Stochastic volatility models on option pricing have received much study following the discovery of the non-at implied surface following the crash of the stock markets in 1987. The most widely used stochastic volatility model is introduced by Heston (1993) because of its ability to generate volatility satisfying the market observations, being non-negative and mean-reverting, and also providing a closed-form solution for the European options. However, little research has been done on Heston model used to price early-exercise options. This presumably is largely due to the absence of a closed-form solution and the increase in computational requirement that complicates the required calibration exercise. This thesis examines the performance of the Heston model versus the Black-Scholes model for the American Style equity option of Microsoft and the index option of S&P 100 index. We employ a finite difference method combined with a Projected Successive Over-relaxation method for pricing an American put option under the Black-Scholes model, while an Alternating Direction Implicit method is utilized to decompose a multi-dimensional partial differential equation into several one dimensional steps under the Heston model. For the calibration of the Heston model, we apply a two step procedure where in the first step we apply an indirect inference method to historical stock prices to estimate diffusion parameters under a probability measure and then use a least squares method to estimate the instantaneous volatility and the market risk premium which are used to switch from working under the probability measure to working under the risk-neutral measure. We find that option price is positively related with the value of the mean reverting speed and the long-term variance. It is not sensitive to the market price of risk and it is negatively related with the risk free rate and the volatility of volatility. By comparing the European put option and the American put option under the Heston model, we observe that their implied volatility generally follow similar patterns. However, there are still some interesting observations that can be made from the comparison of the two put options. First, for the out-of-the-money category, the American and European options have rather comparable implied volatilities with the American options' implied volatility being slightly bigger than the European options. While for the in-the-money category, the implied volatility of the European options is notably higher than the American options and its value exceeds the implied volatility of the American options. We also assess the performance of the Heston model by comparing its result with the result from the Black-Scholes model. We observe that overall the Heston model performs better than the Black-Scholes model. In particular, the Heston model has tendency of underpricing the in-the-money option and overpricing the out-of-the-money option. Whereas, the Black-Scholes model is inclined to underprice both the in-the-money option and the out-of-the-money option.b

Large amounts of hydrocarbon reserves are trapped in fractured reservoirs where fluid flux is far more rapid along fractures than through the porous matrix, even though the volume of the pore space may be a hundred times greater than the volume of the fractures. These are considered extremely challenging in terms of accurate recovery prediction because of their complexity and heterogeneity. Conventional reservoir simulators are generally not suited to naturally fractured reservoirs’ production history simulation, especially when production processes are associated with large pressure and temperature changes that lead to large redistribution of effective stresses, causing natural fracture aperture alterations. In this case, all the effective processes, i.e. hydraulic, thermal and geomechanical, should be considered simultaneously to explain and evaluate the behavior of stress-sensitive reservoirs over the production period. This is called thermo-hydro-mechanical (THM) coupling. In this study, a fully coupled thermo-hydro-mechanical approach is developed to simulate the physical behavior of fractures in a plane strain thermo-poroelastic medium. A hybrid numerical method, which implements both the finite difference method (FDM) and the displacement discontinuity method (DDM), is established to study the pressure, temperature, deformation and stress variations of fractures and surrounding rocks during production processes. This method is straightforward and can be implemented in conventional reservoir simulators to update fracture conductivity as it uses the same grid block as the reservoir grids and requires only discretization of fractures. The hybrid model is then verified with couple of analytical solutions for the fracture aperture variation under different conditions. This model is implemented for some examples to present the behavior of fracture network as well as its surrounding rock under thermal injection and production. The results of this work clearly show the importance of rate, aspect ratio (i.e. geometry) and the coupling effects among fracture flow rate and aperture changes arising from coupled stress, pressure and temperature changes. The outcomes of this approach can be used to study the behavior of hydraulic injection for induced fracturing and promoting of shearing such as hydraulic fracturing of shale gas or shale oil reservoirs as well as massive waste disposal in the porous carbonate rocks. Furthermore, implementation of this technique should be able to lead to a better understanding of induced seismicity in injection projects of all kinds, whether it is for waste water disposal, or for the extraction of geothermal energy.

