Herman Deconinck

Design principles for bounded higher-order convection schemes - a unified approach

The design of bounded, higher-order convection schemes is considered with a view to selecting those discretizations giving good resolution of sharp gradients, while at the same time providing competitive accuracy and convergence behaviour when applied to smooth, recirculating flows. The present work contains a detailed classification and analysis and extensive tables of most non-linear scalar convection schemes so far proposed within the cell-centred, finite-volume framework. The analysis includes a review and comparison of the two most frequently-used non-linear approaches, flux limiters (FL) and normalized variables (NV), and the three major boundedness criteria typically employed: total-variation diminishing (TVD), positivity and the convection-boundedness criterion (CBC). All NV schemes considered are converted to FL form to allow direct comparison and classification of a wide range of schemes. Several specific design principles for positive non-linear schemes are considered and it is shown how these can be applied to understand the relative performance of different approaches. Finally the performance of many existing schemes is compared and ranked on the basis of two scalar convection test cases, one smooth and one discontinuous, which demonstrates the wide variation in both accuracy and convergence behaviour of the various schemes and the benefits of the design principles considered.

A multidimensional generalization of Roe's flux difference splitter for the euler equations

Abstract Upwind methods for the 1-D Euler equations, such as TVD schemes based on Roes approximate Riemann solver, are reinterpreted as residual distribution schemes, assuming continuous piecewise linear space variation of the unknowns defined at the cell vertices. From this analysis three distinct steps are identified, each allowing for a multidimensional generalization without reference to dimensional splitting or 1-D Riemann problems. A key element is the necessity to have continuous piecewise linear variation of the unknowns, requiring linear triangles in two space dimensions and tetrahedra in three space dimensions. Flux differences naturally generalize to flux contour integrals over the triangles. Roes flux difference splitter naturally generalizes to a multidimensional flux balance splitter if one assumes that the parameter vector variable is the primary dependent unknown having linear variation in space. Nonlinear positive and second-order scalar distribution schemes provide a true generalization of the TVD schemes in one space dimension. Although refinements and improvements are still possible for all these elements, computational examples show that these generalizations present a new framework for solving the multidimensional Euler equations.

High-Order Methods for Computational Physics

R. Abgrall, T. Sonar, O. Friedrich, G. Billet: High Order Approximations for Compressible Fluid Dynamics on Unstructured and Cartesian Meshes * B. Cockburn: Discontinuous Galerkin Methods for Convection-Dominated Problems * R.D. Henderson: Adaptive Spectral Element Methods for Turbulence and Transition * C. Schwab:

Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes

hp

Multidimensional upwind schemes based on fluctuation-splitting for systems of conservation laws

-FEM for Fluid Flow Simulation * C * W. Shu: High Order ENO and WENO Schemes for Computational Fluid Dynamics.

The COOLFluiD framework: design solutions for high performance object oriented scientific computing software

We consider the numerical solution of the shallow water equations on unstructured grids. We focus on flows over wet areas. The extension to the case of dry bed will be reported elsewhere. The shallow water equations fall into the category of systems of conservation laws which can be symmetrized thanks to the existence of a mathematical entropy coinciding, in this case, with the total energy. Our aim is to show the application of a particular class of conservative residual distribution (RD) schemes to the discretization of the shallow water equations and to analyze their discrete accuracy and stability properties. We give a review of conservative RD schemes showing relations between different approaches previously published, and recall L^~ stability and accuracy criteria characterizing the schemes. In particular, the accuracy of the RD method in presence of source terms is analyzed, and conditions to construct rth order discretizations on irregular triangular grids are proved. It is shown that the RD approach gives a natural way of obtaining high order discretizations which, moreover, preserves exactly the steady lake at rest solution independently on mesh topology, nature of the variation of the bottom and polynomial order of interpolation used for the unknowns. We also consider more general analytical solutions which are less investigated from the numerical view point. On irregular grids, linearity preserving RD schemes yield a truly second order approximation. We also sketch a strategy to achieve discretizations which preserve exactly some of these solutions. Numerical results on steady and time-dependent problems involving smooth and non-smooth variations of the bottom topology show very promising features of the approach.

Convection algorithms based on a diagonalization procedure for the multidimensional Euler equations

A class of truly multidimensional upwind schemes for the computation of inviscid compressible flows is presented here, applicable to unstructured cell-vertex grids. These methods use very compact stencils and produce sharp resolution of discontinuities with no overshoots.

Error estimation and adaptive discretization methods in computational fluid dynamics

The numerical simulation of complex physical phenomena is a challenging endeavor. Software packages developed for such purpose should combine high performance and extreme flexibility, in order to allow an easy integration of new algorithms, models and functionalities, without penalizing run-time efficiency. COOLFluiD is an object-oriented framework for multi-physics simulations using multiple numerical methods on unstructured grids, aiming at satisfying these needs. To this end, specific design patterns and advanced techniques, combining static and dynamic polymorphism, have been employed to attain modularity and efficiency. Some of the main design and implementation solutions adopted in COOLFluiD are presented in this paper, in particular the Perspective and the Method-Command Patterns, used to implement respectively the physical models and the numerical modules.

A Novel Two-Dimensional Viscous Inverse Design Method for Turbomachinery Blading

Convection algorithms for the multidimensional Euler equations are presented based on characteristic propagation properties, with the aim of following closely the physical transfer of information. As opposed to the one dimensional case, information in two or three dimensional Euler flows is propagated in infinitely many directions, each one corresponding to an arbitrary wave front normal. It is shown that a complete decoupling (diagonalization) of the Euler equations can be obtained by an appropriate choice of wave front propagation directions. Actually,two directions are sufficient, one related to the pressure gradient and another related to the local strain tensor. This new formulation allows the definition of numerical schemes with upwind properties depending only on the local flow properties and not on the mesh geometry.

Implicit upwind residual-distribution Euler and Navier-Stokes solver on unstructured meshes

Adaptive Mesh Generation.- Adjoint Error Correction for Integral Outputs.- Adaptive Finite Element Methods for Incompressible Fluid Flow.- A General Lagrangian Formulation for the Computation of A Posteriori Finite Element Bounds.- Computable Error Estimators and Adaptive Techniques for Fluid Flow Problems.- Adaptive Finite Element Approximation of Hyperbolic Problems.