Dissertations / Theses on the topic 'The finite difference method'
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1
Siam, Mohamed. "The finite difference method in photonics." Thesis, McGill University, 2009. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=32263.
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This thesis explains and implements the Finite Difference Method to simulate for the propagating modes in integrated waveguides. The equidistant and non-equidsitant methods are explained and implemented. A shape recognition engine is implemented to recognize rectangular waveguide structures provided by the user in the form of images. A geometric meshing algorithm is developed to improve accuracy.Cette thèse explique et met en oeuvre la méthode des diffrences finis pour simuler la propagation de modes de guides d'ondes intgré. La méthode équidistante et non-équidistante est expliquée et mise en oeuvre. Un moteur de reconnaissance de formes est mise en oeuvre pour reconnaître la structure des guides d'ondes rectangulaire prévues par l'utilisateur sous forme d'images. Un algorithme géomtrique de maillage est développé pour améliorer l'exactitude.
2
Eng, Ju-Ling. "Higher order finite-difference time-domain method." Connect to resource, 2006. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1165607826.
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Lee, Check Fu. "Finite difference method for electromagnetic scattering problems." Thesis, Massachusetts Institute of Technology, 1990. http://hdl.handle.net/1721.1/14041.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1990.Includes bibliographical references (leaves 184-192).
by Check F. Lee.
Ph.D.
4
Ciydem, Mehmet. "Ray Based Finite Difference Method For Time Domain Electromagnetics." Phd thesis, METU, 2005. http://etd.lib.metu.edu.tr/upload/12606633/index.pdf.
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In this study, novel Ray Based finite difference method for Time Domain electromagnetics(RBTD) has been developed. Instead of solving Maxwell&rsquos hyperbolic partial differential equations directly, Geometrical Optics tools (wavefronts, rays) and Taylor series have been utilized. Discontinuities of electromagnetic fields lie on wavefronts and propagate along rays. They are transported in the computational domain by transport equations which are ordinary differential equations. Then time dependent field solutions at a point are constructed by using Taylor series expansion in time whose coefficients are these transported distincontinuties. RBTD utilizes grid structure conforming to wave fronts and rays and treats all electromagnetic problems, regardless of their dimensions, as one dimensional problem along the rays. Hence CFL stability condition is implemented always at one dimensional eqaulity case on the ray. Accuracy of RBTD depends on the accuracy of grid generation and numerical solution of transport equations. Simulations for isotropic medium (homogeneous/inhomogeneous) have been conducted. Basic electromagnetic phenomena such as propagation, reflection and refraction have been implemented. Simulation results prove that RBTD eliminates numerical dispersion inherent to FDTD and is promising to be a novel method for computational electromagnetics.
5
Kitts, Paula. "Dielectric smoothing in the finite-difference Poisson-Boltzmann method." Thesis, University of Sheffield, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.284963.
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Petit, Frédéric. "Reverberation Chamber Modeling Using Finite-Difference Time-Domain Method." Diss., University of Marne la Vallée, 2002. http://hdl.handle.net/10919/71555.
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Since the last few years, the unprecedented growth of communication systems involving the propagation of electromagnetic waves is particularly due to developments in mobile phone technology. The reverberation chamber is a reliable bench-test, enabling the study of the effects of electromagnetic waves on a specific electronic appliance. However, the operating of a reverberation chamber being rather complicated, development of numerical models are of utmost importance to determine the crucial parameters to be considered.This thesis consists in the modelling and the simulation of the operating principles of a reverberation chamber by means of the Finite-Difference Time-Domain method. After a brief study based on field and power measurements performed in a reverberation chamber, the second chapter deals with the different problems encountered during the modelling. The consideration of losses being a very important factor in the operating of the chamber, two methods of implementation of these losses are set out in this chapter. Chapter~3 consists in the analysis of the influence of the stirrer on the first eigenmodes of the chamber; the latter modes can undergo a frequency shift of several MHz. Chapter~4 shows a comparison of results issued from high frequency simulations and theoretical statistical results. The problem of an object placed in the chamber, resulting in a field disturbance is also tackled. Finally, in the fifth chapter, a comparison of statistical results for stirrers having different shapes is set out.
7
Parvin, S. "Diffusion-convection problems in parabolic equations." Thesis, University of Manchester, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.382761.
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Postell, Floyd Vince. "High order finite difference methods." Diss., Georgia Institute of Technology, 1990. http://hdl.handle.net/1853/28876.
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Druma, Calin. "Formulation of steady-state and transient potential problems using boundary elements." Ohio : Ohio University, 1999. http://www.ohiolink.edu/etd/view.cgi?ohiou1175886094.
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Turer, Ibrahim. "Specific Absorption Rate Calculations Using Finite Difference Time Domain Method." Master's thesis, METU, 2004. http://etd.lib.metu.edu.tr/upload/12605200/index.pdf.
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This thesis investigates the problem of interaction of electromagnetic radiation with
human tissues. A Finite Difference Time Domain (FDTD) code has been developed to
model a cellular phone radiating in the presence of a human head. In order to implement
the code, FDTD difference equations have been solved in a computational domain
truncated by a Perfectly Matched Layer (PML). Specific Absorption Rate (SAR)
calculations have been carried out to study safety issues in mobile communication.
11
Turan, Umut. "Simulation Of A Batch Dryer By The Finite Difference Method." Master's thesis, METU, 2005. http://etd.lib.metu.edu.tr/upload/12606478/index.pdf.
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The objectives of this study are to investigate the dynamic behavior of an apple slab subjected to drying at constant external conditions and under changing in the drying temperatures and to determine the effects of temperature and time combinations at different steps during drying on the process dynamics parameters, time constant and process gain of the system. For this purpose, a semi-batch dryer system was simulated by using integral method of analysis.
