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A time-dependent extension of the first-order hyperbolic system method [J. Comput. Phys., 227 (2007)[315-352] for advection-diffusion problems is introduced. Diffusive/viscous terms are written and discretized as a hyperbolic system, which recovers the original equation in the steady state. The resulting scheme orders advantages over traditional schemes: a dramatic simplification in the discretization, high-order accuracy in the solution gradients, and orders-of-magnitude convergence acceleration. The hyperbolic advection-diffusion system is discretized by the second-order upwind residual-distribution scheme in a unified manner, and the system of implicit-residual-equations is solved by Newton's method over every physical time step. The numerical results are presented for linear and nonlinear advection-diffusion problems, demonstrating solutions and gradients produced to the same order of accuracy, with rapid convergence over each physical time step, typically less than five Newton iterations., Prepared in collaboration with National Aeronautics and Space Administration (NASA), Langley Research Center, Hampton, VA.

In this dissertation the heat flux prediction capabilities of Residual Distribution (RD) schemes for hypersonic flow fields are investigated. Two canonical configurations are considered: the flat plate and the blunt body (cylinder) problems, with a preference for the last one. Both simple perfect gas and more complex thermo-chemical non-equilibrium (TCNEQ) thermodynamic models have been considered.The unexpected results identified early in the investigation lead to a thorough analysis to identify the causes of the unphysical hypersonic heating.The first step taken is the assessment of the quality of flow field and heat transfer predictions obtained with RD methods for subsonic configurations. The result is positive, both for flat plate and cylinder configurations, as RD schemes produce accurate flow solutions and heat flux predictions whenever no shock waves are present, irrespective of the gas model employed.Subsonic results prove that hypersonic heating anomalies are a consequence of the presence of a shock wave in the domain and/or the way it is handled numerically.Regarding hypersonic flows, the carbuncle instability is discarded first as the cause of the erroneous stagnation heating. The anomalies are shown next to be insensitive to the kind and level of dissipation introduced via the (quasi-)positive contribution P to blended B schemes. Additionally, insufficient mesh resolution locally over the region where the shock wave is captured numerically is found to be irrelevant.Capturing the bow shock in a manner that total enthalpy is preserved immediately before and after the numerical shock wave is, on the contrary, important for correct heating prediction.However, a carefully conceived shock capturing term is, by itself, not sufficient to guarantee correct heating predictions, since the LP scheme employed (be it stand-alone in a shock fitting context or combined into a blended scheme for a shock capturing computation) needs to, Doctorat en Sciences de l'ingénieur, info:eu-repo/semantics/nonPublished

This thesis is concerned with the further development and analysis of a class of Computational Fluid Dynamics (CFD) methods for the numerical simulation of compressible flows on unstructured grids, known as Residual Distribution (RD).The RD method constitutes a class of discretization schemes for hyperbolic systems of conservation laws, which forms an attractive alternative to the more classical Finite Volume methods, particularly since it allows better representation of the flow physics by genuinely multi-dimensional upwinding and offers second-order accuracy on a compact stencil.Despite clear advantages of RD schemes, they also have some unexpected anomalies in common with Finite Volume methods and an attempt to resolve them is presented. The most notable anomaly is the violation of the entropy condition, which as a consequence allows unphysical expansion shocks to exist in the numerical solution. In the thesis the genuinely multi-dimensional character of this anomaly is analyzed and a multi-dimensional entropy fix is presented and shown to avoid expansion shocks. Another infamous anomaly is the carbuncle phenomenon, an instability observed in many numerical solutions with strong shocks, such as the bow shock on a blunt body in hypersonic flow. The occurence of the carbuncle phenomenon with RD methods is analyzed and a novel formulation for a shock fix, based on an anisotropic diffusion term added in the shock layer, is presented and shown to cure the anomaly in 2D and 3D hypersonic flow problems.In the present work an effort has been made also to an objective and quantitative assessment of the merits of the RD method for typical aerodynamical engineering applications, such as the transonic flow over airfoils and wings.Validation examples including inviscid, laminar as well as high Reynolds number turbulent flows and comparisons against results from state-of-the-art Finite Volume methods are presented.It is shown that the second-order mu, Doctorat en Sciences de l'ingénieur, info:eu-repo/semantics/nonPublished

In this paper we consider the estimation of the error distribution in a heteroscedastic nonparametric regression model with multivariate covariates. As estimator we consider the empirical distribution function of residuals, which are obtained from multivariate local polynomial fits of the regression and variance functions, respectively. Weak convergence of the empirical residual process to a Gaussian process is proved. We also consider various applications for testing model assumptions in nonparametric multiple regression. The model tests obtained are able to detect local alternatives that converge to zero at an n(-12)-rate, independent of the covariate dimension. We consider in detail a test for additivity of the regression function. (C) 2010 Elsevier Inc. All rights reserved.

Consider a heteroscedastic regression model Y = m(X) + s(X) e, where m(X) = E(Y vertical bar X) and sigma(2)(X) = Var(Y vertical bar X) are unknown, and the error e is independent of the covariate X. We propose a new type of test statistic for testing whether the regression curve m(.) belongs to some parametric family of regression functions. The proposed test statistic measures the distance between the empirical distribution function of the parametric and of the nonparametric residuals. The asymptotic theory of the proposed test is developed, and the proposed testing procedure is illustrated by means of a small simulation study and the analysis of a data set.

In this paper we consider the estimation of the error distribution in a heteroscedastic nonparametric regression model with multivariate covariates. As estimator we consider the empirical distribution function of residuals, which are obtained from multivariate local polynomial fits of the regression and variance functions, respectively. Weak convergence of the empirical residual process to a Gaussian process is proved. We also consider various applications for testing model assumptions in nonparametric multiple regression. The obtained model tests are able to detect local alternatives that converge to zero at n−1/2rate, independent of the covariate dimension. We consider in detail a test for additivity of the regression function.

This thesis presents the author's contribution to the design and implementation of COOLFluiD,an object oriented software platform for the high performance simulation of multi-physics phenomena on unstructured grids. In this context, the final goal has been to provide a reliable tool for handling high speed aerothermodynamic applications. To this end, we introduce a number of design techniques that have been developed in order to provide the framework with flexibilityand reusability, allowing developers to easily integrate new functionalities such as arbitrary mesh-based data structures, numerical algorithms (space discretizations, time stepping schemes, linear system solvers, ),and physical models. Furthermore, we describe the parallel algorithms that we have implemented in order to efficiently read/write generic computational meshes involving millions of degrees of freedom and partition them in a scalable way: benchmarks on HPC clusters with up to 512 processors show their effective suitability for large scale computing. Several systems of partial differential equations, characterizing flows in conditions of thermal and chemical equilibrium (with fixed and variable elemental fractions)and, particularly, nonequilibrium (multi-temperature models) have been integrated in the framework. In order to simulate such flows, we have developed two state-of-the-art flow solvers: 1- a parallel implicit 2D/3D steady and unsteady cell-centered Finite Volume (FV) solver for arbitrary systems of PDE's on hybrid unstructured meshes; 2- a parallel implicit 2D/3D steady vertex-centered Residual Distribution (RD) solver for arbitrary systems of PDE's on meshes with simplex elements (triangles and tetrahedra). The FV~code has been extended to handle all the available physical models, in regimes ranging from incompressible to hypersonic. As far as the RD code is concerned, the strictly conservative variant of the RD method, Doctorat en Sciences de l'ingénieur, info:eu-repo/semantics/nonPublished

This thesis presents the author's contribution to the design and implementation of COOLFluiD,an object oriented software platform for the high performance simulation of multi-physics phenomena on unstructured grids. In this context, the final goal has been to provide a reliable tool for handling high speed aerothermodynamic applications. To this end, we introduce a number of design techniques that have been developed in order to provide the framework with flexibilityand reusability, allowing developers to easily integrate new functionalities such as arbitrary mesh-based data structures, numerical algorithms (space discretizations, time stepping schemes, linear system solvers, ),and physical models. Furthermore, we describe the parallel algorithms that we have implemented in order to efficiently read/write generic computational meshes involving millions of degrees of freedom and partition them in a scalable way: benchmarks on HPC clusters with up to 512 processors show their effective suitability for large scale computing. Several systems of partial differential equations, characterizing flows in conditions of thermal and chemical equilibrium (with fixed and variable elemental fractions)and, particularly, nonequilibrium (multi-temperature models) have been integrated in the framework. In order to simulate such flows, we have developed two state-of-the-art flow solvers: 1- a parallel implicit 2D/3D steady and unsteady cell-centered Finite Volume (FV) solver for arbitrary systems of PDE's on hybrid unstructured meshes; 2- a parallel implicit 2D/3D steady vertex-centered Residual Distribution (RD) solver for arbitrary systems of PDE's on meshes with simplex elements (triangles and tetrahedra). The FV~code has been extended to handle all the available physical models, in regimes ranging from incompressible to hypersonic. As far as the RD code is concerned, the strictly conservative variant of the RD method, Doctorat en Sciences de l'ingénieur, info:eu-repo/semantics/nonPublished

An effective approach is presented for the numerical solution of the equations governing steady laminar and turbulent flow, heat and mass transfer at low Mach number. The approach adopted combines a compact and accurate discretization using the residual-distribution (RD) approach with a fully-coupled implicit solution procedure. The system RD approach adopted employs genuinely-multidimensional upwinding to achieve accurate and stable discrete equations on a highly compact computational stencil. This combines very naturally with the fully-implicit coupled solution procedure for which the number of non-linear iterations required is essentially independent of the grid size. This contrasts with other widely used segregated approaches in which the pressure-velocity system is discretized and solved as a set of scalar equations. A further key distinction from many other implicit methods is that the compact nature of the discretization allows the full convection and diffusion terms in all equations to be treated implicitly without any form of deferred correction. The code developed solves the 2D, axisymmetric and 3D Navier-Stokes equations in incompressible or weakly-compressible form on unstructured grids of triangles or tetrahedra. The RD form of the system Lax-Wendroff scheme is applied to the convection and pressure terms while the viscous terms are treated using the Galerkin finite-element method. The system scheme for convection provides natural stabilization, allowing a collocated variable arrangement to be used. Convection terms in scalar equations are treated using scalar multidimensional RD schemes such as the N and PSI schemes, both of which are positive. The discrete equation system is solved using a fully-coupled implicit approach based on Picard/Newton linearization with the linear system solved using standard Krylov subspace methods (e.g. GMRES or BiCGStab) with ILU(0) preconditioning. A number of practical issues relating to the solution procedure have been

We review some criteria allowing the systematic construction of arbitrary order non-oscillatory residual distribution schemes for the solution of hyperbolic conservation laws on unstructured triangulations. For simplicity, we present the main ideas for the case of scalar conservation laws, and then present results for some representative problems involving the solution of scalar models, of a Cauchy-Riemann system, and of the Euler equations.

An effective approach is presented for the numerical solution of the equations governing steady laminar and turbulent flow, heat and mass transfer at low Mach number. The approach adopted combines a compact and accurate discretization using the residual-distribution (RD) approach with a fully-coupled implicit solution procedure. The system RD approach adopted employs genuinely-multidimensional upwinding to achieve accurate and stable discrete equations on a highly compact computational stencil. This combines very naturally with the fully-implicit coupled solution procedure for which the number of non-linear iterations required is essentially independent of the grid size. This contrasts with other widely used segregated approaches in which the pressure-velocity system is discretized and solved as a set of scalar equations. A further key distinction from many other implicit methods is that the compact nature of the discretization allows the full convection and diffusion terms in all equations to be treated implicitly without any form of deferred correction. The code developed solves the 2D, axisymmetric and 3D Navier-Stokes equations in incompressible or weakly-compressible form on unstructured grids of triangles or tetrahedra. The RD form of the system Lax-Wendroff scheme is applied to the convection and pressure terms while the viscous terms are treated using the Galerkin finite-element method. The system scheme for convection provides natural stabilization, allowing a collocated variable arrangement to be used. Convection terms in scalar equations are treated using scalar multidimensional RD schemes such as the N and PSI schemes, both of which are positive. The discrete equation system is solved using a fully-coupled implicit approach based on Picard/Newton linearization with the linear system solved using standard Krylov subspace methods (e.g. GMRES or BiCGStab) with ILU(0) preconditioning. A number of practical issues relating to the solution procedure have been

We review some criteria allowing the systematic construction of arbitrary order non-oscillatory residual distribution schemes for the solution of hyperbolic conservation laws on unstructured triangulations. For simplicity, we present the main ideas for the case of scalar conservation laws, and then present results for some representative problems involving the solution of scalar models, of a Cauchy-Riemann system, and of the Euler equations.

This thesis presents the construction, the analysis and the verication of compact residual discretizations for the solution of conservation laws on unstructured meshes. The schemes considered belong to the class of residual distribution (RD) or fluctuation splitting (FS) schemes. The methodology presented relies on three main elements: design of compact linear first-order stable schemes for linear hyperbolic PDEs, a positivity preserving procedure mapping stable first-order linear schemes onto nonlinear second-order schemes with non-oscillatory shock capturing capabilities, and a conservative formulation enabling to extend the schemes to nonlinear CLs. These three design steps, and the underlying theoretical tools, are discussed in depth. The nonlinear RD schemes resulting from this construction are tested on a large set of problems involving the solution of scalar models, and systems of CLs. This extensive verification fills the gaps left open, where no theoretical analysis is possible. Numerical results are presented on the Euler equations of a perfect gas, on a two-phase flow model with highly nonlinear thermodynamics, and on the shallow-water equations. On irregular grids, the schemes proposed yield quite accurate and stable solutions even on very difficult computations. Direct comparisone show that these results are more accurate than the ones given by FV and WENO schemes. Moreover, our schemes have a compact nearest-neighbor stencil. This encourages to further develop our approach, toward the design of very high-order schemes, which would represent a very appealing alternative, both in terms of accuracy and efficiency, to now classical FV and ENO/WENO discretizations. These schemes might also be very competitive with respect to very high-order DG schemes., Doctorat en sciences appliquées, info:eu-repo/semantics/nonPublished

In the common nonparametric regression model the problem of testing for the parametric form of the conditional variance is considered. A stochastic process based on the dierence between the empirical processes obtained from the standardized nonparametric residuals under the null hypothesis (of a specic parametric form of the variance function) and the alternative is introduced and its weak convergence established. This result is used for the construction of a Cramer von Mises type statistic for testing the parametric form of the conditional variance. The nite sample properties of a bootstrap version of this test are investigated by means of a simulation study. In particular the new procedure is compared with some of the currently available methods for this problem and its performance is illustrated by means of a data example.

This thesis presents the construction, the analysis and the verication of compact residual discretizations for the solution of conservation laws on unstructured meshes. The schemes considered belong to the class of residual distribution (RD) or fluctuation splitting (FS) schemes. The methodology presented relies on three main elements: design of compact linear first-order stable schemes for linear hyperbolic PDEs, a positivity preserving procedure mapping stable first-order linear schemes onto nonlinear second-order schemes with non-oscillatory shock capturing capabilities, and a conservative formulation enabling to extend the schemes to nonlinear CLs. These three design steps, and the underlying theoretical tools, are discussed in depth. The nonlinear RD schemes resulting from this construction are tested on a large set of problems involving the solution of scalar models, and systems of CLs. This extensive verification fills the gaps left open, where no theoretical analysis is possible. Numerical results are presented on the Euler equations of a perfect gas, on a two-phase flow model with highly nonlinear thermodynamics, and on the shallow-water equations. On irregular grids, the schemes proposed yield quite accurate and stable solutions even on very difficult computations. Direct comparisone show that these results are more accurate than the ones given by FV and WENO schemes. Moreover, our schemes have a compact nearest-neighbor stencil. This encourages to further develop our approach, toward the design of very high-order schemes, which would represent a very appealing alternative, both in terms of accuracy and efficiency, to now classical FV and ENO/WENO discretizations. These schemes might also be very competitive with respect to very high-order DG schemes., Doctorat en sciences appliquées, info:eu-repo/semantics/nonPublished

We are concerned with the solution of time-dependent non-linear hyperbolic partial differential equations. We investigate the combination of residual distribution methods with a consistent mass matrix (discretisation in space) and a Runge–Kutta-type time-stepping (discretisation in time). The introduced non-linear blending procedure allows us to retain the explicit character of the time-stepping procedure. The resulting methods are second order accurate provided that both spatial and temporal approximations are. The proposed approach results in a global linear system that has to be solved at each time-step. An efficient way of solving this system is also proposed. To test and validate this new framework, we perform extensive numerical experiments on a wide variety of classical problems. An extensive numerical comparison of our approach with other multi-stage residual distribution schemes is also given.

We are concerned with the solution of time-dependent non-linear hyperbolic partial differential equations. We investigate the combination of residual distribution methods with a consistent mass matrix (discretisation in space) and a Runge–Kutta-type time-stepping (discretisation in time). The introduced non-linear blending procedure allows us to retain the explicit character of the time-stepping procedure. The resulting methods are second order accurate provided that both spatial and temporal approximations are. The proposed approach results in a global linear system that has to be solved at each time-step. An efficient way of solving this system is also proposed. To test and validate this new framework, we perform extensive numerical experiments on a wide variety of classical problems. An extensive numerical comparison of our approach with other multi-stage residual distribution schemes is also given.

We are concerned with the solution of time-dependent non-linear hyperbolic partial differential equations. We investigate the combination of residual distribution methods with a consistent mass matrix (discretisation in space) and a Runge–Kutta-type time-stepping (discretisation in time). The introduced non-linear blending procedure allows us to retain the explicit character of the time-stepping procedure. The resulting methods are second order accurate provided that both spatial and temporal approximations are. The proposed approach results in a global linear system that has to be solved at each time-step. An efficient way of solving this system is also proposed. To test and validate this new framework, we perform extensive numerical experiments on a wide variety of classical problems. An extensive numerical comparison of our approach with other multi-stage residual distribution schemes is also given.