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陷，并提高了其稳定性和鲁棒性。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，材料科学和许多复杂的多物理问题，需要涉及时间演化的模型的数值模拟。这种时间演化必须以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根本性挑战，并提供了上级数值方法，可以取代目前在计算科学和工程实践中使用的现有的时间步进计划。该项目提供了一个多机构的合作，包括两个早期的职业生涯的研究人员，它涉及到一个博士生的培训。在这个项目中的研究解决了高阶时间步长的两个方面：减少的Runge-Kutta方法和无条件稳定的ImEx线性多步方法。一个具体的重点在于时间步进偏微分方程。对于那些，可以与数值边界层，所造成的多级时间步进格式的订单减少。基于这种几何理解的现象，补救措施减少。这包括弱级阶的概念，以及修改的边界条件。多步法提供了避免降阶的另一种途径。这里的关键挑战是它们相当有限的稳定性行为。基于ImEx多步法的新稳定性理论，该项目开发了新的方案，对于某些问题，可以实现无条件稳定。新的计划可以包括到许多现有的计算代码通过一个简单的修改的时间步进系数，从而使从业者选择的时间步长的基础上，仅仅考虑精度。

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.