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

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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.