Unsteady Stokes solvers, coupling stress and pressure forces, are a key component of accurate free surface simulators for highly viscous fluids. Because of the simultaneous application of stress and pressure terms, this creates a much larger system than the standard decoupled approach. We propose a reduced fluid model wherein interior regions are represented with incompressible polynomial vector fields. Sets of standard grid cells are consolidated into super-cells, each of which are modelled using only 26 degrees of freedom. This reduced model retains desirable behaviour of the full Stokes system with smaller computational cost.

We present the first application of a central scheme in an unstructured meshless code and extend it to limit diffusion in shearing flows.
Some numerical diffusion is required in simulations of compressible fluids to maintain stability and prevent formation of spurious structures. The Kurganov-Tadmor (KT) central scheme uses a signal velocity and a linear reconstruction of fields to limit numerical diffusion away from discontinuities.
We implement the KT scheme as a drop-in replacement for the Riemann solver in the GIZMO hydrodynamics code. Both the original finite-volume version of the KT scheme, which is quasi-Lagrangian in meshless geometry, as well as a new fully-Lagrangian finite-mass variant are presented. In addition, to mitigate excessive diffusion, a new shear-based switch is proposed. The new methods, as well as the default Riemann solver, were applied to a set of test problems. The results show that, although the KT scheme is more diffusive than the Riemann solver, it produces correct results with better convergence. The switch is shown to reduce diffusion in shearing cases, while not compromising stability in the supersonic regime. The fully-Lagrangian variant is shown to behave similarly to its Riemann solver counterpart. We conclude that the new variants of the KT scheme are good alternatives to Riemann solvers in meshless geometry, especially where its simplicity is desirable, such as for a complex equation of state.

Computational hydrodynamics is significant in the study of physical phenomena in astronomy, with the bulk of the universe’s matter being low-density gas.The Kurganov-Tadmor (KT) central scheme is a robust numerical flux scheme that performs well in simulating compressible fluids. It performs admirably at the supersonic regime, but is found to cause excessive diffusion, especially in the case of shearing problems. Diffusion is required to maintain stability and prevent formation of spurious structures, but too much suppresses even physically relevant structures.
A new method of controlling the diffusion in the KT scheme is proposed, based on determining whether the local flow is shearing or not. This is shown to be effective at reducing diffusion while retaining stability through an implementation in the GIZMO hydrodynamics code, which uses an unstructured geometry where fluid is simulated by representing it as a collection of particles whose movement represents the flow.
The method was tested using a range of problems, focusing on its behaviour in both shearing flows as well as in the supersonic regime. Results show that diffusion is noticeably reduced in shearing cases, while stability is not impacted in the supersonic regime. Additionally, it was shown to perform on-par with, or better than, other methods in a difficult test containing both shearing and supersonic components. Thus, the new method achieves the desired reduction of diffusion while maintaining the robustness of the KT scheme.

Numerical hydrodynamics attempts to capture the behaviour of physical fluid flow via numerical computation. The Euler fluid equations are the most commonly used mathematical representation of inviscid, compressible fluids, and one method of computing its solution numerically is called the Kurganov-Tadmor (KT) central scheme. Because of the limits of numerical analysis, however, challenges arise when attempting to solve ill-conditioned problems. Of interest are two such cases: singular gradients found in shocks, and the incompressible limit. Artificial viscosity is often used to stabilize methods in these ill-conditioned situations, but provides a challenge in determining how much is enough for stability, but not so much as to dominate over the physical result. An iterative improvement of the KT scheme will be presented, where a preconditioning method is applied to minimize the viscosity applied at subsonic cases, but keep its current stability in the supersonic regime.

Smoothed particle hydrodynamics (SPH) is a numerical method for solving Euler’s fluid equations. Originally developed for simulating gases in astrophysics, it has found uses in other fields including fluid flows and rigid body physics. SPH defines physical properties – density, velocity, and pressure – at points that follow the fluid field. These physical properties are used to interpolate values for the entire space. This interpolation, however, is inexact and leads to numerical errors in the calculation of momentum and energy fluxes between the particles. Methods for improving this error based off literature was assessed, and planned to be implemented into the SPH code, Gasoline.
A common, accurate class of methods for calculating fluxes are Riemann solvers. Due to their computational expense, however, they render simulations prohibitively slow in some cases – a faster, and possibly sufficiently accurate, alternative is the Kurganov-Tadmor (KT) central scheme. The KT scheme was initially used for grid simulations, thus making the upcoming implementation novel among SPH codes. It has been implemented into the hydrodynamics code, GIZMO, which currently uses a Riemann solver; and its viability will be analyzed via the use of standard test cases, where the KT scheme will be compared against that of a Riemann solver. Initial results are promising, but changes will be made due to the differences between grid codes and SPH. The implementation of the Kurganov-Tadmor central scheme into SPH is expected to yield improved results – accuracy close to exact Riemann solvers at significantly faster speeds.

Smoothed particle hydrodynamics (SPH) is a numeric computational method commonly used in astrophysics simulations, where physical properties are defined at points whose coordinates track the flow of the fluid rather than at a grid as is with a mesh-based method. Physical properties, such as density, pressure, and temperature, are interpolated from the values defined at nearby points using a smoothing function. As with any numerical solution, however, SPH has difficulty with mathematical singularities - of particular importance in hydrodynamics are shocks, discontinuities across which entropy increases. These can be seen in real life as waves produced by explosions and supersonic aircraft. Results for these discontinuities have traditionally been improved via the addition of an artificial viscosity, smearing it into a resolvable continuous curve. The behaviour of SPH on shocks of various strengths is analyzed, using varying degrees of artificial viscosity, as well as the effect of additional shock-specific thermal conduction. It was found that this addition yielded an improvement to shock results, but minor variations to the formulation were necessary to prevent detrimental effects on other cases, particularly shearing flows.

A pioneering and now ubiquitous model in population ecology, the Lotka-Volterra system describes the temporal dynamics of two interacting species as a pair of differential equations. While mathematically elegant, it suffers from being deterministic, demonstrating the periodicity of predator-prey relationships, but little else. Its simplicity prevents more thorough analysis and generalization for more complex ecosystems, calling for a more robust analogue.
The inaccuracies in classical Lotka-Volterra largely stem from its lack of discrete, spatial, and stochastic elements – a more modern model, applying these corrections, is the Stochastic Lattice Lotka-Volterra, which treats the deterministic rates as probabilities applied over a discrete N×N lattice. As the same interactions defined by classical Lotka-Volterra are performed a large number of times, it becomes a Monte Carlo approximation, with probabilities being equivalent to rates at the mean-field limit.
Applying this new model in C, my analysis finds that while it approximates classical Lotka-Volterra, it exhibits more consistency with known ecological theory. In particular, it demonstrates decreasing fluctuation amplitude over time, as well as a bifurcation between the coexistence and extinction states, whereas the classical model implies an inevitable, perfectly periodic, coexistence for any two-species system.
Novel modifications to the simulation provides insight into the spatiotemporal behaviours of different ecosystems, such as species invasiveness and a parity law in trophic interactions. Thus, by applying the Lotka-Volterra model in silica, with the inclusion of discrete, spatial, and stochastic elements, a more accurate and robust model is created, applicable for the study of a wider range of ecosystems.