Initially, the dynamic behavior of the drying temperature was investigated by using first order system dynamic model. Process dynamic parameters, time constant and process gain of the system, for change in drying temperature were determined.
Secondly, investigation of the drying kinetics of the apple slab was carried out under constant external conditions in a semi-batch dryer. A mathematical model for diffusion mechanism assumed in one dimensional transient analysis of moisture distribution was solved by using explicit finite difference method of analysis.
Thirdly, investigation of the drying kinetics of the apple slab was carried out under change in drying temperature at different time steps during drying. Inverse response system model was used for the representation of the dynamic behavior of drying. Process dynamic parameters, time constant and process gain of the system were determined.
Model predicted results for apple slab drying under constant external condition and under step change in the drying temperature were compared with the experimental data.
12
Korkut, Fuat. "Generalized Finite Difference Method In Elastodynamics Using Perfectly Matched Layer." Phd thesis, METU, 2012. http://etd.lib.metu.edu.tr/upload/12614476/index.pdf.
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This study deals with the use of the generalized finite difference method (GFDM) in perfectly matched layer (PML) analysis of the problems in wave mechanics, in particular, in elastodynamics. It is known that PML plays the role of an absorbing layer, for an unbounded domain, eliminating reflections of waves for all directions of incidence and frequencies. The study is initiated for purpose of detecting any possible advantages of using GFDM in PML analysis: GFDM is a meshless method suitable for any geometry of the domain, handling the boundary conditions properly and having an easy implementation for PML analysis. In the study, first, a bounded 2D fictitious plane strain problem is solved by GFDM to determine its appropriate parameters (weighting function, radius of influence, etc.). Then, a 1D semi-infinite rod on elastic foundation is considered to estimate PML parameters for GFDM. Finally, the proposed procedure, that is, the use of GFDM in PML analysis, is assessed by considering the compliance functions (in frequency domain) of surface and embedded rigid strip foundations. The surface foundation is assumed to be supported by three types of soil medium: rigid strip foundation on half space (HS), on soil layer overlying rigid bedrock, and on soil layer overlying HS. For the embedded rigid strip foundation, the supporting soil medium is taken as HS. In addition of frequency space analyses stated above, the direct time domain analysis is also performed for the reaction forces of rigid strip foundation over HS. The results of GFDM for both frequency and time spaces are compared with those of finite element method (FEM) with PML and boundary element method (BEM), when possible, also with those of other studies. The excellent matches observed in the results show the reliability of the proposed procedure in PML analysis (that is, of using GFDM in PML analysis).
13
Sandnes, Pål Grøthe. "Meshfree Least Square-based Finite Difference method in CFD applications." Thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for marin teknikk, 2011. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-15454.
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Most commercial computational fluid dynamics (CFD) packages available today are based on the finite volume- or finite element method. Both of these methods have been proven robust, efficient and appropriate for complex geometries. However, due to their crucial dependence on a well constructed grid, extensive preliminary work have to be invested in order to obtain satisfying results. During the last decades, several so-called meshfree methods have been proposed with the intension of entirely eliminating the grid dependence. Instead of a grid, meshfree methods use the nodal coordinates directly in order to calculate the spatial derivatives. In this master thesis, the meshfree least square-based finite difference (LSFD) method has been considered. The method has initially been thoroughly derived and tested for a simple Poisson equation. With its promising numerical performance, it has further been applied to the full Navier- Stokes equations, describing fluid motions in a continuum media. Several numerical methods used to solve the incompressible Navier-Stokes equations have been proposed, and some of them have also been presented in this thesis. However, the temporal discretization has finally been done using a 1st order semi-implicit projection method, for which the primitive variables (velocity and pressure) are solved directly. In order to verify the developed meshfree LSFD code, in total four flow problems have been considered. All of these cases are well known due to their benchmarking relevance, and LSFD performs well compared to both earlier observations and theory. Even though the developed program in this thesis only supports two dimensional, incompressible and laminar flow regimes, the idea of meshfree LSFD is quite general and may very well be applied to more complex flows, including turbulence
14
Roth, Jacob M. "The Explicit Finite Difference Method: Option Pricing Under Stochastic Volatility." Scholarship @ Claremont, 2013. http://scholarship.claremont.edu/cmc_theses/545.
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This paper provides an overview of the finite difference method and its application to approximating financial partial differential equations (PDEs) in incomplete markets. In particular, we study German’s [6] stochastic volatility PDE derived from indifference pricing. In [6], it is shown that the first order- correction to derivatives valued by indifference pricing can be computed as a function involving the stochastic volatility PDE itself. In this paper, we present three explicit finite difference models to approximate the stochastic volatility PDE and compare the resulting valuations to those generated by an Euler- Maruyama Monte Carlo pricing algorithm. We also discuss the significance of boundary condition choice for explicit finite difference models.
15
Basson, Gysbert. "An explicit finite difference method for analyzing hazardous rock mass." Thesis, Stellenbosch : Stellenbosch University, 2011. http://hdl.handle.net/10019.1/17957.
