In this thesis, we survey some popular pricing methods of average options. They can be classified into three cateogries include approximation, Monte Carlo, and binomial tree approaches. We examine the accuracy of these methods by two cases, exchange rate and stock price. Numerical testing results show the accuracy of approximation and binomial tree are not stable. For the big-size feature of average option, their outputs are doubtful and damaging in pactice. Despite this, they are still valuable. This is because they own the other advantages. For example, the approximation approach can give us a quick formlas to calculate the Greek, and the binomial tree approach can price the American style options.

We present a series of numerical galaxy formation studies which apply new numerical methods to produce increasingly realistic galaxy formation models. We first investigate the metallicity evolution of a large set of idealized hydrodynamical galaxy merger simulations of colliding galaxies. We find that inflows of metal--poor interstellar gas triggered by galaxy tidal interactions can account for the systematically lower central oxygen abundances observed in local interacting galaxies. We show the central metallicity evolution during merger events is determined by a competition between the inflow of low--metallicity gas and enrichment from star formation. We find a time-averaged depression in the galactic nuclear metallicity of ~0.07 dex for gas--poor disk--disk interactions, which explains the observed close pair mass-metallicity and separation-metallicity relationships. / Astronomy

Recent work by a number of researchers has highlighted areas in which conservative numerical methods give poor solutions. One such situation is in the modelling of material interfaces. A number of methods for overcoming this shortfall of conservative numerical methods are developed. The flow situations that are considered include multicomponent gases and systems of gases and liquids. It is shown that the errors associated with conservative methods when applied to model gas-liquid interfaces are considerably larger than those for gas-gas interfaces. The first approach used for overcoming the errors in conservative methods is a hybrid primitive-conservative method. This method is used in conjunction with a number of new Riemann solvers for a liquid ambient to provide accurate solutions to a number of challenging one and two dimensional test problems. These test problems include the interaction of a shock wave with a bubble in a gas and an underwater explo.; ion. The application of these hybrid methods to the problem of the interaction of a shock wave with a gas bubble in aa liquid demonstrate that they are unable to provide an accurate solution. Two one dimensional methods are described that are able to provide solutions to such test problems. These methods are the moving grid-Chimera approach and a cut cell approach. The cut cell approach is extended into two dimensions and is shown to be able to provide solutions to the problem of the interaction of a shock wave with a gas bubble in a liquid. This method is also shown to be able to provide more accurate solutions to multicomponent gas problems than those on a standard Cartesian grid.

Explosions are a common phenomena in the Universe. Beginning with the Big Bang, one could say the history of the Universe is narrated by a series of explosions. Yet no matter how large, small, or complex, all explosions occur through a series of similar physical processes beginning with their initiation to their dynamical interaction with the environment. Of particular interest to this study is how these processes are modified in a magnetized medium. The role of the magnetic field is investigated in two scenarios. The first scenario addresses how a magnetic field alters the propagation of a gaseous detonation where the application of interest is the modification of a condensed-phase explosion. The second scenario is focused on the aftermath of the explosion event and addresses how fluid mixing changes in a magnetized medium. A primary focus of this thesis is the development of a numerical tool capable of simulating explosive phenomenon in a magnetized medium. While the magnetohydrodynamic (MHD) equations share many of the mathematical characteristics of the hydrodynamic equations, numerical methods developed for the conservation equations of a magnetized plasma are complicated by the requirement that the magnetic field must be divergent free. The advantages and disadvantages of the proposed method are discussed in relation to explosion applications.

