The purpose of this paper is solving the European option pricing problem under the Black–Scholes model. Our approach is to use the so-called stochastic Runge–Kutta (SRK) numericalscheme to find the corresponding expectation of the functional to the stochastic differentialequation under the Black–Scholes model. Several numerical solutions were made to study howquickly the result converges to the theoretical value. Then, we study the order of convergenceof the SRK method with the help of MATLAB.

This dissertation focuses on the development of adaptive time-stepping and high-order parallel stages exponential Runge–Kutta methods for discretizing stiff partial differential equations (PDEs). The design of exponential Runge–Kutta methods relies heavily on the existing stiff order conditions available in the literature, primarily up to order 5. It is well-known that constructing higher-order efficient methods that strictly satisfy all the stiff order conditions is challenging. Typically, methods up to order 5 have been derived by relaxing one or more order conditions, depending on the desired accuracy level. Our approach will be based on a comprehensive investigation of these conditions. We will derive novel and efficient exponential Runge–Kutta schemes of orders up to 5, which not only fulfill the stiff order conditions in a strict sense but also support the implementation of variable step sizes. Furthermore, we develop the first-ever sixth-order exponential Runge–Kutta schemes by leveraging the exponential B-series theory. Notably, all the newly derived schemes allow the efficient computation of multiple stages, either simultaneously or in parallel. To establish the convergence properties of the proposed methods, we perform an analysis within an abstract Banach space in the context of semigroup theory. Our numerical experiments are given on parabolic PDEs to confirm the accuracy and efficiency of the newly constructed methods.

Differential-algebraic equations (DAEs) consist of mixed systems of ordinary differential equations (ODEs) coupled with linear or nonlinear equations. Such systems may be viewed as ODEs with integral curves lying in a manifold. DAEs appear frequently in applications such as classical mechanics and electrical circuits. This thesis concentrates on systems of index 2, originally index 3, and mixed index 2 and 3. Fast and efficient numerical solvers for DAEs are highly desirable for finding solutions. We focus primarily on the class of Gauss-Lobatto SPARK methods. However, we also introduce an extension to methods proposed by Murua for solving index 2 systems to systems of mixed index 2 and 3. An analysis of these methods is also presented in this thesis. We examine the existence and uniqueness of the proposed numerical solutions, the influence of perturbations, and the local error and global convergence of the methods. When applied to index 2 DAEs, SPARK methods are shown to be equivalent to a class of collocation type methods. When applied to originally index 3 and mixed index 2 and 3 DAEs, they are equivalent to a class of discontinuous collocation methods. Using these equivalences, (s,s)--Gauss-Lobatto SPARK methods can be shown to be superconvergent of order 2s. Symplectic SPARK methods applied to Hamiltonian systems with holonomic constraints preserve well the total energy of the system. This follows from a backward error analysis approach. SPARK methods and our proposed EMPRK methods are shown to be Lagrange-d'Alembert integrators. This thesis also presents some numerical results for Gauss-Lobatto SPARK and EMPRK methods. A few problems from mechanics are considered.

Delay differential equations (DDEs) are a class of differential equations that have received considerable recent attention and been shown to model many real life problems, traditionally formulated as systems of ordinary differential equations (ODEs), more naturally and more accurately. Ideally a DDE modeling package should provide facilities for approximating the solution, performing a sensitivity analysis and estimating unknown parameters. In this thesis we propose new techniques for efficient simulation, accurate sensitivity analysis and reliable parameter estimation of DDEs. We propose a new framework for designing a delay differential equation (DDE) solver which works with any supplied initial value problem (IVP) solver that is based on a general linear method (GLM) and can provide dense output. This is done by treating a general DDE as a special example of a discontinuous IVP. We identify a precise process for the numerical techniques used when solving the implicit equations that arise on a time step, such as when the underlying IVP solver is implicit or the delay vanishes. We introduce an equation governing the dynamics of sensitivities for the most general system of parametric DDEs. Then, having a similar view as the simulation (DDEs as discontinuous ODEs), we introduce a formula for finding the size of jumps that appear at discontinuity points when the sensitivity equations are integrated. This leads to an algorithm which can compute sensitivities for various kind of parameters very accurately. We also develop an algorithm for reliable parameter identification of DDEs. We propose a method for adding extra constraints to the optimization problem, changing a possibly non-smooth optimization to a smooth problem. These constraints are effectively handled using information from the simulator and the sensitivity analyzer. Finally, we discuss the structure of our evolving modeling package DDEM. We present a process that has been used for incorporating existing codes to reduce the implementation time. We discuss the object-oriented paradigm as a way of having a manageable design with reusable and customizable components. The package is programmed in C++ and provides a user-friendly calling sequences. The numerical results are very encouraging and show the effectiveness of the techniques.