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Thesis (MSc)--Stellenbosch University, 2011.ENGLISH ABSTRACT: FLAC3D is a three-dimensional explicit nite difference program for solving a variety of solid mechanics problems, both linear and non-linear. The development of the algorithm and its initial implementation were performed by Itasca Consulting Group Inc. The main idea of the algorithm is to discritise the domain of interest into a Lagrangian grid where each cell represents an element of the material. Each cell can then deform according to a prescribed stress/strain law together with the equations of motion. An in-depth study of the algorithm was performed and implemented in Java. During the implementation, it was observed that the type of boundary conditions typically used has a major in uence on the accuracy of the results, especially when boundaries are close to regions with large stress variations, such as in mining excavations. To improve the accuracy of the algorithm, a new type of boundary condition was developed where the FLAC3D domain is embedded in a linear elastic material, named the Boundary Node Shell (BNS). Using the BNS shows a signi cant improvement in results close to excavations. The FLAC algorithm is also quite amendable to paralellization and a multi-threaded version that makes use of multiple Central Processing Unit (CPU) cores was developed to optimize the speed of the algorithm. The nal outcome is new non-commercial Java source code (JFLAC) which includes the Boundary Node Shell (BNS) and shared memory parallelism over and above the basic FLAC3D algorithm.
AFRIKAANSE OPSOMMING: FLAC3D is 'n eksplisiete eindige verskil program wat 'n verskeidenheid liniêre en nieliniêre soliede meganika probleme kan oplos. Die oorspronklike algoritme en die implimentasies daarvan was deur Itasca Consulting Group Inc. toegepas. Die hoo dee van die algoritme is om 'n gebied te diskritiseer deur gebruik te maak van 'n Lagrangese rooster, waar elke sel van die rooster 'n element van die rooster materiaal beskryf. Elke sel kan dan vervorm volgens 'n sekere spannings/vervormings wet. 'n Indiepte ondersoek van die algoritme was uitgevoer en in Java geïmplimenteer. Tydens die implementering was dit waargeneem dat die grense van die rooster 'n groot invloed het op die akkuraatheid van die resultate. Dit het veral voorgekom in areas waar stress konsentrasies hoog is, gewoonlik naby areas waar myn uitgrawings gemaak is. Dit het die ontwikkelling van 'n nuwe tipe rand kondisie tot gevolg gehad, sodat die akkuraatheid van die resultate kon verbeter. Die nuwe rand kondisie, genaamd die Grens Node Omhulsel (GNO), aanvaar dat die gebied omring is deur 'n elastiese materiaal, wat veroorsaak dat die grense van die gebied 'n elastiese reaksie het op die stress binne die gebied. Die GNO het 'n aansienlike verbetering in die resultate getoon, veral in areas naby myn uitgrawings. Daar was ook waargeneem dat die FLAC algoritme parralleliseerbaar is en het gelei tot die implentering van 'n multi-SVE weergawe van die sagteware om die spoed van die algoritme te optimeer. Die nale uitkomste is 'n nuwe nie-kommersiële Java weergawe van die algoritme (JFLAC), wat die implimentering van die nuwe GNO randwaardekondisie insluit, asook toelaat vir die gebruik van multi- Sentrale Verwerkings Eenheid (SVE) as 'n verbetering op die basiese FLAC3D algoritme.
16
Lidgate, Simon. "Advanced finite difference - beam propagation : method analysis of complex components." Thesis, University of Nottingham, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.408596.
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Krishnaiah, K. Mohana. "Novel stable subgridding algorithm in finite difference time domain method." Thesis, University of Bristol, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.262808.
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18
lin, zhipeng. "Computational Methods in Financial Mathematics Course Project." Digital WPI, 2009. https://digitalcommons.wpi.edu/etd-theses/1192.
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This course project is made up of two parts. Part one is an investigation and implementation of pricing of financial derivatives using numerical methods for the solution of partial differential equations. Part two is an introduction of Monte Carlo methods in financial engineering. The name of course is MA573:Computational Methods in Financial Mathematics, spring 2009, given by Professor Marcel Blais.
19
Kung, Christopher W. "Development of a time domain hybrid finite difference/finite element method for solutions to Maxwell's equations in anisotropic media." Columbus, Ohio : Ohio State University, 2009. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1238024768.
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20
Wang, Siyang. "Finite Difference and Discontinuous Galerkin Methods for Wave Equations." Doctoral thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-320614.
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Wave propagation problems can be modeled by partial differential equations. In this thesis, we study wave propagation in fluids and in solids, modeled by the acoustic wave equation and the elastic wave equation, respectively. In real-world applications, waves often propagate in heterogeneous media with complex geometries, which makes it impossible to derive exact solutions to the governing equations. Alternatively, we seek approximated solutions by constructing numerical methods and implementing on modern computers. An efficient numerical method produces accurate approximations at low computational cost. There are many choices of numerical methods for solving partial differential equations. Which method is more efficient than the others depends on the particular problem we consider. In this thesis, we study two numerical methods: the finite difference method and the discontinuous Galerkin method. The finite difference method is conceptually simple and easy to implement, but has difficulties in handling complex geometries of the computational domain. We construct high order finite difference methods for wave propagation in heterogeneous media with complex geometries. In addition, we derive error estimates to a class of finite difference operators applied to the acoustic wave equation. The discontinuous Galerkin method is flexible with complex geometries. Moreover, the discontinuous nature between elements makes the method suitable for multiphysics problems. We use an energy based discontinuous Galerkin method to solve a coupled acoustic-elastic problem.
21
Meagher, Timothy P. "A New Finite Difference Time Domain Method to Solve Maxwell's Equations." PDXScholar, 2018. https://pdxscholar.library.pdx.edu/open_access_etds/4389.