Finite-difference numerical methods are developed for the solution of some systems in the biomedical sciences; namely, a predator-prey model and the SEIR (Susceptible/Exposed/ Infectious/Recovered) measles model. First-order methods are developed to solve the predator-prey model and one second-order method is developed to solve the SEIR measles model. The predator-prey model is extended to one-space dimension to incorporate diffusion. The SEIR measles model is extended to one-space dimension to incorporate (i) diffusion, (ii) convection and (iii) diffusion-convection. The SEIR measles model is extended further to model diffusion in two-space dimensions. The reaction terms in these systems of partial differntial equations contain nonlinear expressions. Nevetheless, it is seen that the numerical solutions are obtained by solving a linear algebraic system at each time step, as opposed to solving a nonlinear algebraic systems, which is often required when integrating non-linear partial differential equations. The development of each numerical method is made in the light of experience gained in solving the system of ordinary differential equations for each system. The numerical methods proposed for the solution of the initial-value problem for the predator-prey and measles models are characterized to be implicit. However, in each case it is seen that the numerical solutions are obtained explicitly. In a series of numerical experiments, in which the ordinary differential equations are solved first of all, it is seen that the proposed methods have superior stability properties to those of the well-known, first-order, Euler method to which they are compared. Incorporating the proposed methods into the numerical solution of partial differential equations is seen to lead to economical and reliable methods for solving the systems.

Closed-loop ground source heat pumps (GSHPs) are used to transfer thermal energy between the subsurface and conditioned spaces for heating and cooling applications. A basic GSHP is composed of a ground heat exchanger (GHX), which is a closed loop of pipe buried in the shallow subsurface circulating a heat exchange fluid, connected to a heat pump. These systems offer an energy efficient alternative to conventional heating and cooling systems; however, installation costs are higher due to the additional cost associated with the GHX. By further developing our understanding of how these ground loops interact with the subsurface, it may possible to design them more intelligently, efficiently, and economically. To gain insight into the physical processes occurring between the GHX and the subsurface and to identify efficiencies and inefficiencies in GSHP design and operation, two main research goals were defined: comprehensive monitoring of a fully functioning GSHP and intensive simulation of these systems using computer models. A 6-ton GSHP was installed at a residence in Elora, ON. An array of 64 temperature sensors was installed on and surrounding the GHX and power consumption and temperature sensors were installed on the system inside the residence. The data collected were used to help characterize and understand the function of the system, provide motivation for further investigations, and assess the impact of the time of use billing scheme on GSHP operation costs. To simulate GSHPs, two computer models were utilized. A 3D finite element model was employed to analyse the effects of pipe configuration and pipe spacing on system performance. A unique, transient 1D finite difference heat conduction model was developed to simulate a single pipe in a U-tube shape with inter-pipe interactions and was benchmarked against a tested analytical solution. The model was used to compare quasi-steady state and transient simulation of GSHPs, identify system performance efficiencies through pump schedule optimization, and investigate the effect of pipe length on system performance. A comprehensive comparison of steady state and pulsed simulation concludes that it is possible to simulate transient operation using a steady state assumption for some cases. Optimal pipe configurations are identified for a range of soil thermal properties. Optimized pump schedules are identified and analysed for a specific heat pump and fluid circulation pump. Finally, the effect of pipe spacing and length on system performance is characterized. It was found that there are few design inefficiencies that could be easily addressed to improve general design practice.

In this thesis, the theory of three dimensional numerical modeling of transport and re-suspension of fine sediments is studied and a computer program is develped for simulation of the three dimensional suspended sediment transport. The computer program solves the three dimensional advection-diffusion equation simultaneously with a computer program prepared earlier for the simulation of three dimensional current systems. This computer program computes the velocity vectors, eddy viscosities and water surface elavations which are used as inputs by the program of fine sediment transport. The model is applied to Bay of Izmir for different wind conditions.