Delay differential equations (DDEs) are a class of differential equations that have received considerable recent attention and been shown to model many real life problems, traditionally formulated as systems of ordinary differential equations (ODEs), more naturally and more accurately. Ideally a DDE modeling package should provide facilities for approximating the solution, performing a sensitivity analysis and estimating unknown parameters. In this thesis we propose new techniques for efficient simulation, accurate sensitivity analysis and reliable parameter estimation of DDEs. We propose a new framework for designing a delay differential equation (DDE) solver which works with any supplied initial value problem (IVP) solver that is based on a general linear method (GLM) and can provide dense output. This is done by treating a general DDE as a special example of a discontinuous IVP. We identify a precise process for the numerical techniques used when solving the implicit equations that arise on a time step, such as when the underlying IVP solver is implicit or the delay vanishes. We introduce an equation governing the dynamics of sensitivities for the most general system of parametric DDEs. Then, having a similar view as the simulation (DDEs as discontinuous ODEs), we introduce a formula for finding the size of jumps that appear at discontinuity points when the sensitivity equations are integrated. This leads to an algorithm which can compute sensitivities for various kind of parameters very accurately. We also develop an algorithm for reliable parameter identification of DDEs. We propose a method for adding extra constraints to the optimization problem, changing a possibly non-smooth optimization to a smooth problem. These constraints are effectively handled using information from the simulator and the sensitivity analyzer. Finally, we discuss the structure of our evolving modeling package DDEM. We present a process that has been used for incorporating existing codes to reduce the implementation time. We discuss the object-oriented paradigm as a way of having a manageable design with reusable and customizable components. The package is programmed in C++ and provides a user-friendly calling sequences. The numerical results are very encouraging and show the effectiveness of the techniques.

Aquesta tesi doctoral proposa formulacions de Galerkin discontinu (DG) d'alt ordre per fluxos viscosos incompressibles. Es desenvolupa un nou mètode de DG amb penalti interior (IPM-DG), que condueix a una forma feble simètrica i coerciva pel terme de difusió, i que permet assolir una aproximació espacial d'alt ordre. Aquest mètode s'aplica per resoldre les equacions de Stokes i Navier-Stokes. L'espai d'aproximació de la velocitat es descompon dins de cada element en una part solenoidal i una altra irrotacional, de manera que es pot dividir la forma dèbil IPM-DG en dos problemes desacoblats. El primer permet el càlcul de les velocitats i de les pressions híbrides, mentre que el segon calcula les pressions en l'interior dels elements. Aquest desacoblament permet una reducció important del número de graus de llibertat tant per velocitat com per pressió. S'introdueix també un paràmetre extra de penalti resultant en una formulació DG alternativa per calcular les velocitats solenoidales, on les pressions no apareixen. Les pressions es poden calcular com un post-procés de la solució de les velocitats. Es contemplen altres formulacions DG, com per exemple el mètode Compact Discontinuous Galerkin, i es comparen al mètode IPM-DG. Es proposen mètodes implícits de Runge-Kutta d'alt ordre per problemes transitoris incompressibles, permetent obtenir esquemes incondicionalment estables i amb alt ordre de precisió temporal. Les equacions de Navier-Stokes incompressibles transitòries s'interpreten com un sistema de Equacions Algebraiques Diferencials, és a dir, un sistema d'equacions diferencials ordinàries corresponent a la equació de conservació del moment, més les restriccions algebraiques corresponent a la condició d'incompressibilitat. Mitjançant exemples numèrics es mostra l'aplicabilitat de les metodologies proposades i es comparen la seva eficiència i precisió. / This PhD thesis proposes divergence-free Discontinuous Galerkin formulations providing high orders of accuracy for incompressible viscous flows. A new Interior Penalty Discontinuous Galerkin (IPM-DG) formulation is developed, leading to a symmetric and coercive bilinear weak form for the diffusion term, and achieving high-order spatial approximations. It is applied to the solution of the Stokes and Navier-Stokes equations. The velocity approximation space is decomposed in every element into a solenoidal part and an irrotational part. This allows to split the IPM weak form in two uncoupled problems. The first one solves for velocity and hybrid pressure, and the second one allows the evaluation of pressures in the interior of the elements. This results in an important reduction of the total number of degrees of freedom for both velocity and pressure. The introduction of an extra penalty parameter leads to an alternative DG formulation for the computation of solenoidal velocities with no presence of pressure terms. Pressure can then be computed as a post-process of the velocity solution. Other DG formulations, such as the Compact Discontinuous Galerkin method, are contemplated and compared to IPM-DG. High-order Implicit Runge-Kutta methods are then proposed to solve transient incompressible problems, allowing to obtain unconditionally stable schemes with high orders of accuracy in time. For this purpose, the unsteady incompressible Navier-Stokes equations are interpreted as a system of Differential Algebraic Equations, that is, a system of ordinary differential equations corresponding to the conservation of momentum equation, plus algebraic constraints corresponding to the incompressibility condition. Numerical examples demonstrate the applicability of the proposed methodologies and compare their efficiency and accuracy.

