A RBF partition of unity collocation method based on finite difference for initial-boundary value problems

Meshfree radial basis function (RBF) methods are popular tools used to numerically solve partial differential equations (PDEs). They take advantage of being flexible with respect to geometry, easy to implement in higher dimensions, and can also provide high order convergence. Since one of the main disadvantages of global RBF-based methods is generally the computational cost associated with the solution of large linear systems, in this paper we focus on a localizing RBF partition of unity method (RBF-PUM) based on a finite difference (FD) scheme. Specifically, we propose a new RBF-PUM-FD collocation method, which can successfully be applied to solve time-dependent PDEs. This approach allows to significantly decrease ill-conditioning of traditional RBF-based methods. Moreover, the RBF-PUM-FD scheme results in a sparse matrix system, reducing the computational effort but maintaining at the same time a high level of accuracy. Numerical experiments show performances of our collocation scheme on two benchmark problems, involving unsteady convection-diffusion and pseudo-parabolic equations

Adomian decomposition method, nonlinear equations and spectral solutions of burgers equation

Tese de doutoramento. Ciências da Engenharia. 2006. Faculdade de Engenharia. Universidade do Porto, Instituto Superior Técnico. Universidade Técnica de Lisbo

The Magnus expansion and some of its applications

Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial re-summation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related non-perturbative expansions. Second, to provide a bridge with its implementation as generator of especial purpose numerical integration methods, a field of intense activity during the last decade. Third, to illustrate with examples the kind of results one can expect from Magnus expansion in comparison with those from both perturbative schemes and standard numerical integrators. We buttress this issue with a revision of the wide range of physical applications found by Magnus expansion in the literature.Comment: Report on the Magnus expansion for differential equations and its applications to several physical problem

Stability analysis of linear ODE-PDE interconnected systems

Les systèmes de dimension infinie permettent de modéliser un large spectre de phénomènes physiques pour lesquels les variables d'états évoluent temporellement et spatialement. Ce manuscrit s'intéresse à l'évaluation de la stabilité de leur point d'équilibre. Deux études de cas seront en particulier traitées : l'analyse de stabilité des systèmes interconnectés à une équation de transport, et à une équation de réaction-diffusion. Des outils théoriques existent pour l'analyse de stabilité de ces systèmes linéaires de dimension infinie et s'appuient sur une algèbre d'opérateurs plutôt que matricielle. Cependant, ces résultats d'existence soulèvent un problème de constructibilité numérique. Lors de l'implémentation, une approximation est réalisée et les résultats sont conservatifs. La conception d'outils numériques menant à des garanties de stabilité pour lesquelles le degré de conservatisme est évalué et maîtrisé est alors un enjeu majeur. Comment développer des critères numériques fiables permettant de statuer sur la stabilité ou l'instabilité des systèmes linéaires de dimension infinie ? Afin de répondre à cette question, nous proposons ici une nouvelle méthode générique qui se décompose en deux temps. D'abord, sous l'angle de l'approximation sur les polynômes de Legendre, des modèles augmentés sont construits et découpent le système original en deux blocs : d'une part, un système de dimension finie approximant est isolé, d'autre part, l'erreur de troncature de dimension infinie est conservée et modélisée. Ensuite, des outils fréquentiels et temporels de dimension finie sont déployés afin de proposer des critères de stabilité plus ou moins coûteux numériquement en fonction de l'ordre d'approximation choisi. En fréquentiel, à l'aide du théorème du petit gain, des conditions suffisantes de stabilité sont obtenues. En temporel, à l'aide du théorème de Lyapunov, une sous-estimation des régions de stabilité est proposée sous forme d'inégalité matricielle linéaire et une sur-estimation sous forme de test de positivité. Nos deux études de cas ont ainsi été traitées à l'aide de cette méthodologie générale. Le principal résultat obtenu concerne le cas des systèmes EDO-transport interconnectés, pour lequel l'approximation et l'analyse de stabilité à l'aide des polynômes de Legendre mène à des estimations des régions de stabilité qui convergent exponentiellement vite. La méthode développée dans ce manuscrit peut être adaptée à d'autres types d'approximations et exportée à d'autres systèmes linéaires de dimension infinie. Ce travail ouvre ainsi la voie à l'obtention de conditions nécessaires et suffisantes de stabilité de dimension finie pour les systèmes de dimension infinie.Infinite dimensional systems allow to model a large panel of physical phenomena for which the state variables evolve both temporally and spatially. This manuscript deals with the evaluation of the stability of their equilibrium point. Two case studies are treated in particular: the stability analysis of ODE-transport, and ODE-reaction-diffusion interconnected systems. Theoretical tools exist for the stability analysis of these infinite-dimensional linear systems and are based on an operator algebra rather than a matrix algebra. However, these existence results raise a problem of numerical constructibility. During implementation, an approximation is performed and the results are conservative. The design of numerical tools leading to stability guarantees for which the degree of conservatism is evaluated and controlled is then a major issue. How can we develop reliable numerical criteria to rule on the stability or instability of infinite-dimensional linear systems? In order to answer this question, one proposes here a new generic method, which is decomposed in two steps. First, from the perspective of Legendre polynomials approximation, augmented models are built and split the original system into two blocks: on the one hand, a finite-dimensional approximated system is isolated, on the other hand, the infinite-dimensional truncation error is preserved and modeled. Then, frequency and time tools of finite dimension are deployed in order to propose stability criteria that have high or low numerical load depending on the approximated order. In frequencies, with the aid of the small gain theorem, sufficient stability conditions are obtained. In temporal, with the aid of the Lyapunov theorem, an under estimate of the stability regions is proposed as a linear matrix inequality and an over estimate as a positivity test. Our two case studies have been treated with this general methodology. The main result concerns the case of ODE-transport interconnected systems, for which the approximation and stability analysis using Legendre polynomials leads to exponentially fast converging estimates of stability regions. The method developed in this manuscript can be adapted to other types of approximations and exported to other infinite-dimensional linear systems. Thus, this work opens the way to obtain necessary and sufficient finite-dimensional conditions of stability for infinite-dimensional systems

