Shape preserving C2C^2 interpolatory subdivision schemes

Stationary interpolatory subdivision schemes which preserve shape properties such as convexity or monotonicity are constructed. The schemes are rational in the data and generate limit functions that are at least $C^2$. The emphasis is on a class of six-point convexity preserving subdivision schemes that generate $C^2$ limit functions. In addition, a class of six-point monotonicity preserving schemes that also leads to $C^2$ limit functions is introduced. As the algebra is far too complicated for an analytical proof of smoothness, validation has been performed by a simple numerical methodology

Stability of subdivision schemes

The stability of stationary interpolatory subdivision schemes for univariate data is investigated. If the subdivision scheme is linear, its stability follows from the convergence of the scheme, but for nonlinear subdivision schemes one needs stronger conditions and the stability analysis of nonlinear schemes is more involved. Apart from the fact that it is natural to demand that subdivision schemes are stable, it also has an advantage in a theoretical sense: is it shown that the approximation properties of stable schemes can very easily be determined

A sand wave simulation model

Sand waves form a prominent regular pattern in the offshore seabeds of sandy shallow seas. A two dimensional vertical (2DV) flow and morphological numerical model describing the behaviour of these sand waves has been developed. The model contains the 2DV shallow water equations, with a free water surface and a general bed load formula. The water movement is coupled to the sediment transport equation with a seabed evolution equation. The domain is non-periodic in both directions. The spatial discretisation is performed by a spectral method based on Chebyshev polynomials. A fully implicit method is chosen for the discretisation in time. Firstly, we validate the model mathematically by reproducing the results obtained using a linear stability analysis for infinitely small sand waves. Hereby, we investigate a steady current situation induced by a wind stress applied at the sea surface. The bed forms we find have wavelengths in the order of hundreds of metres when the resistance at the seabed is relatively large. The results show that it is possible to model the initial evolution of sand waves with a numerical simulation model. Next, we investigate the influence of the chosen turbulent viscosity parameterisation by comparing the constant viscosity model with a depth dependent viscosity. This paper forms a part of a study to investigate the intermediate term behaviour of sand waves

Analysis of stabilization operators in a Galerkin least-squares finite element discretization of the incompressible Navier-Stokes equations

In this paper the design and analysis of a dimensionally consistent stabilization operator for a time-discontinuous Galerkin least-squares finite element method for unsteady viscous flow problems governed by the incompressible Navier-Stokes equations, is discussed. The analysis results in a class of stabilization operators which satisfy essential conditions for the stability of the numerical discretization

A linear approach to shape preserving spline approximation

This report deals with approximation of a given scattered univariate or bivariate data set that possesses certain shape properties, such as convexity, monotonicity, and/or range restrictions. The data are approximated for instance by tensor-product B-splines preserving the shape characteristics present in the data. Shape preservation of the spline approximant is obtained by additional linear constraints. Constraints are constructed which are local {\em linear sufficient\/} conditions in the unknowns for convexity or monotonicity. In addition, it is attractive if the objective function of the minimization problem is also linear, as the problem can be written as a linear programming problem then. A special linear approach based on constrained least squares is presented, which reduces the complexity of the problem in case of large data sets in contrast with the $\ell_\infty$ and the $\ell_1$-norms. An algorithm based on iterative knot insertion which generates a sequence of shape preserving approximants is given. It is investigated which linear objective functions are suited to obtain an efficient knot insertion method

Space-time discontinuous Galerkin method for wet-chemical etching of microstructures

In this paper we discuss the application of a space-time discontinuous Galerkin finite element method for convection-diffusion problems to the simulation of wet-chemical etching of microstructures. In the space-time DG method no distinction is made in the discretization between the space and time variables and discontinuous basis functions are used both in space and time. This approach results in an efficient numerical technique to deal with time-dependent flow domains as occur in wet-chemical etching, while maintaining a fully conservative discretization. The method offers great flexibility in mesh adaptation and special attention is given to the generation of an initial solution and mesh when there is no etching cavity yet. Numerical simulations of the etching of a two-dimensional slit are discussed for different regimes, namely diffusion-controlled and convection-dominated etching. These results show good agreement with analytical results in the diffusion-controlled regime. Using a simple model for the fluid velocity the typical asymmetric etching cavities are obtained in the convection dominated regime and the results agree qualitatively well with those obtained from full Navier-Stokes simulations