A Numerical Study of Diagonally Split Runge–Kutta Methods for PDEs with Discontinuities

Diagonally split Runge–Kutta (DSRK) time discretization methods are a class of implicit time-stepping schemes which offer both high-order convergence and a form of nonlinear stability known as unconditional contractivity. This combination is not possible within the classes of Runge–Kutta or linear multistep methods and therefore appears promising for the strong stability preserving (SSP) time-stepping community which is generally concerned with computing oscillation-free numerical solutions of PDEs. Using a variety of numerical test problems, we show that although second- and third-order unconditionally contractive DSRK methods do preserve the strong stability property for all time step-sizes, they suffer from order reduction at large step-sizes. Indeed, for time-steps larger than those typically chosen for explicit methods, these DSRK methods behave like first-order implicit methods. This is unfortunate, because it is precisely to allow a large time-step that we choose to use implicit methods. These results suggest that unconditionally contractive DSRK methods are limited in usefulness as they are unable to compete with either the first-order backward Euler method for large step-sizes or with Crank–Nicolson or high-order explicit SSP Runge–Kutta methods for smaller step-sizes. We also present stage order conditions for DSRK methods and show that the observed order reduction is associated with the necessarily low stage order of the unconditionally contractive DSRK methods. The work of C.B. Macdonald was partially supported by an NSERC Canada PGS-D scholarship, a grant from NSERC Canada, and a scholarship from the Pacific Institute for the Mathematical Sciences (PIMS). The work of S. Gottlieb was supported by AFOSR grant number FA9550-06-1-0255. The work of S.J. Ruuth was partially supported by a grant from NSERC Canada.     

Reduced Collocation Methods: Reduced Basis Methods in the Collocation Framework

In this paper, we present the first reduced basis method well-suited for the collocation framework. Two fundamentally different algorithms are presented: the so-called Least Squares Reduced Collocation Method (LSRCM) and Empirical Reduced Collocation Method (ERCM). This work provides a reduced basis strategy to practitioners who prefer a collocation, rather than Galerkin, approach. Furthermore, the empirical reduced collocation method eliminates a potentially costly online procedure that is needed for non-affine problems with Galerkin approach. Numerical results demonstrate the high efficiency and accuracy of the reduced collocation methods, which match or exceed that of the traditional reduced basis method in the Galerkin framework.     

A nonlinear galerkin method: The two-level fourier-collocation case

In this article we study the implementation of the Nonlinear Galerkin method as a multiresolution method when a two-level Fourier-collocation discretization is used. The set of collocation points with an even number of points is considered as the fine grid and decomposed into two coarse grids containing half as many points. Using these two grids we decompose the unknown into the sum of a large scale component containing only low frequency modes and based on one of the coarse grids and a small scale component containing only high frequency modes and based on the other coarse grid. This produces interesting connections between the physical space and the Fourier space representations of the function. A nonlinear Galerkin scheme is then applied to the Burgers equation; finally, implementation issues are discussed showing the advantages of the method.     

A Fifth Order Flux Implicit WENO Method

The weighted essentially non-oscillatory (WENO) method is an excellent spatial discretization for hyperbolic partial differential equations with discontinuous solutions. However, the time-step restriction associated with explicit methods may pose severe limitations on their use in applications requiring large scale computations. An efficient implicit WENO method is necessary. In this paper, we propose a prototype flux-implicit WENO (iWENO) method. Numerical tests on classical scalar equations show that this is a viable and stable method, which requires appropriate time-stepping methods. Future study will include the examination of such methods as well as extension of iWENO to systems and higher dimensional problems.Sigal Gottlieb - The work of this author supported by NSF grant DMS-0106743.Steven J. Ruuth - The work of this author was partially supported by a grant from NSERC Canada.     

Mutations in the carboxyl terminus of the agouti protein decrease agouti inhibition of ligand binding to the melanocortin receptors

Several mutations that cause ectopic expression of the agouti gene result in obesity, hyperinsulinemia, and yellow coat color. A candidate pathway for agouti induced obesity and hyperinsulinemia is through altered signaling by melanocortin receptors, as agouti normally regulates coat coloration through antagonism of melanocortin receptor 1. Furthermore, melanocortin peptides mediate functions including steroidogenesis, lipolysis, and thermoregulation. We report apparent inhibition dissociation constants for mouse and human agouti protein inhibition of ligand binding to the melanocortin receptors, to determine which of these receptors might be involved in agouti induced diabetes. The similarity in the apparent K(I) values for agouti inhibition of ligand binding to the brain melanocortin receptors 3 and 4 (mouse: K(I) app = 190 +/- 74 and 54 +/- 18 nM; human: K(I) app = 140 +/- 56 and 70 +/- 18 nM, respectively) suggests that the MC3-R is a potential candidate for a receptor mediating the effects of agouti protein overexpression. Agouti residues important for melanocortin receptor inhibition were identified through the analysis of deletion constructs and site-specific variants. Val83 is important for inhibition of binding to MC1-R (K(I) app for Val83Ala agouti increased 13-fold relative to wild-type protein). Arg85, Pro86, and Pro89 are important for selective inhibition of binding between MC1-R and MC3-R and MC4-R as their apparent K(I) values are essentially unchanged at MC1-R, while they have increased 6-10-fold relative to wild-type protein at MC3-R and MC4-R.