Lattice refinement in LQC, its meaning and its necessity are discussed. The rôle of lattice refinement for the realisation of a successful inflationary model is explicitly shown. A simple and effective numerical technique to solve the constraint equation for any choice of lattice refinement model is briefly illustrated. Phenomenological and consistency requirements leading to a particular choic...

A compactly supported scaling function can come from a refinement equation with infinitely many nonzero coefficients (an infinite mask). In this case we prove that the symbol of the mask must have the special rational form ã(Z) = b̃(Z)c̃(Z)/b̃(Z). Any finite combination of the shifts of a refinable function will have such a mask, and will be refinable. We also study compactly supported solutions o...

3 Quick tutorial 4 3.1 Mode 0: Steady-state, fixed spatial grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Mode 1: Linear heat equation, fixed spatial grid and fixed time step size . . . . . . . . . . . . 7 3.3 Mode 2: Linear heat equation, fixed spatial grid and adaptive time stepping . . . . . . . . . . 7 3.4 Mode 3: Linear heat equation, adaptive spatial refinement and...

The refinement equation f(x) = ∑N k=0 ck f(2x− k) plays a key role in wavelet theory and in subdivision schemes in approximation theory. This paper explores the relationship of the refinement equation to the mapping τ(x) = 2x mod 1. A simple necessary condition for the existence of an integrable solution to the refinement equation is obtained by considering the periodic cycles of τ . Another si...

Solutions of the semilinear wave equation are found numerically in three spatial dimensions with no assumed symmetry using distributed adaptive mesh refinement. The threshold of singularity formation is studied for the two cases in which the exponent of the nonlinear term is either p = 5 or p = 7. Near the threshold of singularity formation, numerical solutions suggest an approach to self-simil...

We introduce a finite-difference method to simulate pore scale steady-state creeping fluid flow in porous media. First, a geometrical approximation is invoked to describe the interstitial space of grid-based images of porous media. Subsequently, a generalized Laplace equation is derived and solved to calculate fluid pressure and velocity distributions in the interstitial space domain. We use a ...

It is known that one can improve the accuracy of the finite element solution of partial differential equation (PDE) by uniformly refining a triangulation. Similarly, one can uniformly refine a quadrangulation. Recently a refinement scheme of pentagonal partition was introduced in [4]. It is demonstrated that the numerical solution of Poisson equation based on the pentagonal refinement scheme ou...