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We have constructed a new finite-difference time-domain (FDTD) method in this project. Our new algorithm focuses on the most important and more challenging transverse electric (TE) case. In this case, the electric field is discontinuous across the interface between different dielectric media. We use an electric permittivity that stays as a constant in each medium, and magnetic permittivity that is constant in the whole domain. To handle the interface between different media, we introduce new effective permittivities that incorporates electromagnetic fields boundary conditions. That is, across the interface between two different media, the tangential component, Er(x,y), of the electric field and the normal component, Dn(x,y), of the electric displacement are continuous. Meanwhile, the magnetic field, H(x,y), stays as continuous in the whole domain. Our new algorithm is built based upon the integral version of the Maxwell's equations as well as the above continuity conditions. The theoretical analysis shows that the new algorithm can reach second-order convergence O(∆x2)with mesh size ∆x. The subsequent numerical results demonstrate this algorithm is very stable and its convergence order can reach very close to second order, considering accumulation of some unexpected numerical approximation and truncation errors. In fact, our algorithm has clearly demonstrated significant improvement over all related FDTD methods using effective permittivities reported in the literature. Therefore, our new algorithm turns out to be the most effective and stable FDTD method to solve Maxwell's equations involving multiple media.
22
Kama, Phumezile. "Non-standard finite difference methods in dynamical systems." Thesis, Pretoria : [s.n.], 2009. http://upetd.up.ac.za/thesis/available/etd-07132009-163422.
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Randhawa, Banljinder Singh. "Electromagnetic modelling of curved structures using a hybrid finite-volume finite-difference time-domain method." Thesis, University of York, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.362043.
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Kluge, Tino. "Illustration of stochastic processes and the finite difference method in finance." Universitätsbibliothek Chemnitz, 2003. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200300079.
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The presentation shows sample paths of stochastic processes in form of animations. Those stochastic procsses are usually used to model financial quantities like exchange rates, interest rates and stock prices. In the second part the solution of the Black-Scholes PDE using the finite difference method is illustratedDer Vortrag zeigt Animationen von Realisierungen stochstischer Prozesse, die zur Modellierung von Groessen im Finanzbereich haeufig verwendet werden (z.B. Wechselkurse, Zinskurse, Aktienkurse). Im zweiten Teil wird die Loesung der Black-Scholes Partiellen Differentialgleichung mittels Finitem Differenzenverfahren graphisch veranschaulicht
25
Cai, Ming. "Finite difference time domain method and its application in antenna analysis." Thesis, London South Bank University, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.263739.
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Kluge, Tino. "Pricing derivatives in stochastic volatility models using the finite difference method." [S.l. : s.n.], 2002. http://www.bsz-bw.de/cgi-bin/xvms.cgi?SWB10447116.
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Kıran, Güçoğlu Arzu Tanoğlu Gamze. "The solution of some differential equations by nonstandard finite difference method/." [S.l.] : [s.n.], 2005. http://library.iyte.edu.tr/tezler/master/matematik/T000332.pdf.
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Thesis (Master)--İzmir Institute of Technology, İzmir, 2005Keywords: Nonlinear differential equations, finite difference method, numeric simulation. Includes bibliographical references (leaves. 55-57).
28
Wang, Bohe. "The application of finite difference method and MATLAB in engineering plates." Morgantown, W. Va. : [West Virginia University Libraries], 1999. http://etd.wvu.edu/templates/showETD.cfm?recnum=1037.
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Thesis (M.S.)--West Virginia University, 1999.Title from document title page. Document formatted into pages; contains iv, 87 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 86-87).
29
Tomiso, Nayon. "Modeling electrically small apertures using the finite difference-time domain method." Thesis, Massachusetts Institute of Technology, 1997. http://hdl.handle.net/1721.1/42597.
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Kluge, Tino. "Pricing derivatives in stochastic volatility models using the finite difference method." Master's thesis, Universitätsbibliothek Chemnitz, 2016. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-195504.
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The Heston stochastic volatility model is one extension of the Black-Scholes model which describes the money markets more accurately so that more realistic prices for derivative products are obtained. From the stochastic differential equation of the underlying financial product a partial differential equation (p.d.e.) for the value function of an option can be derived. This p.d.e. can be solved with the finite difference method (f.d.m.). The stability and consistency of the method is examined. Furthermore a boundary condition is proposed to reduce the numerical error. Finally a non uniform structured grid is derived which is fairly optimal for the numerical result in the most interesting pointDas stochastische Volatilitaetsmodell von Heston ist eines der Erweiterungen des Black-Scholes-Modells. Von der stochastischen Differentialgleichung fuer den unterliegenden Prozess kann eine partielle Differentialgleichung fuer die Wertfunktion einer Option abgeleitet werden. Es wird die Loesung mittels Finiter Differenzenmethode untersucht (Konsistenz, Stabilitaet). Weiterhin wird eine Randbedingung und ein spezielles nicht-uniformes Netz vorgeschlagen, was zu einer starken Reduzierung des numerischen Fehlers der Wertfunktion in einem ganz bestimmten Punkt fuehrt
31
Ulimoen, Magnus. "A high-resolution finite difference method for weather and climate models." Thesis, Uppsala universitet, Institutionen för informationsteknologi, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-372172.
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Weather and climate modelling has been one of the major actors in scientific computing. The need for higher resolution for the current models has revealed problems, some of which can be solved by using the SBP-SAT method. Ground effects in the atmosphere, caused by e.g. trees, buildings or mountains may require the discrete grid to be adapted to fit the atmosphere. The work presented here combines grids with different resolution in the vertical direction to solve a simplified model of the atmosphere. The results suggest making minor adjustments to parts of the grid in conjunction with the grid adaptions creates an efficient solver for the heterogeneous atmosphere.
32
Garg, Nimisha. "Analysis of Slot Antennas Using the Finite Difference Time Domain Method." FIU Digital Commons, 2001. https://digitalcommons.fiu.edu/etd/3843.
