General Linear Time-stepping Methods for Large Scale Simulations
Runge-Kutta(RK) and linear multistep (LM) methods have been extensively used for the integration of ordinary and partial differential equations (PDEs). Both families of methods have well known limitations. Stability requirements limit the efficiency attainable by any LM method, whereas RK methods suffer from accuracy reduction in the presence of stiffness and nonhomogeneous boundary and source terms. General linear (GL) time-stepping methods are generalizations of both RK and LM methods and therefore allow the development of new integration schemes with superior properties. However, GL methods have not been extensively studied in the context of time-dependent PDEs, and very little has been done to make this class of methods available for practical use. The proposed research seeks to fill this gap.
This research will investigate theoretically order conditions for a class of general linear methods of practical importance. This theory will be used to develop new high order methods that circumvent the efficiency and accuracy reduction due to boundaries, sources, and stiffness. A rigorous analysis of the stiff behavior will be carried out in a singular perturbation framework, and will be extended to index one differential algebraic systems. The proposed research is the first to address strong stability preserving GL schemes for hyperbolic systems. A framework for partitioned general linear schemes will be developed to address multiphysics problems. The new GL methods will be made available to the science and engineering community at large through a general purpose software package. Their performance will be illustrated on real life, multiscale, multiphysics simulations arising in the prediction of atmospheric pollution.