MODELING AND NUMERICAL ANALYSIS OF SINGLE DROPLET DRYING DALMAZ, Nesip M.Sc., Department of Chemical Engineering Supervisor: Prof. Dr. H. &Ouml / nder &Ouml / ZBELGE Co-Supervisor: Asst. Prof. Dr. Yusuf ULUDAg August 2005, 120 pages A new single droplet drying model is developed that can be used as a part of computational modeling of a typical spray drier. It is aimed to describe the drying behavior of a single droplet both in constant and falling rate periods using receding evaporation front approach coupled with the utilization of heat and mass transfer equations. A special attention is addressed to develop two different numerical solution methods, namely the Variable Grid Network (VGN) algorithm for constant rate period and the Variable Time Step (VTS) algorithm for falling rate period, with the requirement of moving boundary analysis. For the assessment of the validity of the model, experimental weight and temperature histories of colloidal silica (SiO2), skimmed milk and sodium sulfate decahydrate (Na2SO4&amp / #8901 / 10H2O) droplets are compared with the model predictions. Further, proper choices of the numerical parameters are sought in order to have successful iteration loops. The model successfully estimated the weight and temperature histories of colloidal silica, dried at air temperatures of 101oC and 178oC, and skimmed milk, dried at air temperatures of 50oC and 90oC, droplets. However, the model failed to predict both the weight and the temperature histories of Na2SO4&amp / #8901 / 10H2O droplets dried at air temperatures of 90oC and 110oC. Using the vapor pressure expression of pure water, which neglects the non-idealities introduced by solid-liquid interactions, in model calculations is addressed to be the main reason of the model resulting poor estimations. However, the developed model gives the flexibility to use a proper vapor pressure expression without much effort for estimation of the drying history of droplets having highly soluble solids with strong solid-liquid interactions. Initial droplet diameters, which were calculated based on the estimations of the critical droplet weights, were predicted in the range of 1.5-2.0 mm, which are in good agreement with the experimental measurements. It is concluded that the study has resulted a new reliable drying model that can be used to predict the drying histories of different materials.

Realistic predictions of service life of steel catenary risers (SCR) require an accurate characterization of seafloor stiffness in the zone where the riser contacts the seafloor, the so- called touchdown area (TDA). This paper describes the key features of a seafloor-riser interaction model based on the previous experimental model tests. The seafloor is represented in terms of non-linear load-deflection (P-y) relationships, which are also able to account for soil stiffness degradation due to vertical cyclic loading. The P-y approach has some limitations, but simulations show good agreement with experimental data. Hence, stiffness degradation and rate effects during penetration and uplift motion (suction force increase) of the riser are well captured through comparison with previous experimental tests carried out at the Centre for Offshore Foundation Systems (COFS) and Norwegian Geotechnical Institute (NGI). The analytical framework considers the riser-seafloor interaction problem in terms of a pipe resting on a bed of springs, and requires the iterative solution of a fourth-order ordinary differential equation. A series of simulations is used to illustrate the capabilities of the model. Due to the non-linear soil springs with stiffness degradation it is possible to simulate the trench formation process and estimate deflections and moments along the riser length. The seabed model is used to perform parametric studies to assess the effects of stiffness, soil strength, amplitude of pipe displacements, and riser tension on pipe deflections and bending stresses. The input parameters include the material properties (usually pipe and soil), model parameters, and loading conditions such as the amplitude of imposed dis- placements, tension, and moment. Primary outputs from this model include the deflected shape of the riser pipe and bending moments along riser length. The code also provides the location of maximum trench depth and the position where the maximum bending moment occurs and any point where user is interested in.