We present two self-consistent non-parametric models of the local cosmic velocity field based on the density distribution in the PSCz redshift survey of IRAS galaxies. Two independent methods have been applied, both based on the assumptions of gravitational instability and linear biasing. They give remarkably similar results, with no evidence of systematic differences and an r.m.s discrepancy of only ~ 70 kms(^-1) in each Cartesian velocity component. These uncertainties are consistent with a detailed independent error analysis carried out on mock PSCz catalogues constructed from TV-body simulations. The denser sampling provided by the PSCz survey compared to previous IRAS galaxy surveys allows us to reconstruct the velocity field out to larger distances. The most striking feature of the model velocity field is a coherent large-scale streaming motion along a basehne connecting Perseus-Pisces, the Local Supercluster, the Great Attractor, and the Shapley Concentration. We find no evidence for back-infall onto the Great Attractor. Instead, material behind and around the Great Attractor is inferred to be streaming towards the Shapley Concentration, aided by the expansion of two large neighbouring un- derdense regions. The PSCi model velocities compare well with those predicted from the 1.2-Jy redshift survey of IRAS galaxies and, perhaps surprisingly, with those predicted from the distribution of Abell/ACO clusters, out to 140 h(^-1)Mpc. Comparison of the real-space density fields (or, alternatively, the peculiar velocity fields) inferred from the PSCz and cluster catalogues gives a relative (linear) bias parameter between clusters and IRAS galaxies of b(_c) = 4.4 ± 0.6. In addition, we compare the cumulative bulk flows predicted from the PSCz gravity field with those measured from the MarkIII and SFI catalogues of peculiar velocities. A conservative estimate of β = Ω(_0)(^0.6)/b, where b is the bias parameter for IRAS galaxies, gives β= 0.76 ± 0.13 (1-σ), in agreement with other recent determinations. Finally, we perform a detailed comparison of the IRAS PSCz and 1.2-Jy spherical harmonic coefficients of the density and velocity fields in redshift space. Both the monopole terms of the density and velocity fields predicted from the surveys show some inconsistencies. The mismatch in the velocity monopole terms is resolved by masking the 1.2-Jy survey with the PSCz mask and using the galaxies within the PSCz survey for fluxes larger than 1.2 Jy. Davis, Nusser and Willick (1996) have found a discrepancy between the IRAS 1.2-Jy survey gravity field and the MarkIII peculiar velocity field. We conclude that the use of the deeper IRAS PSCz catalogue cannot alone resolve this mismatch.

The resolution of differential equations of diverse degree of complexity is necessary to simulate the phenomena present in the complex turbomachinery flows and in particular, requires accounting for unsteady effects that may have a preponderant role. Today, only the LES (Large Eddy Simulation) fully compressible approach has the required accuracy to predict the physics associated to reactive and turbulent flows in such complex geometries. This work covers the numerical modelling of physics in the near-wall region of a high-pressure turbine blade with special focus on thermal predictions. This work was supported by the European project COPA-GT, dedicated to the numerical multi-physics simulation of a complete gas turbine.

This thesis explores higher order numerical methods for solving fractional differential equations.

Stochastic Differential Equations (SDE) offer a rich framework to model the probabilistic evolution of the state of a system. Numerical approximation methods are typically needed in evaluating relevant Quantities of Interest arising from such models. In this dissertation, we present novel effective methods for evaluating Quantities of Interest relevant to computational finance when the state of the system is described by an SDE.

Shocks form in the solutions of first-order nonlinear hyperbolic PDEs with constant co-efficients. Where solitary waves arise in the solutions of first-order nonlinear hyperbolic PDEs with variable coefficients, those solitary waves occur due to the coupling of nonlinearity and dispersive effects that comes from the medium’s heterogeneity. In this thesis, we study a fluid that propagates in a narrow pipe with periodically-varying cross-sectional area described by a system of first-order nonlinear hyperbolic PDEs. Multiple-scale perturbation theory is applied to derive homogenized effective equations, which take the form of a constant-coefficient system including higher-order dispersive terms. We investigate the behavior of the solution by deriving the linear dispersion relation of the homogenized system. The homogenized equations are solved using a psuedospectral discretization in space and explicit Runge-Kutta method in time. Lastly, we develop a Riemann solver in Clawpack to solve the variable coefficients system and compare the obtained solution with the homogenized equations solution.

Differential equations are frequently used for modeling systems in the physical sciences, biology, and other important real-world disciplines. Oftentimes, however, these equations cannot be solved exactly, so suitable computer algorithms are necessary to provide an approximated solution. While these computational simulations fail to exactly represent all behaviors of the true solution, they can be constructed to exactly, or very closely, reproduce certain properties which are key to the physical or scientific applications of a problem. This paper explores a computational method specifically constructed for modeling the behavior of systems with linear damping, or a reduction of energy, introduced in them. The method was designed to be conformal symplectic, and closely reproduce dissipation of physical properties such as linear and angular momentum, mass, and energy, caused by the damping. The algorithm was constructed in such a way that it maintains low computational cost to implement. Additionally, the method demonstrates favorable accuracy and stability properties in simulation. The method can also handle more complex scenarios, such as systems with forcing terms, and nonlinear systems. In these cases, it has been shown to hold advantages over other commonly used methods in particular circumstances.