment models (full zeroth moment, half higher moments) for a Fokker–Planck equation in one space dimension. Mixed-moment minimum-entropy models are known to overcome the zero net-flux problem of full-moment minimum-entropy Mn models. A realizability theory for these mixed moments of arbitrary order is derived, as well as a new closure, which we refer to as Kershaw closure. They ...

We consider a divergence-free non-dissipative fourth-order explicit staggered nite di erence scheme for the hyperbolic Maxwell's equations. Special one-sided di erence operators are derived in order to implement the scheme near metal boundaries and dielectric interfaces. Numerical results show the scheme is long-time stable, and is fourth-order convergent over complex domains that include diele...

We investigate quasi-explicit time-integration schemes for solving dynamic fracture problems with set-valued cohesive zone models. These schemes combine a central difference time-integration scheme and a partially implicit and lumped treatment of the cohesive forces. At each time step, the displacements of the nodes in the interior of the domain are computed in an explicit way, while the displa...

In this paper, the formulation of a new group explicit method with a fourth order accuracy is described in solving the two dimensional Helmholtz equation. The formulation is based on the nine-point fourth order compact finite difference approximation formula. The complexity analysis of the developed scheme is also presented. Several numerical experiments were conducted to test the feasibility o...

This paper presents an unconditionally stable leapfrog alternating-direction-implicit finite-difference time-domain (ADIFDTD) method for lossy media. Conductivity terms of lossy media are incorporated into the leapfrog ADI-FDTD method in an analogous manner as the conventional explicit FDTD method since the leapfrog ADI-FDTD method is a perturbation of the conventional explicit FDTD method. Imp...

In this paper, we study finite element approximations of the viscosity solution of the fully nonlinear Monge-Ampère equation, det(Du) = f (> 0) using the well-known nonconforming Morley element. Our approach is based on the vanishing moment method, which was recently proposed as a constructive way to approximate fully nonlinear second order equations by the author and Feng in [15]. The vanishin...

Based on the concept of domain decomposition we construct a class of alternating group explicit method for fourth order parabolic equations. Furthermore, an exponential type alternating group explicit method for 2D convection-diffusion equations is derived, which is effective in convection dominant cases. Both of the two methods have the property of unconditional stability and intrinsic paralle...

We have developed a three level implicit method for solution of the Helmholtz equation. Using the cubic spline in space and finite difference in time directions. The approach has been modied to drive Numerov type nite difference method. The method yield the tri-diagonal linear system of algebraic equations which can be solved by using a tri-diagonal solver. Stability and error estimation of the...

In this paper, the problem of the approximation by finite differences of solutions to elliptic problems with rapidly oscillating coefficients and periodic boundary conditions is considered. The mesh-size is denoted by h while ε denotes the period of the rapidly oscillating coefficient. Using Bloch wave decompositions, we analyze the case where the ratio h/ε is rational. We show that if h/ε is k...