In this thesis, high order accurate discretization schemes for partial differential equations are investigated. In the first paper, the linearized two-dimensional Navier-Stokes equations are considered. A special formulation of the boundary conditions is used and estimates for the solution to the continuous problem in terms of the boundary conditions are derived using a normal mode analysis. Similar estimates are achieved for the discretized equations. For the discretization, a second order finite difference scheme on a staggered mesh is used. In Paper II, the analysis for the second order scheme is used to develop a fourth order scheme for the fully nonlinear Navier-Stokes equations. The fully nonlinear incompressible Navier-Stokes equations in two space dimensions are considered on an orthogonal curvilinear grid. Numerical tests are performed with a fourth order accurate Padé type spatial finite difference scheme and a semi-implicit BDF2 scheme in time. In Papers III-V, a class of high order accurate time-discretization schemes based on the deferred correction principle is investigated. The deferred correction principle is based on iteratively eliminating lower order terms in the local truncation error, using previously calculated solutions, in each iteration obtaining more accurate solutions. It is proven that the schemes are unconditionally stable and stability estimates are given using the energy method. Error estimates and smoothness requirements are derived. Special attention is given to the implementation of the boundary conditions for PDE. The scheme is applied to a series of numerical problems, confirming the theoretical results. In the sixth paper, a time-compact fourth order accurate time discretization for the one- and two-dimensional wave equation is considered. Unconditional stability is established and fourth order accuracy is numerically verified. The scheme is applied to a two-dimensional wave propagation problem with discontinuous coefficients.

In this thesis, the numerical solution of time-dependent partial differential equations (PDE) is studied. In particular high-order finite difference methods on Summation-by-parts (SBP) form are analysed and applied to model problems as well as the PDEs governing aerodynamics. The SBP property together with an implementation of boundary conditions called SAT (Simultaneous Approximation Term), yields stability by energy estimates. The first derivative SBP operators were originally derived for Cartesian grids. Since aerodynamic computations are the ultimate goal, the scheme must also be stable on curvilinear grids. We prove that stability on curvilinear grids is only achieved for a subclass of the SBP operators. Furthermore, aerodynamics often requires addition of artificial dissipation and we derive an SBP version. With the SBP-SAT technique it is possible to split the computational domain into a multi-block structure which simplifies grid generation and more complex geometries can be resolved. To resolve extremely complex geometries an unstructured discretisation method must be used. Hence, we have studied a finite volume approximation of the Laplacian. It can be shown to be on SBP form and a new boundary treatment is derived. Based on the Laplacian scheme, we also derive an SBP artificial dissipation for finite volume schemes. We derive a new set of boundary conditions that leads to an energy estimate for the linearised three-dimensional Navier-Stokes equations. The new boundary conditions will be used to construct a stable SBP-SAT discretisation. To obtain an energy estimate for the discrete equation, it is necessary to discretise all the second derivatives by using the first derivative approximation twice. According to previous theory that would imply a degradation of formal accuracy but we present a proof that this is not the case.

In this thesis, stable and efficient hybrid methods which combine high order finite difference methods and unstructured finite volume methods for time-dependent initial boundary value problems have been developed. The hybrid methods make it possible to combine the efficiency of the finite difference method and the flexibility of the finite volume method. We carry out a detailed analysis of the stability of the hybrid methods, and in particular the stability of interface treatments between structured and unstructured blocks. Both the methods employ so called summation-by-parts operators and impose boundary and interface conditions weakly, which lead to an energy estimate and stability. We have constructed and analyzed first-, second- and fourth-order Laplacian based artificial dissipation operators for finite volume methods on unstructured grids. The first-order artificial dissipation can handle shock waves, and the fourth-order artificial dissipation eliminates non-physical numerical oscillations efficiently. A stable hybrid method for hyperbolic problems has been developed. It is shown that the stability at the interface can be obtained by modifying the dual grid of the unstructured finite volume method close to the interface. The hybrid method is applied to the Euler equation by the coupling of two stand-alone CFD codes. Since the coupling is administered by a third separate coupling code, the hybrid method allows for individual development of the stand-alone codes. It is shown that the hybrid method is an accurate, efficient and practically useful computational tool that can handle complex geometries and wave propagation phenomena. Stable and accurate interface treatments for the linear advection–diffusion equation have been studied. Accurate high-order calculation are achieved in multiple blocks with interfaces. Three stable interface procedures — the Baumann–Oden method, the “borrowing” method and the local discontinuous Galerkin method, have been investigated. The analysis shows that only minor differences separate the different interface handling procedures. A conservative stable and efficient hybrid method for a parabolic model problem has been developed. The hybrid method has been applied to the full Navier–Stokes equations. The numerical experiments support the theoretical conclusions and show that the interface coupling is stable and converges at the correct order for the Navier–Stokes equations.