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The objective of this thesis was to analyze a Coplanar Waveguide (CPW)-fed folded slot antenna using the Finite Difference Time Domain (FDTD) method. Important parameters such as S-parameters and the input impedance of the antenna were simulated using XFDTD software and were analyzed. An important goal of this thesis was to provide design information about the folded slot antenna. For this purpose the effects of antenna layout on the resonant frequency and the bandwidth of the antenna were investigated. First the effect of adding more number of slots to the basic CPW-fed folded slot antenna geometry on the S-parameters, the input impedance and the radiation patterns of the antenna were studied. Next the width of the slot was varied and the effect of changing this design parameter of the antenna was analyzed. Finally the slot separation was varied and its effect on the antenna parameters is studied. This work concluded that, by including additional slots, the input impedance of the antenna can be controlled over a wide range.
33
Abalenkovs, Maksims. "Huygens subgridding for the frequency-dependent/finite-difference time-domain method." Thesis, University of Manchester, 2011. https://www.research.manchester.ac.uk/portal/en/theses/huygens-subgridding-for-the-frequencydependentfinitedifference-timedomain-method(45581358-ff4d-4699-b3db-5bf76a021601).html.
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Computer simulation of electromagnetic behaviour of a device is a common practice in modern engineering. Maxwell's equations are solved on a computer with help of numerical methods. Contemporary devices constantly grow in size and complexity. Therefore, new numerical methods should be highly efficient. Many industrial and research applications of numerical methods need to account for the frequency dependent materials. The Finite-Difference Time-Domain (FDTD) method is one of the most widely adopted algorithms for the numerical solution of Maxwell's equations. A major drawback of the FDTD method is the interdependence of the spatial and temporal discretisation steps, known as the Courant-Friedrichs-Lewy (CFL) stability criterion. Due to the CFL condition the simulation of a large object with delicate geometry will require a high spatio-temporal resolution everywhere in the FDTD grid. Application of subgridding increases the efficiency of the FDTD method. Subgridding decomposes the simulation domain into several subdomains with different spatio-temporal resolutions. The research project described in this dissertation uses the Huygens Subgridding (HSG) method. The frequency dependence is included with the Auxiliary Differential Equation (ADE) approach based on the one-pole Debye relaxation model. The main contributions of this work are (i) extension of the one-dimensional (1D) frequency-dependent HSG method to three dimensions (3D), (ii) implementation of the frequency-dependent HSG method, termed the dispersive HSG, in Fortran 90, (iii) implementation of the radio environment setting from the PGM-files, (iv) simulation of the electromagnetic wave propagating from the defibrillator through the human torso and (v) analysis of the computational requirements of the dispersive HSG program.
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Kluge, Tino. "Pricing derivatives in stochastic volatility models using the finite difference method." Master's thesis, Universitätsbibliothek Chemnitz, 2003. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200300086.
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The Heston stochastic volatility model is one extension of the Black-Scholes model which describes the money markets more accurately so that more realistic prices for derivative products are obtained. From the stochastic differential equation of the underlying financial product a partial differential equation (p.d.e.) for the value function of an option can be derived. This p.d.e. can be solved with the finite difference method (f.d.m.). The stability and consistency of the method is examined. Furthermore a boundary condition is proposed to reduce the numerical error. Finally a non uniform structured grid is derived which is fairly optimal for the numerical result in the most interesting pointDas stochastische Volatilitaetsmodell von Heston ist eines der Erweiterungen des Black-Scholes-Modells. Von der stochastischen Differentialgleichung fuer den unterliegenden Prozess kann eine partielle Differentialgleichung fuer die Wertfunktion einer Option abgeleitet werden. Es wird die Loesung mittels Finiter Differenzenmethode untersucht (Konsistenz, Stabilitaet). Weiterhin wird eine Randbedingung und ein spezielles nicht-uniformes Netz vorgeschlagen, was zu einer starken Reduzierung des numerischen Fehlers der Wertfunktion in einem ganz bestimmten Punkt fuehrt
35
Rouf, Hasan. "Unconditionally stable finite difference time domain methods for frequency dependent media." Thesis, University of Manchester, 2010. https://www.research.manchester.ac.uk/portal/en/theses/unconditionally-stable-finite-difference-time-domain-methods-for-frequency-dependent-media(50e4adf1-d1e4-4ad2-ab2d-70188fb8b7b6).html.
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The efficiency of the conventional, explicit finite difference time domain (FDTD)method is constrained by the upper limit on the temporal discretization, imposed by the Courant–Friedrich–Lewy (CFL) stability condition. Therefore, there is a growing interest in overcoming this limitation by employing unconditionally stable FDTD methods for which time-step and space-step can be independently chosen. Unconditionally stable Crank Nicolson method has not been widely used in time domain electromagnetics despite its high accuracy and low anisotropy. There has been no work on the Crank Nicolson FDTD (CN–FDTD) method for frequency dependent medium. In this thesis a new three-dimensional frequency dependent CN–FDTD (FD–CN–FDTD) method is proposed. Frequency dependency of single–pole Debye materials is incorporated into the CN–FDTD method by means of an auxiliary differential formulation. In order to provide a convenient and straightforward algorithm, Mur’s first-order absorbing boundary conditions are used in the FD–CN–FDTD method. Numerical tests validate and confirm that the FD–CN–FDTD method is unconditionally stable beyond the CFL limit. The proposed method yields a sparse system of linear equations which can be solved by direct or iterative methods, but numerical experiments demonstrate that for large problems of practical importance iterative solvers are to be used. The FD–CN–FDTD sparse matrix is diagonally dominant when the time-stepis near the CFL limit but the diagonal dominance of the matrix deteriorates with the increase of the time-step, making the solution time longer. Selection of the matrix solver to handle the FD–CN–FDTD sparse system is crucial to fully harness the advantages of using larger time-step, because the computational costs associated with the solver must be kept as low as possible. Two best–known iterative solvers, Bi-Conjugate Gradient Stabilised (BiCGStab) and Generalised Minimal Residual (GMRES), are extensively studied in terms of the number of iteration requirements for convergence, CPU time and memory requirements. BiCGStab outperforms GMRES in every aspect. Many of these findings do not match with the existing literature on frequency–independent CN–FDTD method and the possible reasons for this are pointed out. The proposed method is coded in Fortran and major implementation techniques of the serial code as well as its parallel implementation in Open Multi-Processing (OpenMP) are presented. As an application, a simulation model of the human body is developed in the FD–CN–FDTD method and numerical simulation of the electromagnetic wave propagation inside the human head is shown. Finally, this thesis presents a new method modifying the frequency dependent alternating direction implicit FDTD (FD–ADI–FDTD) method. Although the ADI–FDTD method provides a computationally affordable approximation of the CN–FDTD method, it exhibits a loss of accuracy with respect to the CN-FDTD method which may become severe for some practical applications. The modified FD–ADI–FDTD method can improve the accuracy of the normal FD–ADI–FDTD method without significantly increasing the computational costs.
