UC San Diego

This thesis comprises of the study of two physical problems. In the first half, we look at molecular dynamics (MD) simulation, which is widely used to study the motion and thermodynamic properties of molecules, and is applicable to a variety of problems in biochemistry, physics, and other fields. Computational limitations and the complexity of problems, however, result in the need for error quantification. We examine the inherent two-scale nature of MD to construct a large-scale dynamics approximation where internal motion of the atoms are approximated. This approximation is useful for evaluating the differences between full, classical MD simulations and those based on large-scale approximation schemes. We provide numerical results examining error in momenta, energy, and the macroscopic variables used in these large-scale dynamics as a means of error estimation. Accuracy of the macroscopic variables varies depending on the particular user-chosen approximation for the individual atomic motion. Our approximation conserves momenta and, although energy is not explicitly set to be conserved, fractional error is small.

In the second half, we turn our attention to another differential equation -- the level set equation, which is a popular approach to modeling evolving interfaces. The solver's ability to implicitly track moving fronts lends itself to a number of applications; in particular, our focus is on modeling high-explosive (HE) burn and detonation shock dynamics (DSD). We present a level set advection solver in two and three dimensions using the discontinuous Galerkin method with high-order finite elements. During evolution, the level set function is reinitialized to a signed distance function to maintain accuracy. Our approach leads to stable front propagation and convergence on high-order, curved, unstructured meshes. We provide results for two- and three-dimensional benchmark problems as well as applications to DSD.