Parallel Implicit-Explicit General Linear Methods

High-order discretizations of partial differential equations(PDEs)necessitate high-order time integration schemes capable of handling both stiff and nonstiff operators in an efficient manner.Implicit-explicit(IMEX)integration based on general linear methods(GLMs)offers an attractive solution due to their high stage and method order,as well as excellent stability properties.The IMEX characteristic allows stiff terms to be treated implicitly and nonstiff terms to be efficiently integrated explicitly.This work develops two systematic approaches for the development of IMEX GLMs of arbitrary order with stages that can be solved in parallel.The first approach is based on diagonally implicit multi-stage integration methods(DIMSIMs)of types 3 and 4.The second is a parallel generalization of IMEX Euler and has the interesting feature that the linear stability is independent of the order of accuracy.Numerical experiments confirm the theoretical rates of convergence and reveal that the new schemes are more efficient than serial IMEX GLMs and IMEX Runge-Kutta methods.

GIScience research challenges for realizing discrete global grid systems as a Digital Earth

Increasing data resources are available for documenting and detecting changes in environmental,ecological,and socioeconomic processes.Currently,data are distributed across a wide variety of sources(e.g.data silos)and published in a variety of formats,scales,and semantic representations.A key issue,therefore,in building systems that can realize a vision of earth system monitoring remains data integration.Discrete global grid systems(DGGSs)have emerged as a key technology that can provide a common multi-resolution spatial fabric in support of Digital Earth monitoring.However,DGGSs remain in their infancy with many technical,conceptual,and operational challenges.With renewed interest in DGGS brought on by a recently proposed standard,the demands of big data,and growing needs for monitoring environmental changes across a variety of scales,we seek to highlight current challenges that we see as central to moving the field(s)and technologies of DGGS forward.For each of the identified challenges,we illustrate the issue and provide a potential solution using a reference DGGS implementation.Through articulation of these challenges,we hope to identify a clear research agenda,expand the DGGS research footprint,and provide some ideas for moving forward towards a scaleable Digital Earth vision.Addressing such challenges helps the GIScience research community to achieve the real benefits of DGGS and provides DGGS an opportunity to play a role in the next generation of GIS.