36
Egorova, Vera. "Finite Difference Methods for nonlinear American Option Pricing models: Numerical Analysis and Computing." Doctoral thesis, Universitat Politècnica de València, 2016. http://hdl.handle.net/10251/68501.
Full textAbstract:
[EN] The present PhD thesis is focused on numerical analysis and computing of finite difference schemes for several relevant option pricing models that generalize the Black-Scholes model. A careful analysis of desirable properties for the numerical solutions of option pricing models as the positivity, stability and consistency, is provided.
In order to handle the free boundary that arises in American option pricing problems, various transformation techniques based on front-fixing method are applied and studied. Special attention is paid to multi-asset option pricing, such as exchange or spread option. Appropriate transformation allows eliminating of the cross derivative term. Transformation techniques of partial differential equations to remove convection and reaction terms are studied in order to simplify the models and avoid possible troubles of stability.
This thesis consists of six chapters. The first chapter is an introduction containing definitions of option and related terms and derivation of the Black-Scholes equation as well as general aspects of theory of finite difference schemes, including preliminaries on numerical analysis.
Chapter 2 is devoted to solve linear Black-Scholes model for American put and call options. A Landau transformation and a new front-fixing transformation are applied to the free boundary value problem. It leads to non-linear partial differential equation (PDE) in a fixed domain. Stable and consistent explicit numerical schemes are proposed preserving positivity and monotonicity of the solution in accordance with the behaviour of the exact solution.
Efficiency of the front-fixing method demonstrated in Chapter 2 has motivated us to apply the method to some more complicated nonlinear models. A new change of variables resulting in a time dependent boundary instead of fixed one, is applied to nonlinear Black-Scholes model for American options, such as Barles and Soner and Risk Adjusted Pricing models.
Chapter 4 provides a new alternative approach for solving American option pricing problem based on rationality of investor. There exists an intensity function that can be reduced in the simplest case to penalty approach.
Chapter 5 deals with multi-asset option pricing. Appropriate transformation allows eliminating of the cross derivative term avoiding computational drawbacks and possible troubles of stability.
Concluding remarks are given in Chapter 6. All the considered models and numerical methods are accompanied by several examples and simulations. The convergence rate is computed confirming the theoretical study of consistency. Stability conditions are tested by numerical examples. Results are compared with known relevant methods in the literature showing efficiency of the proposed methods.[ES] La presente tesis doctoral se centra en la construcción de esquemas en diferencias finitas y el análisis numérico de relevantes modelos de valoración de opciones que generalizan el modelo de Black-Scholes. Se proporciona un análisis cuidadoso de las propiedades de las soluciones numéricas tales como la positividad, la estabilidad y la consistencia. Con el fin de manejar la frontera libre que surge en los problemas de valoración de opciones Americanas, se aplican y se estudian diversas técnicas de transformación basadas en el método de fijación de las fronteras (front-fixing). Se presta especial atención a la valoración de opciones de múltiples activos, como son las opciones ''exchange'' y ''spread''. Esta tesis se compone de seis capítulos. El primer capítulo es una introducción que contiene las definiciones de opción y términos relacionados y la derivación de la ecuación de Black-Scholes, así como aspectos generales de la teoría de los esquemas en diferencias finitas, incluyendo preliminares de análisis numérico. El capítulo 2 está dedicado a resolver el modelo lineal de Black-Scholes para opciones Americanas put y call. Para fijar las fronteras del problema de frontera libre se aplican transformaciones como la de Landau y un nuevo cambio de variable propuesto. La eficiencia del método front-fixing mostrada en el capítulo 2 ha motivado el estudio de su aplicación a algunos modelos no lineales más complicados. En particular, se propone un cambio de variables que lleva a una nueva frontera dependiente del tiempo en lugar de una fija. Este cambio se aplica a modelos no lineales de Black-Scholes para opciones Americanas, como son el de Barles y Soner y el modelo RAPM (Risk Adjusted Pricing Methodology). El capítulo 4 ofrece una nueva técnica para la resolución de problemas de valoración de opciones Americanas basada en la racionalidad de los inversores. Aparece una función de la intensidad que se puede reducir en el caso más simple a la técnica de penalización (penalty method). Este enfoque tiene en cuenta el posible comportamiento irracional de los inversores. En la sección 4.2 se aplica esta técnica al modelo de cambio de regímenes lo que lleva a un nuevo modelo que tiene en cuenta el posible ejercicio irracional, así como varios estados del mercado. El enfoque del parámetro de racionalidad junto con una transformación logarítmica permiten construir un esquema numérico eficiente sin aplicar el método front-fixing o la conocida formulación de LCP (Linear Complementarity Problem). El capítulo 5 se dedica a la valoración de opciones de activos múltiples. Una transformación apropiada permite la eliminación del término de derivadas cruzadas evitando inconvenientes computacionales y posibles problemas de estabilidad. Las conclusiones se muestran en el capítulo 6. Se pone en relieve varios aspectos de la presente tesis. Todos los modelos considerados y los métodos numéricos van acompañados de varios ejemplos y simulaciones. Se estudia la convergencia numérica que confirma el estudio teórico de la consistencia. Las condiciones de estabilidad son corroboradas con ejemplos numéricos. Los resultados se comparan con métodos relevantes de la bibliografía mostrando la eficiencia de los métodos propuestos.
