Numerical Methods for Partial Differential Equations.

A new higher order finite element method for elliptic partial differential equations on a stationary smooth surface $\Gamma$ is introduced and analyzed. We assume that $\Gamma$ is characterized as the zero level of a level set function $\phi$ and only a finite element approximation $\phi_h$ (of degree $k \geq 1$) of $\phi$ is known. For the discretization of the partial differential equation, finite elements (of degree $m\geq 1$) on a piecewise linear approximation of $\Gamma$ are used. The discretization is lifted to $\Gamma_h$, which denotes the zero level of $\phi_h$, using a quasi-orthogonal coordinate system that is constructed by applying a gradient recovery technique to $\phi_h$. A complete discretization error analysis is presented in which the error is split into a geometric error, a quadrature error, and a finite element approximation error. The main result is a $H^1(\Gamma)$-error bound of the form $c(h^m + h^{k+1})$. Results of numerical experiments illustrate the higher order convergence of t...

A pseudospectral based method of lines for solving integro-differential boundary-layer equations. Application to the mixed convection over a heated horizontal plate

The method of lines refers to an approximation of one or more partial differential equations with ordinary differential equations in just one of the independent variables. The assumption is that the ordinary differential equations are easier to analyze and solve than the partial differential equations. The approximation can be based on finite differences, finite elements, collocation or Fourier series like expansions. A method of lines obtained with finite differences seems easiest to apply and will be used exclusively in these notes…

Freezing time of a slab using the method of lines

A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions

(English) This thesis deals with finite element methods to solve compressible flow problems, an important branch of Computational Fluid Dynamics (CFD) whose applications are widespread in many areas of engineering and science. In spite of the increasing amount of computational resources made available for the scientific and engineering research communities, the numerical simulation of complex compressible phenomena in many practical applications is still a challenge. These type of flow problems are extremely demanding in what concerns numerical computations and memory requirements. In particular, in this thesis we investigate the possibility to solve the underlying algebraic systems in a decoupled manner, a technique usually called fractional step or segregation method. Although segregation techniques have been broadly studied and analyzed for the incompressible Navier-Stokes equations, allowing for the separate resolution of velocity and pressure unkonwns, much less has been explored for compressible problems. The interest on this type of technique not only comes from the fact that it permits a segregated calculation of the problem unkonwns (usually leading to better conditioned systems) but from the associated reduction of the computational cost. We study three different problems inside the compressible CFD research branch in separated chapters: the isentropic Navier-Stokes equations, the Navier-Stokes problem written in primitive variables (velocity, pressure and temperature), and the navier-Stokes problem written in the classical formulation with conservative variables (momentum, density, total energy). For each of these problems, first we propose a finite element stabilized formulation framed within the Variational MultiScale concept, which allows to use equal interpolation spaces for all the variables in play. Second, and once space and time discretizations are selected, we derive fractional step methods up to second order in time. Finally, all the schemes are implemented in a parallel multiphysics code and representative simulations are carried out in order to analyze the performance of the proposed techniques. (Español) Esta tesis se centra en la aplicación del método de elementos finitos a problemas de flujo compresible, unas de las ramas más importantes dentro de la mecánica de fluidos computacional (CFD, por sus siglas en inglés), y cuya aplicación es amplia en distintas áreas de ciencia e ingeniería. A pesar del aumento de los recursos computacionales que se ponen a disposición de la comunidad científica, la solución numérica de problemas complejos de flujo compresible para distintas aplicaciones prácticas es aún un reto. Este tipo de problemas de flujo son extremadamente exigentes en lo que se refiere al cálculo numérico y los requisitos de memoria computacionales. En particular, en esta tesis se estudia la posibilidad de resolver los sistemas algebraicos que subyacen tras la aplicación del método de elementos finitos de una forma desacoplada, una técnica que usualmente se denomina de paso fraccionado o método de segregación. Aunque las técnicas de segregación han sido ampliamente estudiadas y analizadas para el problema incompresible de Navier-Stokes, lo que permite un cálculo separado de las variables velocidad y presión, poco se ha explorado su aplicación a problemas de flujo compresible. El interés en este tipo de técnicas radica no solo en el hecho de que permite un cálculo segregado de las incógnitas del problema (generalmente dando lugar además a sistemas algebraicos mejor condicionados), sino también en la reducción de coste computacional que ofrecen. Se estudian tres problemas distintos dentro de la rama de investigación en CFD de flujos compresibles: las ecuaciones isentrópicas de Navier-Stokes, el problema compresible de Navier-Stokes escrito en variables primitivas (velocidad, presión y temperatura) y el problema compresible de Navier-Stokes planteado de manera clásica en variables conservativas (momento, densidad y energía total). Para cada uno de estos casos, primero se propone una formulación de elementos finitos, estabilizada bajo el marco de trabajo de las multiescalas variacionales (VMS, por sus siglas en inglés), lo que en la práctica permite utilizar la misma interpolación para las variables del problema. En segundo lugar, y una vez las discretizaciones en tiempo y espacio han sido seleccionadas, se proponen métodos de paso fraccionado para cada problema de hasta segundo orden en tiempo. Finalmente, todos los algoritmos se implementan en una plataforma de código paralelo para problemas multifísicos y se llevan a cabo simulaciones representativas con el fin de analizar el rendimiento de las técnicas propuestas.

