Location of Repository

This thesis presents the construction, the analysis and the verification of a new form of higher than second order fluctuation splitting discretisation for the solution of steady conservation laws on unstructured meshes. This is an alternative approach to the two existing higher than second order fluctuation splitting schemes, which use submesh reconstruction (developed by Abgrall and Roe) and gradient recovery (developed by Caraemi) to obtain the loacl higher degree polynomials used to evaluate the fluctuation. The new higher than second order approach constructs the polynomial interpolant of the values of the dependent variables at an appropriate number of carefully chosen mesh nodes.\ud \ud As they stand, none of the higher than second order methods can guarantee the absence of spurious oscillations from the flow without the application of an additional smoothing stage. The implementation of a technique that removes unphysical oscillations (devised by Hubbard) as part of a new higher than second order approach will be outlined. The design steps and theoretical bases are discussed in depth.\ud \ud The new higher than second order approach is examined and analysed through application to a series of linear and nonlinear scalar problems, using a pseudo-time-stepping technique to reach steady state solution on two-dimensional structured and unstructured meshes. The results demonstrate its effectiveness in approximating the linear and nolinear scalar problems.\ud \ud This thesis also addresses the development and examination of a multistage high order (in space and time) fluctuation splitting scheme for two-dimensional unsteady scalar advection on triangular unstructured meshes. the method is similar in philosophy to that of multistep high order (in space and time) fluctuation splitting scheme for the approximation of time-dependent hyperbolic conservation laws. The construction and implementation of the high order multistage time-dependent method are discussed in detail and its performance is illustrated using several standard test problems. The multistage high order time-dependent method is evaluated in the context of existing fluctuation splitting approaches to modelling time-dependent problems and some suggestions for their future development are made. Results presented indicate that the multistage high orer method can produce a slightly more accurate solution than the multistep high order method.\u

Publisher: School of Computing (Leeds)

