In this paper, we analyze a multiscale method developed under the heterogeneous multiscale method (HMM) framework for numerical approximation of multiscale wave propagation problems in periodic media. In particular, we are interested in the long time O(ε−2) wave propagation, where ε represents the size of the microscopic variations in the media. In large time scales, the solutions of multiscale...

We give a further examination of the stabilized Residual Distribution schemes for the solution of the shallow water equations proposed in (Ricchiuto and Bollermann, J.Comp.Phys., available online 25 October 2008). Based on a non-linear variant of a Lax-Friedrichs scheme, the scheme is wellbalanced, able to handle dry areas and, for smooth regions of the solution, obtains second order of accurac...

A coupled system of two singularly perturbed linear reaction–diffusion two-point boundary value problems is examined. The leading term of each equation is multiplied by a small positive parameter, but these parameters may have different magnitudes. The solutions to the system have boundary layers that overlap and interact. The structure of these layers is analysed, and this leads to the constru...

The order of accuracy of the node-centered finite volume methods is analyzed, and the analysis is based on an exact derivation of the numerical errors in one dimension. The accuracy for various types of grids are considered. Numerical simulations and analysis are performed for both a hyperbolic and a eliptic case, and the results agree. The impact of weakly imposed boundary conditions is analyz...

Two-point boundary value problems for a system of nonlinear first order ordinary differential equations are considered. It was shown in former papers of the authors that starting from the two-point exact difference scheme (EDS) one can derive a so-called truncated difference scheme (TDS) which possesses a prescribed order of accuracy O(|h|m) w.r.t. the maximal step size |h|. This m-TDS represen...

We present a method for solving nonlinear reaction-diiusion equations, s @p @t ? r (Krp) = f (p); using a mixed nite-element method. To linearize the mixed-method equations , we use a two-grid scheme that relegates all of the Newton-like iterations to grids much coarser than the original one, with no loss in order of accuracy. The use of a multigrid-based solver for the indeenite linear systems...

Two techniques for reliably controlling the defect (residual) in the numerical solution of nonstiff initial value problems were given in [D. This work describes an alternative approach based on Hermite-Birkhoff interpolation. The new approach has two main advantagesmit is applicable to Runge-Kutta schemes of any order, and it gives rise to a defect of the optimum asymptotic order of accuracy. F...

Keywords: Elliptic equations Nonlocal boundary value problems Difference schemes Stability a b s t r a c t The well-posedness of the Bitsadze–Samarskii type nonlocal boundary value problem in Hölder spaces with a weight is established. The coercivity inequalities for the solution of the nonlocal boundary value problem for elliptic equations are obtained. The stable second order of accuracy diff...

We develop two-grid schemes for solving nonlinear reaction-diiusion systems , @p @t ? r (Krp) = f(x; p); where p = (p; q) is an unknown vector-valued function. The schemes use discretizations based on a mixed nite-element method. The two-grid approach yields iterative procedures for solving the nonlinear discrete equations. The idea is to relegate all of the Newton-like iterations to grids much...

The possibility that Lorentz symmetry is violated in gravitational processes is relatively unconstrained by experiment, in stark contrast with the level of accuracy to which Lorentz symmetry has been confirmed in the matter sector. One model of Lorentz violation in the gravitational sector is Einstein-aether theory, in which Lorentz symmetry is broken by giving a vacuum expectation value to a d...