Mathematical modeling of physical processes often leads to systems of differential and algebraic equations involving quantities of interest. A computer model created based on these equations can be numerically integrated to predict future states of the system and its evolution in time. This thesis investigates current methods in numerical time-stepping schemes, identifying a number of important features needed to speed up and increase the accuracy of the solutions. The focus is on developing new methods suitable for large-scale applications with multiple physical processes, potentially with significant differences in their time-scales. Various families of new methods are introduced with special attention to multirating, low computational cost implicitness, high order of convergence, and robustness. For each family, the order condition theory is discussed and a number of examples are derived. The accuracy and stability of the methods are investigated using standard analysis techniques and numerical experiments are performed to verify the abilities of the new methods.
Doctor of Philosophy
Mathematical descriptions of physical processes are often in the form of systems of differential equations describing the time-evolution of a phenomenon. Computer simulations are realizations of these equations using well-known discretization schemes. Numerical time-stepping methods allow us to advance the state of a computer model using a sequence of time-steps. This thesis investigates current methods in time-stepping schemes, identifying a number of additional features needed to improve the speed and accuracy of simulations, and devises new methods suitable for large-scale applications where multiple processes of different physical nature drive the equations, potentially with significant differences in their time-scales. Various families of new methods are introduced with proper mathematical formulations provided for creating new ones on demand. The accuracy and stability of the methods are investigated using standard analysis techniques. These methods are then used in numerical experiments to investigate their abilities.