Pole Placement and Reduced-Order Modelling for Time-Delayed Systems Using Galerkin Approximations

The dynamics of time-delayed systems (TDS) are governed by delay differential equa- tions (DDEs), which are infinite dimensional and pose computational challenges. The Galerkin approximation method is one of several techniques to obtain the spectrum of DDEs for stability and stabilization studies. In the literature, Galerkin approximations for DDEs have primarily dealt with second-order TDS (second-order Galerkin method), and the for- mulations have resulted in spurious roots, i.e., roots that are not among the characteristic roots of the DDE. Although these spurious roots do not affect stability studies, they never- theless add to the complexity and computation time for control and reduced-order modelling studies of DDEs. A refined mathematical model, called the first-order Galerkin method, is proposed to avoid spurious roots, and the subtle differences between the two formulations (second-order and first-order Galerkin methods) are highlighted with examples. For embedding the boundary conditions in the first-order Galerkin method, a new pseudoinverse-based technique is developed. This method not only gives the exact location of the rightmost root but also, on average, has a higher number of converged roots when compared to the existing pseudospectral differencing method. The proposed method is combined with an optimization framework to develop a pole-placement technique for DDEs to design closed-loop feedback gains that stabilize TDS. A rotary inverted pendulum system apparatus with inherent sensing delays as well as deliberately introduced time delays is used to experimentally validate the Galerkin approximation-based optimization framework for the pole placement of DDEs. Optimization-based techniques cannot always place the rightmost root at the desired location; also, one has no control over the placement of the next set of rightmost roots. However, one has the precise location of the rightmost root. To overcome this, a pole- placement technique for second-order TDS is proposed, which combines the strengths of the method of receptances and an optimization-based strategy. When the method of receptances provides an unsatisfactory solution, particle swarm optimization is used to improve the location of the rightmost pole. The proposed approach is demonstrated with numerical studies and is validated experimentally using a 3D hovercraft apparatus. The Galerkin approximation method contains both converged and unconverged roots of the DDE. By using only the information about the converged roots and applying the eigenvalue decomposition, one obtains an r-dimensional reduced-order model (ROM) of the DDE. To analyze the dynamics of DDEs, we first choose an appropriate value for r; we then select the minimum value of the order of the Galerkin approximation method system at which at least r roots converge. By judiciously selecting r, solutions of the ROM and the original DDE are found to match closely. Finally, an r-dimensional ROM of a 3D hovercraft apparatus in the presence of delay is validated experimentally

On new and improved semi-numerical techniques for solving nonlinear fluid flow problems.

Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.Most real world phenomena is modeled by ordinary and/or partial differential equations. Most of these equations are highly nonlinear and exact solutions are not always possible. Exact solutions always give a good account of the physical nature of the phenomena modeled. However, existing analytical methods can only handle a limited range of these equations. Semi-numerical and numerical methods give approximate solutions where exact solutions are impossible to find. However, some common numerical methods give low accuracy and may lack stability. In general, the character and qualitative behaviour of the solutions may not always be fully revealed by numerical approximations, hence the need for improved semi-numerical methods that are accurate, computational efficient and robust. In this study we introduce innovative techniques for finding solutions of highly nonlinear coupled boundary value problems. These techniques aim to combine the strengths of both analytical and numerical methods to produce efficient hybrid algorithms. In this work, the homotopy analysis method is blended with spectral methods to improve its accuracy. Spectral methods are well known for their high levels of accuracy. The new spectral homotopy analysis method is further improved by using a more accurate initial approximation to accelerate convergence. Furthermore, a quasi-linearisation technique is introduced in which spectral methods are used to solve the linearised equations. The new techniques were used to solve mathematical models in fluid dynamics. The thesis comprises of an introductory Chapter that gives an overview of common numerical methods currently in use. In Chapter 2 we give an overview of the methods used in this work. The methods are used in Chapter 3 to solve the nonlinear equation governing two-dimensional squeezing flow of a viscous fluid between two approaching parallel plates and the steady laminar flow of a third grade fluid with heat transfer through a flat channel. In Chapter 4 the methods were used to find solutions of the laminar heat transfer problem in a rotating disk, the steady flow of a Reiner-Rivlin fluid with Joule heating and viscous dissipation and the classical von Kάrmάn equations for boundary layer flow induced by a rotating disk. In Chapter 5 solutions of steady two-dimensional flow of a viscous incompressible fluid in a rectangular domain bounded by two permeable surfaces and the MHD viscous flow problem due to a shrinking sheet with a chemical reaction, were solved using the new methods

2nd International Conference on Numerical and Symbolic Computation

The Organizing Committee of SYMCOMP2015 – 2nd International Conference on Numerical and Symbolic Computation: Developments and Applications welcomes all the participants and acknowledge the contribution of the authors to the success of this event. This Second International Conference on Numerical and Symbolic Computation, is promoted by APMTAC - Associação Portuguesa de Mecânica Teórica, Aplicada e Computacional and it was organized in the context of IDMEC/IST - Instituto de Engenharia Mecânica. With this ECCOMAS Thematic Conference it is intended to bring together academic and scientific communities that are involved with Numerical and Symbolic Computation in the most various scientific area