UC Riverside

Numerical simulations commonly store field data on a regular Eulerian grid, but the grid alone is rarely enough. For instance, particle tracing and semi-Lagrangian advection must evaluate the field at arbitrary locations. In those settings, accuracy depends on how faithfully discrete data and the continuum are connected, a connection most fragile where irregular boundaries intersect the grid data. This dissertation develops methods for two problems of this kind.The first subject is the interpolation of divergence-free vector fields from MAC grid data over irregular domains. Preserving this property under interpolation ensures that the reconstructed field remains consistent with the incompressibility of the underlying flow. Spline-based divergence-free interpolation schemes are efficient on regular grids but break down near irregular boundaries, leaving a gap region in which the divergence-free property is lost. We construct interpolation schemes that fill this gap: over triangles and squares in two dimensions, and over tetrahedra and and cubes in three dimensions, completing the incomplete divergence-free polynomial basis with rational functions. In three dimensions, cut cells are divided by a specially constructed marching-cubes table producing non-inverted tetrahedra that match continuously across grid cells. The resulting interpolation is analytically divergence-free, C0 continuous everywhere, local, and compatible with prescribed domain boundary velocities.The second subject is the initialization of particle states for the Material Point Method, where inconsistent particle volumes and positions produce spurious startup motion which we refer to as the boiling effect. We express the inconsistency as a residual between particle-rasterized nodal volumes and target nodal volumes of the domain discretization and eliminate it before the first time step, either by repositioning the particles through a Gauss-Newton iteration or by adjusting their volumes through a regularized least-squares solve. The target volumes are computed in three ways: a radially symmetric closed-form approximation, exact tensor-product integration for box domains, and numerical quadrature. Experiments on square and cube domains show that their accuracy where walls meet is decisive: the exact and quadrature targets stop the boiling effect in two and three dimensions, whereas the radial approximation succeeds only in two.