Solution of the 2D Navier-Stokes equations by a new FE fractional step method.

In this work, a mathematical and numerical approach for the solution of the 2D Navier-Stokes equations for incompressible fluid flow problems is investigated. A new flux conservative technique for the solution of the elliptic part of the equations is formulated. In the new model, the non linear convective terms of the momentum equations are approximated by means of characteristics and the spatial approximations, of equal order, are obtained by polynomials of degree two. The advancing in time is afforded by a fractional step method combined with a suitable stabilization technique so that the Inf-Sup condition is respected. In order to keep down the computational cost, the algebraic systems are solved by an iterative solver (Bi-CGSTAB) preconditioned by means of Schwarz additive scalable preconditioners. The properties of the new method are verified carrying out several numerical tests. At first, some elliptic, parabolic and convective-diffusive problems are solved and discussed, then the results of some time dependent and stationary 2D Navier-Stokes problems (in particular the well known benchmark problem of the natural convection in a square cavity) are discussed and compared to those found in the literature. Another, potentially very important application of the numerical tools developed, regards the solution of 1D Shallow-Water equations. In fact the use of the fractional steps scheme for advancing in time and the finite elements (of different polynomial degrees) for the spatial approximation, makes the above mentioned approach computationally profitable and convenient for real applications. The efficiency and accuracy of the numerical model have been checked by solving a theoretical test. Finally, a brief description of the software suitably developed and used in the tests conclude the thesis

Solution of the 2D Navier-Stokes equations by a new FE fractional step method.

In this work, a mathematical and numerical approach for the solution of the 2D Navier-Stokes equations for incompressible fluid flow problems is investigated. A new flux conservative technique for the solution of the elliptic part of the equations is formulated. In the new model, the non linear convective terms of the momentum equations are approximated by means of characteristics and the spatial approximations, of equal order, are obtained by polynomials of degree two. The advancing in time is afforded by a fractional step method combined with a suitable stabilization technique so that the Inf-Sup condition is respected. In order to keep down the computational cost, the algebraic systems are solved by an iterative solver (Bi-CGSTAB) preconditioned by means of Schwarz additive scalable preconditioners. The properties of the new method are verified carrying out several numerical tests. At first, some elliptic, parabolic and convective-diffusive problems are solved and discussed, then the results of some time dependent and stationary 2D Navier-Stokes problems (in particular the well known benchmark problem of the natural convection in a square cavity) are discussed and compared to those found in the literature. Another, potentially very important application of the numerical tools developed, regards the solution of 1D Shallow-Water equations. In fact the use of the fractional steps scheme for advancing in time and the finite elements (of different polynomial degrees) for the spatial approximation, makes the above mentioned approach computationally profitable and convenient for real applications. The efficiency and accuracy of the numerical model have been checked by solving a theoretical test. Finally, a brief description of the software suitably developed and used in the tests conclude the thesis

Pricing European and American Options under Heston Model using Discontinuous Galerkin Finite Elements

This paper deals with pricing of European and American options, when the underlying asset price follows Heston model, via the interior penalty discontinuous Galerkin finite element method (dGFEM). The advantages of dGFEM space discretization with Rannacher smoothing as time integrator with nonsmooth initial and boundary conditions are illustrated for European vanilla options, digital call and American put options. The convection dominated Heston model for vanishing volatility is efficiently solved utilizing the adaptive dGFEM. For fast solution of the linear complementary problem of the American options, a projected successive over relaxation (PSOR) method is developed with the norm preconditioned dGFEM. We show the efficiency and accuracy of dGFEM for option pricing by conducting comparison analysis with other methods and numerical experiments

On Discontinuous Galerkin Methods for Singularly Perturbed and Incompressible Miscible Displacement Problems

This thesis is concerned with the numerical approximation of problems of fluid flow, in particular the stationary advection diffusion reaction equations and the time dependent, coupled equations of incompressible miscible displacement in a porous medium. We begin by introducing the continuous discontinuous Galerkin method for the singularly perturbed advection diffusion reaction problem. This is a method which coincides with the continuous Galerkin method away from internal and boundary layers and with a discontinuous Galerkin method in the vicinity of layers. We prove that this consistent method is stable in the streamline diffusion norm if the convection field flows non-characteristically from the region of the continuous Galerkin to the region of the discontinuous Galerkin method. We then turn our attention to the equations of incompressible miscible displacement for the concentration, pressure and velocity of one fluid in a porous medium being displaced by another. We show a reliable a posteriori error estimator for the time dependent, coupled equations in the case where the solution has sufficient regularity and the velocity is bounded. We remark that these conditions may not be attained in physically realistic geometries. We therefore present an abstract approach to the stationary problem of miscible displacement and investigate an a posteriori error estimator using weighted spaces that relies on lower regularity requirements for the true solution. We then return to the continuous discontinuous Galerkin method. We prove in an abstract setting that standard (continuous) Galerkin finite element approximations are the limit of interior penalty discontinuous Galerkin approximations as the penalty parameter tends to infinity. We then show that by varying the penalization parameter on only a subset of the domain we reach the continuous discontinuous method in the limit. We present numerical experiments illustrating this approach both for equations of non-negative characteristic form (closely related to advection diffusion reaction equations) and to the problem of incompressible miscible displacement. We show that we may practically determine appropriate discontinuous and continuous regions, resulting in a significant reduction of the number of degrees of freedom required to approximate a solution, by using the properties of the discontinuous Galerkin approximation to the advection diffusion reaction equation. We finally present novel code for implementing the continuous discontinuous Galerkin method in C++

