A rational spectral collocation method with adaptively transformed Chebyshev grid points

A spectral collocation method based on rational interpolants and adaptive grid points is presented. The rational interpolants approximate analytic functions with exponential accuracy by using prescribed barycentric weights and transformed Chebyshev points. The locations of the grid points are adapted to singularities of the underlying solution, and the locations of these singularities are approximated by the locations of poles of Chebyshev-Padé approximants. Numerical experiments on two time-dependent problems, one with finite time blow-up and one with a moving front, indicate that the method far outperforms the standard Chebyshev spectral collocation method for problems whose solutions have singularities in the complex plan close to [-1,1]

IBVPs for Scalar Conservation Laws with Time Discontinuous Fluxes

The initial boundary value problem for a class of scalar non autonomous conservation laws in one space dimension is proved to be well posed and stable with respect to variations in the flux. Targeting applications to traffic, the regularity assumptions on the flow are extended to a merely $\mathbf{L}^{\infty}$ dependence on time. These results ensure, for instance, the well posedness of a class of vehicular traffic models with time dependent speed limits. A traffic management problem is then shown to admit an optimal solution

The use of spline and singular functions in an integral equation method for conformal mapping

We consider the integral equation method of Symm for the conformal mapping of simply-connected domains. For the numerical solution, we examine the use of spline functions of various degrees for the approximation of the source density o. In particular, we consider ways for overcoming the difficulties associated with corner singularities. For this we modify the spline approximation and in the neighbourhood of each corner, where a boundary singularity occurs, we approximate σ by a function which reflects the main singular behaviour of the source density. The singular functions are then blended with the splines, which approximate σ on the remainder of the boundary, so that the global approximating function has continuity of appropriate order at the transition points between the two types of approximation. We show, by means of numerical examples, that such approximations overcome the difficulties associated with corner singularities and lead to numerical results of high accuracy

Small amplitude lateral sloshing in a cylindrical tank with a hemispherical bottom under low gravitational conditions Summary report

Small amplitude lateral sloshing in cylindrical tank with hemispherical bottom under low gravitational condition

Conformal Maps to Multiply-Slit Domains and Applications

By exploiting conformal maps to vertically slit regions in the complex plane, a recently developed rational spectral method [Tee and Trefethen, 2006] is able to solve PDEs with interior layer-like behaviour using significantly fewer collocation points than traditional spectral methods. The conformal maps are chosen to 'enlarge the region of analyticity' in the solution: an idea which can be extended to other numerical methods based upon global polynomial interpolation. Here we show how such maps can be rapidly computed in both periodic and nonperiodic geometries, and apply them to some challenging differential equations

An efficient implementation of an implicit FEM scheme for fractional-in-space reaction-diffusion equations

Fractional differential equations are becoming increasingly used as a modelling tool for processes with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues, which impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids, and robust techniques for computing the fractional power of a matrix times a vector. Our approach is show-cased by solving the fractional Fisher and fractional Allen-Cahn reaction-diffusion equations in two and three spatial dimensions, and analysing the speed of the travelling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator