Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Computational fluid dynamics. --- Galerkin methods. --- Sinc-Galerkin methods --- Sinc methods --- Numerical analysis --- CFD (Computational fluid dynamics) --- Fluid dynamics --- Computer simulation --- Data processing

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This monograph presents computational techniques and numerical analysis to study conservation laws under uncertainty using the stochastic Galerkin formulation. With the continual growth of computer power, these methods are becoming increasingly popular as an alternative to more classical sampling-based techniques. The approach described in the text takes advantage of stochastic Galerkin projections applied to the original conservation laws to produce a large system of modified partial differential equations, the solutions to which directly provide a full statistical characterization of the effect of uncertainties. Polynomial Chaos Methods of Hyperbolic Partial Differential Equations focuses on the analysis of stochastic Galerkin systems obtained for linear and non-linear convection-diffusion equations and for a systems of conservation laws; a detailed well-posedness and accuracy analysis is presented to enable the design of robust and stable numerical methods. The exposition is restricted to one spatial dimension and one uncertain parameter as its extension is conceptually straightforward. The numerical methods designed guarantee that the solutions to the uncertainty quantification systems will converge as the mesh size goes to zero. Examples from computational fluid dynamics are presented together with numerical methods suitable for the problem at hand: stable high-order finite-difference methods based on summation-by-parts operators for smooth problems, and robust shock-capturing methods for highly nonlinear problems. Academics and graduate students interested in computational fluid dynamics and uncertainty quantification will find this book of interest. Readers are expected to be familiar with the fundamentals of numerical analysis. Some background in stochastic methods is useful but not necessary.

Engineering. --- Engineering Fluid Dynamics. --- Numerical Analysis. --- Fluid- and Aerodynamics. --- Numerical analysis. --- Hydraulic engineering. --- Ingénierie --- Analyse numérique --- Technologie hydraulique --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Civil Engineering --- Differential equations, Hyperbolic --- Galerkin methods. --- Numerical solutions. --- Sinc-Galerkin methods --- Sinc methods --- Fluids. --- Fluid mechanics. --- Numerical analysis --- Mathematical analysis --- Engineering, Hydraulic --- Engineering --- Fluid mechanics --- Hydraulics --- Shore protection --- Mechanics --- Physics --- Hydrostatics --- Permeability --- Hydromechanics --- Continuum mechanics

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This volume offers researchers the opportunity to catch up with important developments in the field of numerical analysis and scientific computing and to get in touch with state-of-the-art numerical techniques. The book has three parts. The first one is devoted to the use of wavelets to derive some new approaches in the numerical solution of PDEs, showing in particular how the possibility of writing equivalent norms for the scale of Besov spaces allows to develop some new methods. The second part provides an overview of the modern finite-volume and finite-difference shock-capturing schemes for systems of conservation and balance laws, with emphasis on providing a unified view of such schemes by identifying the essential aspects of their construction. In the last part a general introduction is given to the discontinuous Galerkin methods for solving some classes of PDEs, discussing cell entropy inequalities, nonlinear stability and error estimates.

Differential equations, Partial -- Numerical solutions -- Congresses. --- Electronic books. -- local. --- Galerkin methods -- Congresses. --- Wavelets (Mathematics) -- Congresses. --- Calculus --- Applied Mathematics --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Differential equations, Partial --- Wavelets (Mathematics) --- Galerkin methods --- Numerical solutions --- Sinc-Galerkin methods --- Sinc methods --- Mathematics. --- Partial differential equations. --- Numerical analysis. --- Numerical Analysis. --- Partial Differential Equations. --- Numerical analysis --- Differential equations, partial. --- Partial differential equations --- Mathematical analysis

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This monograph provides a compendium of established and novel error estimation procedures applied in the field of Computational Mechanics. It also includes detailed derivations of these procedures to offer insights into the concepts used to control the errors obtained from employing Galerkin methods in finite and linearized hyperelasticity. The Galerkin methods introduced are considered advanced methods because they remedy certain shortcomings of the well-established finite element method, which is the archetypal Galerkin (mesh-based) method. In particular, this monograph focuses on the systematical derivation of the shape functions used to construct both Galerkin mesh-based and meshfree methods. The mesh-based methods considered are the (conventional) displacement-based, (dual-)mixed, smoothed, and extended finite element methods. In addition, it introduces the element-free Galerkin and reproducing kernel particle methods as representatives of a class of Galerkin meshfree methods. Including illustrative numerical examples relevant to engineering with an emphasis on elastic fracture mechanics problems, this monograph is intended for students, researchers, and practitioners aiming to increase the reliability of their numerical simulations and wanting to better grasp the concepts of Galerkin methods and associated error estimation procedures.

Mechanics. --- Mechanics, Applied. --- Computer mathematics. --- Computational intelligence. --- Solid Mechanics. --- Computational Science and Engineering. --- Computational Intelligence. --- Intelligence, Computational --- Artificial intelligence --- Soft computing --- Computer mathematics --- Electronic data processing --- Mathematics --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Galerkin methods. --- Sinc-Galerkin methods --- Sinc methods --- Numerical analysis --- Solids. --- Data processing. --- Solid state physics --- Transparent solids

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This work describes the propagation properties of the so-called symmetric interior penalty discontinuous Galerkin (SIPG) approximations of the 1-d wave equation. This is done by means of linear approximations on uniform meshes. First, a careful Fourier analysis is constructed, highlighting the coexistence of two Fourier spectral branches or spectral diagrams (physical and spurious) related to the two components of the numerical solution (averages and jumps). Efficient filtering mechanisms are also developed by means of techniques previously proved to be appropriate for classical schemes like finite differences or P1-classical finite elements. In particular, the work presents a proof that the uniform observability property is recovered uniformly by considering initial data with null jumps and averages given by a bi-grid filtering algorithm. Finally, the book explains how these results can be extended to other more sophisticated conforming and non-conforming finite element methods, in particular to quadratic finite elements, local discontinuous Galerkin methods and a version of the SIPG method adding penalization on the normal derivatives of the numerical solution at the grid points.  This work is the first publication to contain a rigorous analysis of the discontinuous Galerkin methods for wave control problems. It will be of interest to a range of researchers specializing in wave approximations.

Waves --- Galerkin methods. --- Approximation theory. --- Mathematics. --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Sinc-Galerkin methods --- Sinc methods --- Numerical analysis --- Cycles --- Hydrodynamics --- Benjamin-Feir instability --- Numerical analysis. --- Fourier analysis. --- Differential equations, partial. --- Algorithms. --- Numerical Analysis. --- Fourier Analysis. --- Approximations and Expansions. --- Partial Differential Equations. --- Applications of Mathematics. --- Algorism --- Algebra --- Arithmetic --- Partial differential equations --- Math --- Science --- Analysis, Fourier --- Mathematical analysis --- Foundations --- Partial differential equations. --- Applied mathematics. --- Engineering mathematics. --- Engineering --- Engineering analysis --- Mathematics