Simulation, computation and prediction in science and engineering (for instance, fluid flow, elasticity, electromagnetics, weather prediction etc.) often require the solution of large-scale partial differential equation (PDE) models.  High-order accurate methods for solving PDEs are of increasing importance as a means to reduce overall computational cost. This project produced new approaches for solving the time variable component of a PDE by addressing two fundamental problems in existing approaches.   

Firstly, Runge-Kutta (RK) methods, which constitute some of the most widely used time-integration methods for differential equations suffer from a reduction in accuracy known as order reduction. Order reduction seriously limits the use of RK schemes for PDEs with time-dependent input data. This project identified the origin of order reduction as a singular perturbation problem, and devised new formulas for the associated computational error.  This resulted in new theoretical remedies (known as weak stage order) that avoid the order reduction error, as well as the development of new Runge-Kutta methods that remedy order reduction.  The new schemes may now be used by others to avoid order reduction in certain classes of problems, and may easily be incorporated into existing codes.  

Secondly, this project developed a new stability theory for a large class of time-integration methods known as IMEX schemes for linear multistep methods (LMMs).  Unlike RK methods, LMMs do not suffer from order reduction, however have less desirable stability properties.  Stability is a required condition that is necessary for a computer to accurately solve a differential equation model.  IMEX schemes are particularly important as they admit desirable stability properties through their IMplicit part, while still maintaining efficiency from their EXplicit part. The new theory identified exactly how one may satisfy the stability requirement while maximizing design flexibility.  New time-integration schemes with desirable stability properties were also developed and applied to models including the Navier-Stokes equations in fluid mechanics, and nonlinear diffusion equations in porous media.  Together the new theory, methods, and applications have extended the practical benefits of IMEX schemes. 

This project was a collaboration between three institutions (MIT, Temple, and NJIT), and led to the education of multiple undergraduate and graduate students in computational science, including both theoretical aspects and high-performance computing.  Developments from the grant have also influenced advanced undergraduate and graduate level courses at the three institutions. New computational codes have been made publicly available through journals, and the (freely available) academic Arxiv database.