Método de Galerkin descontínuo para dois problemas de convecção-difusão

Orientador : Prof. Dr. JinYun YuanCo-orientador : Prof. Dr. Yujiang WuTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Matemática. Defesa: Curitiba, 03/09/2015Inclui referências : f. 82-86Resumo: Nesta tese consideramos dois tipos de problemas de convecção-difusão, a saber, as equações de Navier-Stokes para meios incompressíveis e dependentes do tempo e as equações de convecção-difusão espaço-fracionária em duas dimensões. Para as equações de Navier-Stokes usamos o método das características para linearizar equações não-lineares e introduzimos uma variável auxiliar para reduzir a equação de ordem alta a um sistema de primeira ordem. Escolhendo-se cuidadosamente os fluxos numéricos e adicionando os termos de penalização propomos um método de Galerkin descontínuo característico local (CLDG) simétrico e estável. Com essa simetria, é fácil provar estabilidade numérica e estimativas de erros. Experimentos numéricos são realizados para verificar os resultados teóricos. Para os problemas de convecção-difusão espaço-fracionária ainda utilizamos o método das características para tratar a derivada no tempo e os termos convectivos conjuntamente. Para o termo fracionário introduzimos algumas variáveis auxiliares para decompor a derivada de Riemann-Liouville na integral de Riemann-Liouville e na derivada de ordem inteira. Em seguida um método de Galerkin descontínuo hibridizado (HDG) 'e proposto. Finalmente usamos os métodos analíticos para realizar a análise de estabilidade e estimativas de convergência do esquema HDG. Pelo nosso conhecimento, este é o primeiro trabalho que combina o método de Galerkin descontínuo característico às equações de Navier-Stokes e às equações convecção-difusão espaço-fracionária em 2D. Estes esquemas também podem ser aplicados e estudados em outros problemas. Os resultados numéricos são consistentes com os resultados teóricos. Palavras-chave: método das características; método de Galerkin descontínuo; equações de Navier-Stokes; equações de convecção-difusão espaço-fracionária.Abstract: In this thesis, we consider two kinds of convection-diffusion problems, namely the classical time-dependent incompressible Navier-Stokes equations and the space-fractional convection-diffusion equations in two dimensions. For Navier-Stokes equations, we use the method of characteristics to make nonlinear equations linear, and we introduce an auxiliary variable to reduce high-order equation to one order system. Carefully choosing numerical fluxes and adding penalty terms, a stable and symmetric characteristic local discontinuous Galerkin (CLDG) method is proposed. With this symmetry, it is easy to perform numerical stability and error estimates. Numerical experiments are performed to verify theoretical results. For the space-fractional convection-diffusion problems, we still use the method of characteristics to tackle the time derivative and convective terms together. For the fractional term, we introduce some auxiliary variables to split the Riemann-Liouville derivative into Riemann-Liouville integral and integer order derivative. Thus a hybridized discontinuous Galerkin method (HDG) is proposed. Finally we use general analytic methods to perform the stability analysis and convergence estimates of the HDG scheme. As far as we know, this is the first time the discontinuous Galerkin method and the method of characteristics are combined to numerically solve the Navier-Stokes equations and space-fractional convection-diffusion equations in 2D. These schemes can be applied and further studied into other problems as well. The numerical results are consistent with theoretical results. Keywords: method of characteristics; discontinuous Galerkin method; Navier-Stokes equations; space-fractional convection-diffusion equations

A discontinuous Galerkin method for the Vlasov-Poisson system

A discontinuous Galerkin method for approximating the Vlasov-Poisson system of equations describing the time evolution of a collisionless plasma is proposed. The method is mass conservative and, in the case that piecewise constant functions are used as a basis, the method preserves the positivity of the electron distribution function and weakly enforces continuity of the electric field through mesh interfaces and boundary conditions. The performance of the method is investigated by computing several examples and error estimates associated system's approximation are stated. In particular, computed results are benchmarked against established theoretical results for linear advection and the phenomenon of linear Landau damping for both the Maxwell and Lorentz distributions. Moreover, two nonlinear problems are considered: nonlinear Landau damping and a version of the two-stream instability are computed. For the latter, fine scale details of the resulting long-time BGK-like state are presented. Conservation laws are examined and various comparisons to theory are made. The results obtained demonstrate that the discontinuous Galerkin method is a viable option for integrating the Vlasov-Poisson system.Comment: To appear in Journal for Computational Physics, 2011. 63 pages, 86 figure