[CAT] La present tesi doctoral se centra en la construcció d'esquemes en diferències finites i l'anàlisi numèrica de rellevants models de valoració d'opcions que generalitzen el model de Black-Scholes. Es proporciona una anàlisi cuidadosa de les propietats de les solucions numèri-ques com ara la positivitat, l'estabilitat i la consistència. A fi de manejar la frontera lliure que sorgix en els problemes de valoració d'opcions Americanes, s'apliquen i s'estudien diverses tècniques de transformació basades en el mètode de fixació de les fronteres (front-fixing). Es presta especial atenció a la valoració d'opcions de múltiples actius, com són les opcions ''exchange'' i ''spread''. Esta tesi es compon de sis capítols. El primer capítol és una introducció que conté les definicions d'opció i termes relacionats i la derivació de l'equació de Black-Scholes, així com aspectes generals de la teoria dels esquemes en diferències finites, incloent aspectes preliminars d'anàlisi numèrica. El 2n capítol està dedicat a resoldre el model lineal de Black-Scholes per a opcions Americanes ''put'' i ''call''. Per a fixar les fronteres del problema de frontera lliure s'apliquen transformacions com la de Landau i s'ha proposat un nou canvi de variable proposat. Açò porta a una equació diferencial en derivades parcials no lineal en un domini fix. L'eficiència del mètode front-fixing mostrada en el 2n capítol ha motivat l'estudi de la seua aplicació a alguns models no lineals més complicats. En particular, es proposa un canvi de variables que porta a una nova frontera dependent del temps en compte d'una fixa. Este canvi s'aplica a models no lineals de Black-Scholes per a opcions Americanes, com són el de Barles i Soner i el model RAPM (Risk Adjusted Pricing Methodology). El 4t capítol oferix una nova tècnica per a la resolució de problemes de valoració d'opcions Americanes basada en la racionalitat dels inversors. Apareix una funció de la intensitat que es pot reduir en el cas més simple a la tècnica de penalització (penal method) . Este enfocament té en compte el possible comportament irracional dels inversors. En la secció 4.2 s'aplica esta tècnica al model de canvi de règims el que porta a un nou model que té en compte el possible exercici irracional, així com diversos estats del mercat. L'enfocament del paràmetre de racionalitat junt amb una transformació logarítmica permeten construir un esquema numèric eficient sense aplicar el mètode front-fixing o la coneguda formulació de LCP (Linear Complementarity Problem). El 5é capítol es dedica a la valoració d'opcions d'actius múltiples. Una transformació apropiada permet l'eliminació del terme de derivades mixtes evitant inconvenients computacionals i possibles problemes d' estabilitat. Les conclusions es mostren al 6é capítol. Es posa en relleu diversos aspectes de la present tesi. Tots els models considerats i els mètodes numèrics van acompanyats de diversos exemples i simulacions. S'estu-dia la convergència numèrica que confirma l'estudi teòric de la consistència. Les condicions d'estabilitat són corroborades amb exemples numèrics. Els resultats es comparen amb mètodes rellevants de la bibliografia mostrant l'eficiència dels mètodes proposats.
Egorova, V. (2016). Finite Difference Methods for nonlinear American Option Pricing models: Numerical Analysis and Computing [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/68501
TESIS
Premiado
37
Heger, Walter. "Using the finite difference and the finite element method to solve an electric current diffusion problem." Thesis, McGill University, 1987. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=66150.
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Kuzu, Lokman. "Electromagnetic scattering from chiral materials using the finite difference frequency domain method." Related electronic resource: Current Research at SU : database of SU dissertations, recent titles available full text, 2006. http://proquest.umi.com/login?COPT=REJTPTU0NWQmSU5UPTAmVkVSPTI=&clientId=3739.
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Pegoraro, Adrian. "Modelling heterogeneous nonlinear subwavelength systems with the finite difference time domain method." Thesis, University of Ottawa (Canada), 2005. http://hdl.handle.net/10393/27007.
Full textAbstract:
Predicting the response of heterogeneous nonlinear microscopic systems to laser excitation is very difficult using analytical techniques and is only feasible under simplifying assumptions. However, using numerical methods, it is possible to analyze arbitrary systems and make predictions about their behaviour. This information may be used to develop new techniques and a better understanding of measurements. One area which stands to benefit from such methods is nonlinear microscopy. We use finite difference time domain methods to explain experimental measurements and also develop a new nonlinear microscopy technique which shows a significant improvement in axial resolution over traditional techniques. We then explore the behaviour of Maxwell Garnett nanocomposites and illustrate the limitations of the current theoretical models for these systems.
40
Hayman, Kenneth John. "Finite-difference methods for the diffusion equation." Title page, table of contents and summary only, 1988. http://web4.library.adelaide.edu.au/theses/09PH/09phh422.pdf.