The pseudopotential method has grown as a powerful tool for multiphase fluid flow simulations within the lattice Boltzmann method framework. We consider that due to its simplicity and computational efficiency, the pseudopotential method could be explored also inside the framework of more traditional Computational Fluid Dynamic methods such as Finite Difference, Finite Volume or Finite Element methods. Following this idea, in this work we start from the macroscopic equations resulting from the pseudopotential lattice Boltzmann method and discretize it by a simple Finite Difference scheme. This pseudopotential based finite difference method is then tested in different benchmark problems such as a planar interface, a smooth droplet oscillation, and a single droplet evaporation. Excellent results were obtained in all tests. One of the advantages of the proposed method is that mesh refinement is straightforward, and converged solutions can be used as a tool of validation for the lattice Boltzmann method. Results indicate that the pseudopotential method is suitable to be used with standard discretization methods such as Finite Difference and that in future works more robust discretizations can be used to further enhance the pseudopotential method application.

The described special methods are applicable for various mathematical physics problems with second-order differential equations involving periodic boundary conditions (PBCs) and first-order homogenous boundary conditions (FBCs). Solutions of some linear and nonlinear problems for parabolic type partial differential equations (PDEs) with FBCs are obtained, using the method of lines (MOL) to approach the PDEs in the time and the discretization in space applying the finite difference scheme with exact spectrum (FDSES). For PBCs we use the finite difference scheme (FDS) for locally approximating periodic function's derivatives in a 2n+1, n>=1 -point stencil, obtaining higher order accuracy approximation. This method in the uniform grid with N mesh points is used to approximate the differential operator of the second and the first-order derivatives in the space. In this paper, we show that the approximation using the FDSES method is equivalent to the spectral differentiation matrix method based on trigonometric (Fourier) interpolant. Considering, that the solutions obtained in solving nonlinear problems can be very significantly different from classical solutions, for example, mathematical modelling of processes where temperature or energy is concentrated in a very narrow interval or around a point, again causes increased interest in such areas of application as laser technology, military sphere, etc. In this regard, also in the given publication, the solution of the "blow-up" phenomenon of the boundary problem of the nonlinear heat conduction equation has been studied and obtained with the above-mentioned high-accuracy solving methods.

Convergence analysis of constraint energy minimizing generalized multiscale finite element method for a linear stochastic parabolic partial differential equation driven by additive noises

In this paper, the extension of the generalized finite difference method to the solution of anisotropic elliptical and parabolic partial differential equations is given in the case of considering two space dimensions. The explicit finite difference formulae and the criterion of stability for the anisotropic parabolic equations are given. The stability condition has been expressed in function of the coefficients of the star equation for irregular clouds of nodes. Various anisotropic cases of simulation of cardiac tissue, including elliptic and parabolic equations, have been solved, and the results show the accuracy of the method.

A comparison of the method of lines to finite difference techniques in solving time-dependent partial differential equations

: Research on the numerical solution of free boundary problems for partial differential equations with locally or sequentially one-dimensional methods has been supported by the U.S. Army Research Office through two consecutive 3-year research contracts. The time and resources provided have made it possible to develop a reasonably comprehensive mathematical theory and flexible numerical algorithms on which to base current computational methods and future research. During the first 3-year period, the method of fractional steps and the method of lines were applied to elliptic and parabolic free boundary problems. During the past three years work was directed toward demonstrating the flexibility of the method of lines for increasingly complex problems, on examining the behavior of certain ill-posed elliptic free boundary problems, and on establishing a mathematical theory for sequentially one-dimensional algorithms. (Author)

A numerical method used for solving a two-phase flow problem as found in typical oil recovery is investigated in the setting of physics-based two-level operator splitting. The governing equations involve an elliptic differential equation coupled with a parabolic convection-dominated equation which poses a severe restriction for obtaining fully implicit numerical solutions. Furthermore, strong heterogeneity of the porous medium over many length scales adds to the complications for effectively solving the system. One viable approach is to split the system into three sub-systems: the elliptic, the hyperbolic, and the parabolic equation, respectively. In doing so, we allow for the use of appropriate numerical discretization for each type of equation and the careful exchange of information between them. We propose to use the multiscale finite volume element method (MsFVEM) for the elliptic and parabolic equations, and a nonoscillatory difference scheme for the hyperbolic equation. Performance of this procedure is confirmed through several numerical experiments.

In this paper, the weighted average-based differential quadrature method is presented for solving one-dimensional homogeneous first-order non-linear parabolic partial differential equation. First, the given solution domain is discretized by using uniform discretization grid point. Next, by using Taylor series expansion we obtain central finite difference discretization of the partial differential equation involving with temporal variable associated with weighted average of partial derivative concerning spatial variable. From this, we obtain the system of nonlinear ordinary differential equations and it is linearized by using the quasilinearization method. Then by using the polynomial-based differential quadrature method for approximating derivative involving with spatial variable at specified grid point, we obtain the system of linear equation. Then they obtained linear system equation is solved by using the LU matrix decomposition method. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm and maximum absolute error norm and the local behavior of the solution is captured exactly. Numerical versus exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root mean square error norm and maximum absolute error norm presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving the non-linear parabolic equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution.

Finite Element and Finite Difference Methods for Hyperbolic Partial Differential Equations