This thesis explores the application of mimetic operators in solving the Navier-Stokes equations, focusing on the Boussinesq approximation and its implications for computational fluid dynamics. Mimetic operators, acclaimed for their unique ability to preserve crucial properties of the underlying physics, present a promising avenue for accurate and efficient numerical simulations.

The work begins by explaining mimetic operators, highlighting their mathematical foundations and significance in discretizing partial differential equations while maintaining fundamental properties such as conservation laws and divergence-free constraints. It then shows that solving the Navier-Stokes equations with mimetic operators provides a computationally simplified yet effective framework for modeling buoyancy-driven flows in stratified fluids, underlining the mimetic operator's adaptability to complex geometries.

Furthermore, the thesis addresses the parallelization of computational codes leveraging mimetic operators, aiming to exploit modern high-performance computing architectures for efficient simulations. Parallelization strategies and optimizations are discussed, focusing on scalability and computational efficiency.

A series of test cases are investigated to validate the proposed methodology, encompassing canonical flow problems and applications relevant to ocean modeling. The thesis presents detailed numerical experiments, comparing simulation results with analytical solutions, empirical data, and existing numerical benchmarks. In particular, emphasis is placed on the steps toward validating the ocean model, highlighting the capability of mimetic operators to capture complex oceanic phenomena while maintaining computational efficiency accurately.

Overall, this thesis not only contributes to the advancement of computational fluid dynamics but also offers practical insights. It integrates mimetic operators in solving the Navier-Stokes equations, providing a deeper understanding of their application, parallelization, and validation in the context of buoyancy-driven flows and ocean modeling.

Dynamical graph grammars (DGGs) are capable of modeling and simulating the dynamics of complex biological systems using an exact simulation algorithm derived from a master equation; however, the exact method is slow for large systems. To accelerate the simulations of DGGs we have developed an approximate simulation algorithm that is compatible with the DGG formalism. The approximate simulation algorithm uses a spatial decomposition of the domain at the level of the system's time-evolution operator, to gain efficiency at the cost of some rules or reactions firing out of order, which may introduce errors. The decomposition is coarsely partitioned by effective dimension (d = 0 to 2 or 0 to 3), to expose the potential for exact parallelism between different subdomains within a dimension, where most computing will happen, and to confine errors to interactions between adjacent subdomains of different effective dimensions. Additional efficiency can be achieved through maintaining an incrementally updated match data structure for all possible rule matches. To demonstrate these principles we have developed the Dynamical Graph Grammar Modeling Library (DGGML), and two DGG models for the plant cell cortical microtubule array (CMA). In the first model, we find evidence indicating that the initial formulation of the approximate algorithm is substantially faster than the exact algorithm, and one experiment leads to network formation in the long-time behavior, whereas another leads to a long-time behavior of local alignment. In the second model, we restrict ourselves to the CMA in the periclinal face of a plant cell and explore the effects that different face shapes and boundary conditions have on local and global alignment. In the case of a square face shape, we find the array orientation to be multi-modal, and in the case of a rectangular face shape, we find that different boundary conditions reorient the array mainly between the long and short axes. The periclinal CMA DGG demonstrates the flexibility and utility of DGGML and highlights its viability to be used as a computational means of testing, screening or inventing hypotheses to explain emergent phenomena.