Adaptive Discontinuous Galerkin Methods of Transient Partial Differential Equations

研究人员发展了非常有效的自适应间断Galerkin（DG）方法来求解瞬态双曲和奇摄动对流扩散问题。 这些方法使用不连续的基地，因此是非常有效的捕捉间断。 此外，它们简化了自适应h-，p-和hp-细化，易于在具有不规则边界的非结构化网格上实现，并行化非常好，并满足局部（元素级）守恒原则。 研究人员研究渐近正确的误差估计下的离散化误差h-orp-细化，提供了非常可靠的措施solutionaccuracy。 对线性、非线性、双曲和奇摄动对流扩散问题给出了后验误差估计。 研究人员在从先进的超级计算机到工作站集群的现代并行系统上实现这些DG方法来解决大规模问题，并行程序必须解决处理器、内存和通信层面的异质性，因为自适应性使问题变得非常复杂。 一个平衡的并行计算在自适应h-或p-细化下可能会停止或保持平衡。 计算负载必须在解决方案过程中动态地重新分配，以恢复和保持平衡。 在这样一个异构的系统中，为了使计算时间最小化而在处理器之间进行工作迁移的程序几乎没有得到解决。 复杂的现实物理问题的计算机模拟，如车辆的流动，天气预报和材料处理，需要在最快的超级计算机上花费很长时间。 研究人员开发了非常有效和可靠的计算技术的基础上不连续伽辽金方法，应该有并行和网络计算的优势。 结果的准确性和可靠性由作为计算的一部分自动获得的解决方案误差估计指导。 这些被用来创建自适应程序，将计算资源分配到最需要的区域。 自适应提供的效率导致更精确的计算机模拟，更好的产品和更短的设计周期。

The investigators develop very efficient adaptivediscontinuous Galerkin (DG) methods for transient hyperbolic andsingularly perturbed convection-diffusion problems. Thesemethods use discontinuous bases and thus are very effective atcapturing discontinuities. In addition, they simplify adaptiveh-, p-, and hp-refinement, are simple to implement onunstructured meshes with irregular boundaries, parallelize verywell, and satisfy local (element level) conservation principles.Adaptive enrichment by h-, p-, and r-refinement or combinationsthereof has typically been guided by a posteriori estimates ofdiscretization errors. The investigators study asymptoticallycorrect error estimates of discretization errors under h- orp-refinement that provide very reliable measures of solutionaccuracy. A posteriori error estimates are developed for linearand nonlinear hyperbolic and singularly perturbedconvection-diffusion problems. The investigators implement theseDG methods for large-scale problems on modern parallel systemsranging from advanced supercomputers to clusters of workstations.Parallel procedures must address heterogeneity at the processor,memory, and communications levels as adaptivity greatlycomplicates matters. A balanced parallel computation may ceaseto remain so under adaptive h- or p-refinement. Computationalloads must dynamically be redistributed during the solutionprocess to restore and maintain balance. Procedures to migratework between the processors so as to minimize the computationaltime have hardly been addressed in such a heterogeneousenvironment. Complex realistic computer simulations of physical problemslike the flow about a vehicle, weather prediction, and materialsprocessing require long times on the fastest supercomputers. Theinvestigators develop very efficient and reliable computationaltechniques based on the discontinuous Galerkin method that shouldhave advantages for parallel and network computation. Accuracyand reliability of the results are guided by estimates ofsolution errors that are automatically obtained as part of thecomputation. These are used to create adaptive procedures thatassign computational resources to regions where they are neededmost. The efficiency provided by adaptivity leads to moreaccurate computer simulations, better products, and shorterdesign cycles.