Year: 2007

OAI identifier:
oai:etheses.whiterose.ac.uk:1346

Provided by:
White Rose E-theses Online

Downloaded from
http://etheses.whiterose.ac.uk/1346/1/mebrate.pdf

- (1986). A basis for upwind differencing of the two-dimensional unsteady Eurer equation.
- (2000). A consistent upwind residual scheme for scalar unsteady advection problems. Conference AMIF,
- (1959). A ﬁnite difference method for the numerical computation of discontinuous solutions of the equations of ﬂuid dynamics.
- (2002). A Lax-Wendroff type theorem for residual schemes. Innovative methods for numerical solutions of partial differential equations, World Scientiﬁc,
- (1981). A multiple grid scheme for solving the Euler equations.
- (2000). A new compact high-order multidimensional upwind discretization.
- (1986). A new ﬁnite element formulation for CFD: a discontinuity-capturing operator for multidimensional advective-diffusive systems.
- A new ﬁnite element formulation for CFD: beyond SUPG.
- (1986). A new ﬁnite element formulation for CFD: the generalised streamline operator for multidimensional advective-diffusive systems.
- (2001). A parallel multidimensional upwind algorithm for LES.
- (2005). A third order residual distribution method for steady/unsteady simulations: Formulation and 2D/3D benchmarking.
- (2001). A third-order ﬂuctuation splitting scheme that preserves potential ﬂow.
- Achieving high-order ﬂuctuation splitting schemes by extending the stencil.
- (1981). Approximate Riemann solvers, parameter vectors, and difference schemes.
- (1994). Aspects of unstructured grids and ﬁnite volume solvers for the Euler and Navier-Stokes equations.
- (1993). Compact advection scheme on unstructured grids.
- (1993). Compact advection schemes on unstructured meshes.
- (1997). Compact cell-vertex convection schemes on unstructured meshes.
- (2002). Compact third-order multidimensional upwind scheme for Navier Stokes simulations. In:
- (1998). Compressible ﬂow simulation on unstructured grids using multidimensional upwind schemes.
- (1993). Computationsofinviscidcompressibleﬂowsusingﬂuctuation splittingon triangular meshes. AIAA paper 93-3301,
- (1997). Conservative multidimensional upwinding for steady two-dimensional shallow-water equation.
- (2006). Construction of high order residual distributiveschemes for scalar problems.
- (2004). Construction of second order accurate monotone and stable residual distributive schemes: The steady case.
- (2003). Construction of very high order residual distribution schemes for unsteady advection: preliminary result.
- (2003). Constructionof second orderaccurate monotoneand stable residual distributive schemes: The unsteady case.
- (2002). Deconinck and S.Poedts. Performance comparison of multidimensional upwind residual distribution and dimensionally split ﬁnite volume Roe schemes on the steady solution of conservation laws. Finite Volumes for Complex Applications III,
- (1999). Diffusion characteristics of ﬁnite volume and ﬂuctuation splitting schemes.
- (1999). DiscontinuousGalerkin methods for convection-dominatedproblems.
- (2005). Discretizations that exploit the structure of the governing equations. VKI LS 2006-01, 33rd Computational Fluid Dynamics course, Von Karman Institute for Fluid Dynamics,
- (1989). Dutrado Carmo. A consistentapproximationupwindPetrovGalerkin method for convection dominated problems.
- (1988). Efﬁcient implementation of essentially non-oscillatory shockcapturing schemes.
- (2003). Error estimation and adaptive discretization methods in CFD.
- (1998). Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws.
- (1997). Euler and Navier-Stokes solvers using multidimensional upwind schemes and multigrid acceleration.
- (2002). Finite volume methods for hyperbolic problems,
- (2004). Finite volume methods: foundation and analysis. Encyclopedia of Computational Mechanics,
- (1982). Fluctuation and signals - a framework for numerical evolution problems.
- High order ﬂuctuation schemes on triangular meshes.
- (1999). High order methods for computational physics.
- High resolution shock capturing cell vertex advection schemes on unstructured grids.
- (1999). High-order ENO and WENO schemes for computational ﬂuid dynamics .
- (1995). Hyperbolic systems of conservation laws. Applied Mathematical Sciences,
- (1996). Improving time accuracy for residual distribution schemes.
- (1989). Introduction to ﬁnite element analysis.
- (2005). Introduction to residual distribution methods.
- (2003). Introduction to residual distribution schemes and stabilised ﬁnite elements.
- (2000). LES using a parallel multidimensional upwind solver. ICCFD,
- (1987). Linear advection schemes on triangular meshes.
- (1983). Matrix Computations,
- (2003). Monotone and stable residualdistributionschemesonprismaticspace-timeelements forunsteadyconservation laws.
- (1995). Multidimensional upwind and SUPG methods for the solution of the compressible ﬂow equations on unstructured grids.
- (1995). Multidimensional upwind methods for inviscid and viscous compressible ﬂow.
- (1995). Multidimensional upwind residual discretization schemes for the Euler and Navier-Stokes equation on unstructured meshes.
- (1999). Multidimensional upwind residual distribution schemes and applications. FiniteVolumes forComplex ApplicationsII. Hermes Science Publications,
- (1996). Multidimensional upwinding and grid adaptation for conservation laws.
- (1998). Multidimensionalupwindﬂuctuationdistributionschemesforscalar time dependent problems. Project report,
- Multidimensionalupwinding: its relation to ﬁnite elements.
- (2007). Non-oscillatory third order ﬂuctuation splitting schemes for steady scalar conservation laws.
- (2005). Non-oscillatory third order ﬂuctuation splitting schemes, Finite Volumes for Complex Applications IV,
- (1989). Numerical computation of internal and external ﬂows,
- (1992). Numerical methods for conservation laws. Birkh¨ auser,
- (2003). Numerical methods forconservation laws on structured and unstructured meshes.
- (1996). Numerical Recipes in Fortran 77. Press Syndicate of the Universityof Cambridge,
- (1987). Numerical solution of partial differential equations by the ﬁnite element method,
- (1992). Optimum positive linear schemes for advection in two and three dimensions,
- Optimum” upwind advection on a triangular mesh.
- (1996). Positive matrix distribution schemes for hyperbolic systems.
- (1992). Progress in multidimensional upwinding.
- Residual distribution for general timedependent conservation laws.
- (2001). Residual distributive schemes for conservation laws via adaptive quadrature.
- (2003). Residual distributive schemes for steady problems.
- (2001). Space time residual distribution schemes for hyperbolic conservation laws on unstructured linear ﬁnite elements.
- (2003). Space-time residual distribution schemes for hyperbolicconservation laws overlinear and bilinearelements.
- (2000). Status of multidimensionalupwind residual distribution schemes and application in aeronautics. Fluids
- (1982). Streamline upwind Petrov-Galerkin formulations for convection dominated ﬂows with particular emphasis on the incompressible NavierStokes equations.
- (2004). Study ofa3rd orderresidual distributive scheme for advection-diffusion equations.
- (1989). The design and application of upwind schemes on unstructured meshes.
- (1991). Time dependent calculations using multigrid with applications to unsteady ﬂows past airfoils and wings.
- (2001). Toward the ultimate conservative scheme: Following the quest.
- (2004). Toward very high order accurate schemes for unsteady convection problems on unstructured meshes.
- (2002). Upwind multidimensional residual schemes for steady and unsteady ﬂows.
- (1997). Upwind residual distribution methods for compressible ﬂow: an alternative to ﬁnite volume and ﬁnite element methods.
- (1999). Upwind residual distributionschemes for chemical non-equilibrium ﬂows,
- (1997). Wave propagation algorithms for multi-dimensional hyperbolic systems.

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.