604 research outputs found
Unsteady incompressible Navier-Stokes simulations on deforming domains
Mesh generation has been an important topic of research for the past four decades, primarily because it is one of the critical elements in the numerical simulation of fluid flows. One of the main current issues in this regard is mesh generation and flow solution on domains with moving boundaries. In this research, a novel scheme has been proposed for mesh generation on domains with moving boundaries, with the location of boundary nodes known at any particular time. A new set of linearized equations is derived based on a full nonlinear elliptic grid generation system. The basic assumption in deriving these new equations is that each node experiences only a small amount of disturbance when the mesh moves from one time to the next. Comparison with grids generated by the full elliptic system shows that this new method can generate high quality grids with significantly less computational cost. Inherently, the flow on such a domain will be unsteady. The Navier-Stokes equations for unsteady 2D laminar incompressible flow are expressed in the primitive variables formulation. A SIMPLE-like scheme is applied to link the pressure and velocity fields and ensure conservation of mass is satisfied. The equations are discretized in a pure finite difference formulation and solved by implicitly marching in time. The flow solver is validated against results in the literature for flow through a channel with a moving indentation along one wall
Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs
This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic PDE. We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on with , and that the PDE operator may depend non-affinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [CDS1, CDS2], we show that the state and the control are analytic as functions depending on these parameters . We
establish sparsity of generalized polynomial chaos (gpc) expansions of both, state and control, in terms of the stochastic coordinate sequence of the random inputs, and prove convergence rates of best -term truncations of these expansions. Such truncations are the key for subsequent computations since they do {\em not} assume that the stochastic input data has a finite expansion. In the follow-up paper [KS2], we explain two methods how such best -term truncations can practically be computed, by greedy-type algorithms
as in [SG, Gi1], or by multilevel Monte-Carlo methods as in
[KSS]. The sparsity result allows in conjunction with adaptive wavelet Galerkin schemes for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [DK, GK, K], see [KS2]
Splitting and composition methods in the numerical integration of differential equations
We provide a comprehensive survey of splitting and composition methods for
the numerical integration of ordinary differential equations (ODEs). Splitting
methods constitute an appropriate choice when the vector field associated with
the ODE can be decomposed into several pieces and each of them is integrable.
This class of integrators are explicit, simple to implement and preserve
structural properties of the system. In consequence, they are specially useful
in geometric numerical integration. In addition, the numerical solution
obtained by splitting schemes can be seen as the exact solution to a perturbed
system of ODEs possessing the same geometric properties as the original system.
This backward error interpretation has direct implications for the qualitative
behavior of the numerical solution as well as for the error propagation along
time. Closely connected with splitting integrators are composition methods. We
analyze the order conditions required by a method to achieve a given order and
summarize the different families of schemes one can find in the literature.
Finally, we illustrate the main features of splitting and composition methods
on several numerical examples arising from applications.Comment: Review paper; 56 pages, 6 figures, 8 table
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
Analytical and Numerical Methods for Differential Equations and Applications
The book is a printed version of the Special issue Analytical and Numerical Methods for Differential Equations and Applications, published in Frontiers in Applied Mathematics and Statistic
The exponentially convergent trapezoidal rule
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators
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Computational Multiscale Methods
Many physical processes in material sciences or geophysics are characterized by inherently complex interactions across a large range of non-separable scales in space and time. The resolution of all features on all scales in a computer simulation easily exceeds today's computing resources by multiple orders of magnitude. The observation and prediction of physical phenomena from multiscale models, hence, requires insightful numerical multiscale techniques to adaptively select relevant scales and effectively represent unresolved scales. This workshop enhanced the development of such methods and the mathematics behind them so that the reliable and efficient numerical simulation of some challenging multiscale problems eventually becomes feasible in high performance computing environments
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