Em problemas de viscoelasticidade computacional, a discretização espacial para a solução global das equações de equilíbrio é acoplada à discretização temporal para a solução de um problema de valor inicial local do fluxo viscoelástico. É demonstrado que este acoplamento espacial-temporal (ou global-local) éconsistente se o tensor de deformação total, agindo como elemento acoplador, tem uma aproximação de ordem p ao longo do tempo igual à ordem de convergência do método de integração de Runge-Kutta (RK). Para a interpolação da deformação foram utilizados polinômios baseados em soluções obtidas nos tempos tn+1, tn, . . ., tn+2−p, p ≥ 2, fornecendo dados consistentes de deformação nos estágios do RK. Em uma situação onde tal regra para a interpolação da deformação não é satisfeita, a integração no tempo apresentará, consequentemente, redução de ordem, baixa precisão e, por conseguinte, eficiência inferior. Em termos gerais, o propósito é generalizar esta condição de consistência proposta pela literatura, formalizando-a matematicamente e o demonstrando através da utilização de métodos de Runge-Kutta diagonalmente implícitos (DIRK) até ordem p = 4, aplicados a modelos viscoelásticos não-lineares sujeitos a deformações finitas. Através de exemplos numéricos, os algoritmos de integração temporal adaptados apresentaram ordem de convergência nominal e, portanto, comprovam a validade da formalização do conceito de interpolação consistente da deformação. Comparado com o método de integração de Euler implícito, é demonstrado que os métodos DIRK aqui aplicados apresentam um ganho considerável em eficiência, comprovado através dos fatores de aceleração atingidos. / In computational viscoelasticity, spatial discretization for the solution of the weak form of the balance of linear momentum is coupled to the temporal discretization for solving a local initial value problem (IVP) of the viscoelastic flow. It is shown that this spatial- temporal (or global-local) coupling is consistent if the total strain tensor, acting as the coupling agent, exhibits the same approximation of order p in time as the convergence order of the Runge-Kutta (RK) integration algorithm. To this end we construct interpolation polynomials based on data at tn+1, tn, . . ., tn+2−p, p ≥ 2, which provide consistent strain data at the RK stages. If this novel rule for strain interpolation is not satisfied, time integration shows order reduction, poor accuracy and therefore less efficiency. Generally, the objective is to propose a generalization of this consistency idea proposed in the literature, formalizing it mathematically and testing it using diagonally implicit Runge-Kutta methods (DIRK) up to order p = 4 applied to a nonlinear viscoelasticity model subjected to finite strain. In a set of numerical examples, the adapted time integrators obtain full convergence order and thus approve the novel concept of consistency. Substantially high speed-up factors confirm the improvement in the efficiency compared with Backward Euler algorithm.

Em problemas de viscoelasticidade computacional, a discretização espacial para a solução global das equações de equilíbrio é acoplada à discretização temporal para a solução de um problema de valor inicial local do fluxo viscoelástico. É demonstrado que este acoplamento espacial-temporal (ou global-local) éconsistente se o tensor de deformação total, agindo como elemento acoplador, tem uma aproximação de ordem p ao longo do tempo igual à ordem de convergência do método de integração de Runge-Kutta (RK). Para a interpolação da deformação foram utilizados polinômios baseados em soluções obtidas nos tempos tn+1, tn, . . ., tn+2−p, p ≥ 2, fornecendo dados consistentes de deformação nos estágios do RK. Em uma situação onde tal regra para a interpolação da deformação não é satisfeita, a integração no tempo apresentará, consequentemente, redução de ordem, baixa precisão e, por conseguinte, eficiência inferior. Em termos gerais, o propósito é generalizar esta condição de consistência proposta pela literatura, formalizando-a matematicamente e o demonstrando através da utilização de métodos de Runge-Kutta diagonalmente implícitos (DIRK) até ordem p = 4, aplicados a modelos viscoelásticos não-lineares sujeitos a deformações finitas. Através de exemplos numéricos, os algoritmos de integração temporal adaptados apresentaram ordem de convergência nominal e, portanto, comprovam a validade da formalização do conceito de interpolação consistente da deformação. Comparado com o método de integração de Euler implícito, é demonstrado que os métodos DIRK aqui aplicados apresentam um ganho considerável em eficiência, comprovado através dos fatores de aceleração atingidos. / In computational viscoelasticity, spatial discretization for the solution of the weak form of the balance of linear momentum is coupled to the temporal discretization for solving a local initial value problem (IVP) of the viscoelastic flow. It is shown that this spatial- temporal (or global-local) coupling is consistent if the total strain tensor, acting as the coupling agent, exhibits the same approximation of order p in time as the convergence order of the Runge-Kutta (RK) integration algorithm. To this end we construct interpolation polynomials based on data at tn+1, tn, . . ., tn+2−p, p ≥ 2, which provide consistent strain data at the RK stages. If this novel rule for strain interpolation is not satisfied, time integration shows order reduction, poor accuracy and therefore less efficiency. Generally, the objective is to propose a generalization of this consistency idea proposed in the literature, formalizing it mathematically and testing it using diagonally implicit Runge-Kutta methods (DIRK) up to order p = 4 applied to a nonlinear viscoelasticity model subjected to finite strain. In a set of numerical examples, the adapted time integrators obtain full convergence order and thus approve the novel concept of consistency. Substantially high speed-up factors confirm the improvement in the efficiency compared with Backward Euler algorithm.