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Ezertas, Ahmet Alper. "Sensitivity Analysis Using Finite Difference And Analytical Jacobians." Master's thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/12611067/index.pdf.
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The Flux Jacobian matrices, the elements of which are the derivatives of the flux vectors with respect to the flow variables, are needed to be evaluated in implicit flow solutions and in analytical sensitivity analyzing methods. The main motivation behind this thesis study is to explore the accuracy of the numerically evaluated flux Jacobian matrices and the effects of the errors in those matrices on the convergence of the flow solver, on the accuracy of the sensitivities and on the performance of the design optimization cycle. To perform these objectives a flow solver, which uses exact Newton&rsquos method with direct sparse matrix solution technique, is developed for the Euler flow equations. Flux Jacobian is evaluated both numerically and analytically for different upwind flux discretization schemes with second order MUSCL face interpolation. Numerical flux Jacobian matrices that are derived with wide range of finite difference perturbation magnitudes were compared with analytically derived ones and the optimum perturbation magnitude, which minimizes the error in the numerical evaluation, is searched. The factors that impede the accuracy are analyzed and a simple formulation for optimum perturbation magnitude is derived. The sensitivity derivatives are evaluated by direct-differentiation method with discrete approach. The reuse of the LU factors of the flux Jacobian that are evaluated in the flow solution enabled efficient sensitivity analysis. The sensitivities calculated by the analytical Jacobian are compared with the ones that are calculated by numerically evaluated Jacobian matrices. Both internal and external flow problems with varying flow speeds, varying grid types and sizes are solved with different discretization schemes. In these problems, when the optimum perturbation magnitude is used for numerical Jacobian evaluation, the errors in Jacobian matrix and the sensitivities are minimized. Finally, the effect of the accuracy of the sensitivities on the design optimization cycle is analyzed for an inverse airfoil design performed with least squares minimization.
42
Trojan, Alice von. "Finite difference methods for advection and diffusion." Title page, abstract and contents only, 2001. http://web4.library.adelaide.edu.au/theses/09PH/09phv948.pdf.
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Includes bibliographical references (leaves 158-163). Concerns the development of high-order finite-difference methods on a uniform rectangular grid for advection and diffuse problems with smooth variable coefficients. This technique has been successfully applied to variable-coefficient advection and diffusion problems. Demonstrates that the new schemes may readily be incorporated into multi-dimensional problems by using locally one-dimensional techniques, or that they may be used in process splitting algorithms to solve complicatef time-dependent partial differential equations.
43
Persson, Jonas. "Accurate Finite Difference Methods for Option Pricing." Doctoral thesis, Uppsala : Acta Universitatis Upsaliensis, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-7097.
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Steinle, Peter John. "Finite difference methods for the advection equation /." Title page, table of contents and abstract only, 1993. http://web4.library.adelaide.edu.au/theses/09PH/09phs8224.pdf.
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Filipovic, Zlatko. "Finite difference methods for pricing financial derivatives." Thesis, Imperial College London, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.420931.
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Demir, Ismail. "Seismic wave modelling using finite difference methods." Thesis, University of South Wales, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.284896.
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Lee, Chi-Liang, and 李志良. "Generalized Finite Difference Formulas and Adaptive Parameter Scheme in Finite Difference Method." Thesis, 1997. http://ndltd.ncl.edu.tw/handle/71188752822206895206.
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博士大同工學院
機械工程學系
85
The purpose of the present study is to derive a general form which includesfinite difference formulas and truncation errors for each derivative. In thebeginning, the equations of Taylor series expansion are written with respectto discrete grid point. Multiplying a different coefficient to each equationand then adding these equations together, a general form of finite differenceequation for any derivative has been thus generated. Meanwhile, the truncationerror can be calculated to show the effectiveness of each finite differencerepresentation. Two kinds of application are introduced where more accurateresults can be achieved. The first kind is to incorporate higher-order terms oftruncation error into finite difference equation. The second kind is to generatean optimum finite difference equation by minimizing total truncation error ofthe whole equation and adaptive parameter. By implementing the generalized finite difference formulas, adaptiveparameter scheme of finite difference method is proposed which is wellpertinent to the problems of one-dimensional advection-diffusion equationincluding a variable source term and linear nonhomogeneous second orderdifferential equation. The adaptive parameter, truncation error, and correctionterm have been derived with respect to higher-order approximation. Here, theoptimum value of this adaptive parameter can be easily acquired by exactsolution. It is found that the incorporation of correction term which is produceddue to the existence of variable source term can obtain very accurate results.
48
Luo, Rui-Ming, and 羅瑞明. "A differential quadrature finite difference method." Thesis, 1999. http://ndltd.ncl.edu.tw/handle/35399145666214933828.
Full textAbstract:
碩士國立成功大學
造船及船舶機械工程學系
87
The differential quadrature finite difference method (DQFDM) is used to analyze the flexural deflection of composite non-uniform plates . The finite difference operators are derived by the differential quadrature (DQ). They can be obtained by using the weighting coefficients for DQ discretizations. The derivation is straight and easy . By using different order or the same order but different grid differential quadrature discretizations for the same derivative or partial derivative , various finite difference operators for the same differential or partial differential operators can be obtained . Finite difference operators for unequally spaced and irregular grids can also be generated . The derivations of higher order finite difference operators is also easy .
49
Mo, Chia-Cheng, and 莫嘉政. "Finite Difference Method for Surface Diffusion." Thesis, 2011. http://ndltd.ncl.edu.tw/handle/75606861006569889798.
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Iseri, Shellie M. "High order finite difference methods." Thesis, 1996. http://hdl.handle.net/1957/34637.
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