The Galerkin discretization of a Fredholm integral equation of the second kind on a closed, piecewise analytic surface $\Gamma \subset \hbox {R}^3$ is analyzed. High order, $hp$-boundary elements on grids which are geometrically graded toward the edges and vertices of the surface give exponential convergence, similar to what is known in the $hp$ Finite Element Method. A quadrature strategy is developed which gives rise to a fully discrete ...

In this paper, we provide analytical stability estimates for the method of transport. We first prove stability of the second-order method of transport applied to the linear advection equation with constant coefficient in one dimension by using the von Neumann method and with the positive operator technique. In a second step, we extend the proof to the linear advection equation with variable coefficient. Finally, we investigate and compare ...

An initial-boundary value problem for a system of nonlinear partial differential equations, which consists of a hyperbolic and a parabolic part, is taken into consideration. Spacial derivatives are discretised by third order consistent difference operators, which are constructed such that a summation by parts formula holds. Therefore the space discretisation is energy bounded and algebraically stable implicit Runge-Kutta methods can be ...

An optimal finite difference method for the numerical solution of the Cauchy problem for a given partial differential equation is, by definition, the scheme that minimises the local truncation error after one step. In this paper we conduct a study of certain extremal problems that are closely related to optimal finite difference schemes for finding numerical solutions of such problems.