Collaborative Research: Overcoming Order Reduction and Stability Restrictions in High-Order Time-Stepping

该项目开发了新的计算方法，弥补了现有时间步进方法的基本精度缺点，并增加了它们的稳定性和鲁棒性。各种各样的实际应用，包括流体流动、量子物理、热和中子输运、材料科学以及许多复杂的多物理场问题，都需要涉及时间演化的模型的数值模拟。这一次的进化必须以一种保持现代计算方法的高精度的方式进行。该项目解决了在这种情况下出现的基本挑战，并提供了优越的数值方法，可以取代目前在计算科学和工程实践中使用的现有时间步进方案。该项目提供多机构合作，包括两名早期职业研究人员，并涉及培养一名博士生。本课题主要研究高阶时间步进中的两个方面：龙格-库塔方法的降阶；无条件稳定ImEx线性多步方法。一个特别的焦点在于偏微分方程的时间步进。对于这些问题，阶数的降低可能与数值边界层有关，这是由多阶段时间步进方案引起的。基于对这一现象的几何理解，我们开发了降低阶数的补救措施。这包括弱阶序的概念，以及修正的边界条件。多步法提供了避免订单减少的另一种途径。这里的关键挑战是它们的限制性稳定性行为。基于ImEx多步骤方法的新稳定性理论，本项目开发了新的方案，可以在某些问题上实现无条件稳定性。通过对时间步进系数的简单修改，新方案可以包含在许多现有的计算代码中，从而使从业者能够仅基于精度考虑来选择时间步进。

This project develops new computational approaches that remedy fundamental accuracy shortcomings of existing time-stepping methods, and increase their stability and robustness. A wide variety of practical applications, including fluid flows, quantum physics, heat and neutron transport, materials science, and many complex multi-physics problems, require the numerical simulation of models that involve a time evolution. This time evolution must be performed in a way that the high accuracy of modern computational methods is retained. This project addresses fundamental challenges that arise in this context, and delivers superior numerical methods that could replace existing time-stepping schemes currently used in computational science and engineering practice. This project provides a multi-institution collaboration, including two early-career researchers, and it involves the training of a PhD student.The research in this project addresses two aspects in high-order time-stepping: order reduction in Runge-Kutta methods; and unconditionally stable ImEx linear multistep methods. A specific focus lies on time-stepping for partial differential equations. For those, order reduction can be associated with numerical boundary layers, caused by multi-stage time-stepping schemes. Based on this geometric understanding of the phenomenon, remedies for order reduction are developed. This includes the concept of weak stage order, as well as modified boundary conditions. An alternative avenue to avoid order reduction is provided by multistep methods. The key challenge here is their rather restrictive stability behavior. Based on a new stability theory for ImEx multistep methods, this project develops novel schemes that can, for certain problems, achieve unconditional stability. The new schemes can be included into many existing computational codes via a simple modification of the time-stepping coefficients, thus enabling practitioners to select the time step based solely on accuracy considerations.