An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method

In this paper we propose a collocation method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. They are categorized as singular initial value problems. The proposed approach is based on a Hermite function collocation (HFC) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The new method reduces the solution of a problem to the solution of a system of algebraic equations. Hermite functions have prefect properties that make them useful to achieve this goal. We compare the present work with some well-known results and show that the new method is efficient and applicable.Comment: 34 pages, 13 figures, Published in "Computer Physics Communications

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

On adomian based numerical schemes for euler and navier-stokes equations, and application to aeroacoustic propagation

140 p.En esta tesis se ha desarrollado un nuevo método de integración en tiempo de tipo derivadas sucesivas (multiderivative), llamado ABS y basado en el algoritmo de Adomian. Su motivación radica en la reducción del coste de simulación para problemas en aeroacústica, muy costosos por su naturaleza transitoria y requisitos de alta precisión. El método ha sido satisfactoriamente empleado en ambas partes de un sistema híbrido, donde se distinguen la parte aerodinámica y la acústica.En la parte aerodinámica las ecuaciones de Navier-Stokes incompresibles son resueltas con unmétodo de proyección clásico. Sin embargo, la fase de predicción de velocidad ha sido modificadapara incluir el método ABS en combinación con dos métodos: una discretización espacial MAC devolúmenes finitos, y también con un método de alto orden basado en ADER. El método se ha validado respecto a los problemas (en 2D) de vórtices de Taylor-Green, y el desarrollo de vórticesde Karman en un cilindro cuadrado. La parte acústica resuelve la propagación de ondas descritaspor las ecuaciones linearizadas de Euler, empleando una discretización de Galerkin discontinua. El método se ha validado respecto a la ecuación de Burgers.El método ABS es sencillo de programar con una formulación recursiva. Los resultados demuestran que su sencillez junto con sus altas capacidades de adaptación lo convierten en un método fácilmente extensible a órdenes altos, a la vez que reduce el coste comparado con otros métodos clásicos

On Adomian Based Numerical Schemes for Euler and Navier-Stokes Equations, and Application to Aeroacoustic Propagation

In this thesis, an Adomian Based Scheme (ABS) for the compressible Navier-Stokes equations is constructed, resulting in a new multiderivative type scheme not found in the context of fluid dynamics. Moreover, this scheme is developed as a means to reduce the computational cost associated with aeroacoustic simulations, which are unsteady in nature with high-order requirements for the acoustic wave propagation. We start by constructing a set of governing equations for the hybrid computational aeroacoustics method, splitting the problem into two steps: acoustic source computation and wave propagation. The first step solves the incompressible Navier-Stokes equation using Chorin's projection method, which can be understood as a prediction-correction method. First, the velocity prediction is obtained solving the viscous Burgers' equation. Then, its divergence-free correction is performed using a pressure Poisson type projection. In the velocity prediction substep, Burgers' equation is solved using two ABS variants: a MAC type implementation, and a ``modern'' ADER method. The second step in the hybrid method, related to wave propagation, is solved combining ABS with the discontinuous Galerkin high-order approach. Described solvers are validated against several test cases: vortex shedding and Taylor-Green vortex problems for the first step, and a Gaussian wave propagation in the second case. Although ABS is a multiderivative type scheme, it is easily programmed with an elegant recursive formulation, even for the general Navier-Stokes equations. Results show that its simplicity combined with excellent adaptivity capabilities allows for a successful extension to very high-order accuracy at relatively low cost, obtaining considerable time savings in all test cases considered.Predoc Gobierno Vasc

Reduced Differential Transform Method for (2+1) Dimensional type of the Zakharov-Kuznetsov ZK(n,n) Equations

In this paper, reduced differential transform method (RDTM) is employed to approximate the solutions of (2+1) dimensional type of the Zakharov-Kuznetsov partial differential equations. We apply these method to two examples. Thus, we have obtained numerical solution partial differential equations of Zakharov-Kuznetsov. These examples are prepared to show the efficiency and simplicity of the method