DIRK Schemes with High Weak Stage Order

Runge-Kutta time-stepping methods in general suffer from order reduction: the observed order of convergence may be less than the formal order when applied to certain stiff problems. Order reduction can be avoided by using methods with high stage order. However, diagonally-implicit Runge-Kutta (DIRK) schemes are limited to low stage order. In this paper we explore a weak stage order criterion, which for initial boundary value problems also serves to avoid order reduction, and which is compatible with a DIRK structure. We provide specific DIRK schemes of weak stage order up to 3, and demonstrate their performance in various examples.Comment: 10 pages, 5 figure

Staircase-free finite-difference time-domain formulation for general materials in complex geometries

A stable Cartesian grid staircase-free finite-difference time-domain formulation for arbitrary material distributions in general geometries is introduced. It is shown that the method exhibits higher accuracy than the classical Yee scheme for complex geometries since the computational representation of physical structures is not of a staircased nature, Furthermore, electromagnetic boundary conditions are correctly enforced. The method significantly reduces simulation times as fewer points per wavelength are needed to accurately resolve the wave and the geometry. Both perfect electric conductors and dielectric structures have been investigated, Numerical results are presented and discussed

Convergent Cartesian grid methods for Maxwell's equations in complex geometries

A convergent second-order Cartesian grid finite difference scheme for the solution of Maxwell's equations is presented. The scheme employs a staggered grid in space and represents the physical location of the material and metallic boundaries correctly, hence eliminating problems caused by staircasing, and, contrary to the popular Yee scheme, enforces the correct jump-conditions on the field components across material interfaces. A detailed analysis of the accuracy of the new embedding scheme is presented, confirming its second-order global accuracy. Furthermore, the scheme is proven to be a bounded error scheme and thus convergent. Conditions for fully discrete stability is furthermore established. This enables the derivation of bounds for fully discrete stability with CFL-restrictions being almost identical to those of the much simpler Yee scheme. The analysis exposes that the effects of staircasing as well as a lack of properly enforced jump-conditions on the field components have significant consequences for the global accuracy. It is, among other things, shown that for cases in which a field component is discontinuous along a grid line, as happens at general two- and three-dimensional material interfaces, the Yee scheme may exhibit local divergence and loss of global convergence, To validate the analysis several one- and two-dimensional test cases are presented, showing an improvement of typically 1 to 2 orders of accuracy at little or no additional computational cost over the Yee scheme, which in most cases exhibits First order accuracy. (C) 2001 Academic Press

FDTD method for Maxwells equations in complex geometries

A stable second order Cartesian grid finite difference scheme for the solution of Maxwells equations is presented. The scheme employs a staggered grid in space and represents the physical location of the material and metallic boundaries correctly, hence eliminating problems caused by staircasing, and, contrary to the popular Yee scheme, enforces the correct jump-conditions on the field components across material interfaces. To validate the analysis several test cases are presented, showing an improvement of typically 1-2 orders of accuracy at little or none additional computational cost over the Yee scheme, which in most cases exhibits first order accuracy