Mathematical Sciences: Propagating Fronts By A Consistent Primitive Algorithm With Application To Bubble Dynamics

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The investigator will develop new numerical methods to model weak solutions to nonlinear hyperbolic differential equations derived from a set of conservation laws. Conservation form of the system is widely held as imperative for consistent numerical capturing of shocks. In practice, while this may be true for flows dominated by strong shocks, other flow phenomena may be better modelled by different, not necessarily conservative flow descriptions. In such cases, one may be prepared to sacrifice strict conservation in favor of a nonstrictly conservative model that describes better the dominant flow features. To be reliable, the model should generate acceptably small conservation errors. The investigator shows that propagating fronts is one such phenomenon. Capturing propagating fronts by conservative models gives rise to oscillations and other inaccuracies near material fronts, and is in general unsatisfactory. These are not the common oscillations associated with high order schemes, but rather a result of the simple wave model (i.e. eigenstructure) of the conservative systems. A remedy is to use a nonconservative flow description, which offers a different eigenstructure. It becomes exact near contact surfaces and produces clean oscillation-free interfaces. Near shocks, the model is not exact and conservation errors are committed. A technique recently developed by the investigator offers a cure to this problem. It uses viscous perturbations that render the algorithm conservative to the order of numerical approximation and generate acceptably small conservation errors. The technique has been successfully applied to computing propagating fronts in 1D. The investigator will study several theoretical aspects of this nonstrictly conservative algorithm and its application to the computation of bubble dynamics in several space dimensions. The work of the project involves theoretical analysis and computer simulation of flows. These arise in physical phenomena involving mixing of fluids, the initiation and evolution of vortices, the ignition and evolution of chemical reactions in fluids, and the propagation of flames. The aim of the project is to develop analytical and computational tools to enhance the understanding of the complicated physical phenomena involved. Insight gained this way will supplement experimental data, which are often expensive, difficult or even dangerous to obtain. For example, wind tunnel simulations of space conditions in the upper stratosphere are difficult and expensive to perform and may give unreliable results; studying gun ballistics requires experimenting with strong explosions, which are both expensive and dangerous. Potential impact of this project extends to a variety of industries including the space and aircraft industries (design of space vehicles and of aicraft wings), marine industry (design of boats with improved hydrodynamic properties, the effect of bubbles on submarine hydrodynamics, etc.) and the weapon industries (e.g., internal ballistics of guns).