Efficient high-order accurate numerical schemes for stationary statistical behavior of complex systems

Complex systems arising in nature and society often exhibit chaotic behavior, making it impossible to accurately predict their long-term evolution from individual trajectories. Nevertheless, many important questions, including extreme weather patterns, oceanic phenomena, traffic congestion, sudden changes in electric power systems, and complex fluid dynamics, depend on understanding their long-term statistical behavior rather than precise prediction of every future state. This project develops new computational approaches that enable reliable and efficient prediction of such statistical patterns in complex systems. The outcomes will advance the mathematical and computational foundations for studying turbulence, geophysical flows, and other complex systems where long-term uncertainty is unavoidable. The project will also support education and workforce development through the training of undergraduate and graduate students, the development of educational materials, and the dissemination of computational tools that can benefit the broader scientific community. The project develops efficient, high-order accurate numerical methods for computing stationary statistical behavior of infinite-dimensional chaotic dissipative dynamical systems. The research focuses on designing mean-reverting scalar auxiliary variable methods combined with superconvergent hybridizable discontinuous Galerkin spatial discretizations to achieve unconditional long-time stability and preserve the essential dynamical structures of the underlying systems. The methods will be analyzed to establish convergence of discrete global attractors and invariant measures toward those of the continuous systems. The project further develops higher-order time integration methods, ensemble algorithms for uncertainty quantification and parameter-dependent computations, and adaptive time-stepping strategies for long-time simulations. These developments address fundamental challenges in numerical analysis of chaotic systems, including efficient computation of invariant statistics, preservation of long-time dynamics, and reliable simulation of intermittent and extreme events in complex fluid models. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.