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)方法来求解暂态双曲型和奇摄动对流扩散问题。这些方法使用不连续的碱基，因此在捕捉不连续时非常有效。此外，它们简化了自适应h-、p-和hp-加密，易于在边界不规则的非结构网格上实现，非常好地并行化，并满足局部(单元级别)守恒原则。研究人员研究了h-orp精化下离散化误差的渐近校正误差估计，它提供了非常可靠的解精度度量。给出了线性、非线性双曲型和奇摄动对流扩散问题的后验误差估计。研究人员在从高级超级计算机到工作站集群的现代并行系统上实现了这些大规模问题的eDG方法。并行过程必须解决处理器、内存和通信级别的异构性，因为适应性使问题变得非常复杂。在自适应h-或p-精化下，平衡并行计算可能会停止。在求解过程中必须动态地重新分配计算负载，以恢复和维护平衡。在这种异质环境中，很难解决在处理器之间迁移工作以最小化计算时间的过程。复杂逼真的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，在最快的超级计算机上需要很长的时间。研究人员开发了基于不连续Galerkin方法的非常高效和可靠的计算技术，这种方法在并行和网络计算方面应该具有优势。结果的准确性和可靠性是由作为计算的一部分自动获得的解误差估计来指导的。它们被用来创建自适应程序，将计算资源分配给最需要它们的地区。适应性提供的效率导致更准确的计算机模拟、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.