Numerical modelling of elastic wave propagation: honoring fluid-solid interfaces in finite-difference and pseudospectral methods

Mathematical models of physical processes are strongly affected by how real-world media are parameterized, leading to numerical error. In models of elastic-wave propagation, interfaces between different media are a notorious source of error. For instance, when complex interfaces are represented by a regular (Cartesian) grid, slopes are turned into "staircases", resulting in spurious reflections that do not reflect any real-world wave propagation phenomenon. These are more severe the sharper the interface, and particularly so at interfaces between fluid and solid materials. Finite-difference and pseudospectral solvers inevitably suffer from such effects, and yet are widely used in the literature, based on the assumption that errors can be arbitrarily reduced by enhancing numerical grid resolution. My thesis addresses these errors quantitatively, first in two dimensions, using both time-domain and pseudospectral methods, then in three dimensions, limited to the pseudospectral method. We first generated "ground-truth" data, modeling one flat interface separating a uniform fluid from a uniform solid medium, and choosing our Cartesian reference frame and numerical grid to be perfectly aligned with the interface -- resulting in no "staircase" effects. We then repeated the same simulation iteratively, each time rotating both reference frame and numerical grid by a growing angle. Differences between simulation results could be entirely ascribed to the inadequacy of interface parameterization. From our analysis, a pattern emerged, showing a clear dependence of numerical error on the angle between the interface and the horizontal plane in the unperturbed reference frame. Despite the pseudospectral method being more accurate and performing better than simple finite differences, errors associated with fluid-solid interfaces have a deep impact on the final results, not only quantitatively but also qualitatively. This problem can be addressed by increasing resolution, but this results in a very quick growth in computational costs.