In this paper we consider implicit Runge–Kutta methods for the time-integration of linear Maxwell’s equations. We first present error bounds for the abstract Cauchy problem which respect the unboundedness of the differential operators using energy techniques. The error bounds hold for algebraically stable and coercive methods such as Gauß and Radau collocation methods. The results for the abstr...

In this paper we present a new quadrature method for computing Galerkin stiiness matrices arising from the discretisation of 3D boundary integral equations using continuous piecewise linear boundary elements. This rule takes as points some subset of the nodes of the mesh and can be used for computing non-singular Galerkin integrals corresponding to pairs of basis functions with non-intersecting...

The objective of this paper is to present a comparative study of fitted-mesh finite difference method, Ritz-Galerkin finite element method and B-spline collocation method for a two-parameter singularly perturbed boundary value problems. Due to the small parameters ε and μ, the boundary layers arise. We have taken a piecewise-uniform fittedmesh to resolve the boundary layers and shown that fitte...

Boundary element methods (BEM) are used for forward computation of bioelectromagnetic fields in multi-compartment volume conductor models. Most BEM approaches assume that each compartment is in contact with at most one external compartment. In this work, I present a general surface integral equation and BEM discretization that remove this limitation and allow BEM modeling of general piecewise-h...

A STUDY OF THE FAST SOLUTION OF THE OCCLUDED RADIOSITY EQUATION KENDALL ATKINSON AND DAVID CHIEN Abstract. Consider the numerical solution of the radiosity equation over an occluded surface using a collocation method based on piecewise polynomial interpolation. Hackbusch and Nowak [12] proposed a fast method of solution for boundary integral equations, and we consider the application of their m...

We describe the methods and implementation of a general-purpose code, COLDAE. This code can solve boundary value problems for nonlinear systems of semi-explicit diierential-algebraic equations (DAEs) of index at most 2. Fully implicit index-1 boundary value DAE problems can be handled as well. The code COLDAE is an extension of the package COLNEW (COLSYS) for solving boundary value ODEs. The im...

A boundary integral operator method for stochastic ray tracing in billiards was recently proposed in [1]. In particular, a phase-space boundary integral model for propagating uncertain ray or particle flows was described and shown to interpolate between deterministic and random models of the flow propagation. In this work we describe discretisation schemes for this class of boundary integral op...

We discuss a possibility to construct high order methods on uniform or mildly graded grids for the numerical solution of linear Volterra integro-differential equations with weakly singular or other nonsmooth kernels. Using an integral equation reformulation of the initial value problem, we apply to it a smoothing transformation so that the exact solution of the resulting equation does not conta...

Most boundary element methods for two-dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions. Here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order. The equations are collocated at the br...