Em problemas de viscoelasticidade computacional, a discretização espacial para a solução global das equações de equilíbrio é acoplada à discretização temporal para a solução de um problema de valor inicial local do fluxo viscoelástico. É demonstrado que este acoplamento espacial-temporal (ou global-local) éconsistente se o tensor de deformação total, agindo como elemento acoplador, tem uma aproximação de ordem p ao longo do tempo igual à ordem de convergência do método de integração de Runge-Kutta (RK). Para a interpolação da deformação foram utilizados polinômios baseados em soluções obtidas nos tempos tn+1, tn, . . ., tn+2−p, p ≥ 2, fornecendo dados consistentes de deformação nos estágios do RK. Em uma situação onde tal regra para a interpolação da deformação não é satisfeita, a integração no tempo apresentará, consequentemente, redução de ordem, baixa precisão e, por conseguinte, eficiência inferior. Em termos gerais, o propósito é generalizar esta condição de consistência proposta pela literatura, formalizando-a matematicamente e o demonstrando através da utilização de métodos de Runge-Kutta diagonalmente implícitos (DIRK) até ordem p = 4, aplicados a modelos viscoelásticos não-lineares sujeitos a deformações finitas. Através de exemplos numéricos, os algoritmos de integração temporal adaptados apresentaram ordem de convergência nominal e, portanto, comprovam a validade da formalização do conceito de interpolação consistente da deformação. Comparado com o método de integração de Euler implícito, é demonstrado que os métodos DIRK aqui aplicados apresentam um ganho considerável em eficiência, comprovado através dos fatores de aceleração atingidos. / In computational viscoelasticity, spatial discretization for the solution of the weak form of the balance of linear momentum is coupled to the temporal discretization for solving a local initial value problem (IVP) of the viscoelastic flow. It is shown that this spatial- temporal (or global-local) coupling is consistent if the total strain tensor, acting as the coupling agent, exhibits the same approximation of order p in time as the convergence order of the Runge-Kutta (RK) integration algorithm. To this end we construct interpolation polynomials based on data at tn+1, tn, . . ., tn+2−p, p ≥ 2, which provide consistent strain data at the RK stages. If this novel rule for strain interpolation is not satisfied, time integration shows order reduction, poor accuracy and therefore less efficiency. Generally, the objective is to propose a generalization of this consistency idea proposed in the literature, formalizing it mathematically and testing it using diagonally implicit Runge-Kutta methods (DIRK) up to order p = 4 applied to a nonlinear viscoelasticity model subjected to finite strain. In a set of numerical examples, the adapted time integrators obtain full convergence order and thus approve the novel concept of consistency. Substantially high speed-up factors confirm the improvement in the efficiency compared with Backward Euler algorithm.

This paper investigates the competitiveness of semi-implicit Runge-Kutta (RK) and spectral deferred correction (SDC) time-integration methods up to order six for incompressible Navier-Stokes problems in conjunction with a high-order discontinuous Galerkin method for space discretization. It is proposed to harness the implicit and explicit RK parts as a partitioned scheme, which provides a natural basis for the underlying projection scheme and yields a straight-forward approach for accommodating nonlinear viscosity. Numerical experiments on laminar flow, variable viscosity and transition to turbulence are carried out to assess accuracy, convergence and computational efficiency. Although the methods of order 3 or higher are susceptible to order reduction due to time-dependent boundary conditions, two third-order RK methods are identified that perform well in all test cases and clearly surpass all second-order schemes including the popular extrapolated backward difference method. The considered SDC methods are more accurate than the RK methods, but become competitive only for relative errors smaller than ca .