DSpace Collection:
http://hdl.handle.net/2381/4255
2016-02-14T12:41:52ZStable multispeed lattice Boltzmann methods
http://hdl.handle.net/2381/4278
Title: Stable multispeed lattice Boltzmann methods
Authors: Brownlee, R.A.; Gorban, Alexander N.; Levesley, Jeremy
Abstract: We demonstrate how to produce a stable multispeed lattice Boltzmann method (LBM) for a wide range of velocity sets, many of which were previously thought to be intrinsically unstable. We use non-Gauss--Hermitian cubatures. The method operates stably for almost zero viscosity, has second-order accuracy, suppresses typical spurious oscillation (only a modest Gibbs effect is present) and introduces no artificial viscosity. There is almost no computational cost for this innovation.
DISCLAIMER: Additional tests and wide discussion of this preprint show that the claimed property of coupled steps: no artificial dissipation and the second-order accuracy of the method are valid only on sufficiently fine grids. For coarse grids the higher-order terms destroy coupling of steps and additional dissipation appears.
The equations are true.2009-03-02T12:55:44ZStability and stabilisation of the lattice Boltzmann method Magic steps and salvation operations
http://hdl.handle.net/2381/4277
Title: Stability and stabilisation of the lattice Boltzmann method Magic steps and salvation operations
Authors: Brownlee, R.A.; Gorban, Alexander N.; Levesley, Jeremy
Abstract: We revisit the classical stability versus accuracy dilemma for the lattice Boltzmann methods (LBM). Our goal is a stable method of second-order accuracy for fluid dynamics based on the lattice Bhatnager–Gross–Krook method (LBGK).
The LBGK scheme can be recognised as a discrete dynamical system generated by free-flight and entropic involution. In this framework the stability and accuracy analysis are more natural. We find the necessary and sufficient conditions for second-order accurate fluid dynamics modelling. In particular, it is proven that in order to guarantee second-order accuracy the distribution should belong to a distinguished surface – the invariant film (up to second-order in the time step). This surface is the trajectory of the (quasi)equilibrium distribution surface under free-flight.
The main instability mechanisms are identified. The simplest recipes for stabilisation add no artificial dissipation (up to second-order) and provide second-order accuracy of the method. Two other prescriptions add some artificial dissipation locally and prevent the system from loss of positivity and local blow-up. Demonstration of the proposed stable LBGK schemes are provided by the numerical simulation of a 1D shock tube and the unsteady 2D-flow around a square-cylinder up to Reynolds number O(10000).2009-03-02T12:48:57ZA Geometric Description of the m-Cluster Categories of Type D_n
http://hdl.handle.net/2381/4276
Title: A Geometric Description of the m-Cluster Categories of Type D_n
Authors: Baur, Karin; Marsh, Robert J.
Abstract: We show that the m-cluster category of type Dn is equivalent to a certain geometrically defined category of arcs in a punctured regular nm – m + 1-gon. This generalises a result of Schiffler for m=1. We use the notion of the mth power of translation quiver to realise the m-cluster category in terms of the cluster category.2009-03-02T12:33:56ZDiscontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems on Anisotropically Refined Meshes
http://hdl.handle.net/2381/4275
Title: Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems on Anisotropically Refined Meshes
Authors: Georgoulis, Emmanuil H.; Hal, Edward; Houston, Paul
Abstract: In this paper we consider the a posteriori and a priori error analysis of discontinuous Galerkin interior penalty methods for second order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes. In particular, we discuss the question of error estimation for linear target functionals, such as the outow flux and the local average of the solution. Based on our a posteriori error bound we design and implement the corresponding
adaptive algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local isotropic and anisotropic mesh refinement. The theoretical results are illustrated by a series of numerical experiments.2009-03-02T12:14:20ZHigh Order WENO Finite Volume Schemes Using Polyharmonic Spline Reconstruction
http://hdl.handle.net/2381/4274
Title: High Order WENO Finite Volume Schemes Using Polyharmonic Spline Reconstruction
Authors: Aboiyar, Terhemen; Georgoulis, Emmanuil H.; Iske, Armin
Abstract: Polyharmonic splines are utilized in the WENO reconstruction of finite volume discretizations, yielding a numerical method for scalar conservation laws of arbitrary high order. The resulting WENO reconstruction method is, unlike previous WENO schemes using polynomial reconstructions, numerically stable and very flexible. Moreover, due to the theory of polyharmonic splines, optimal reconstructions are obtained in associated native Sobolev-type spaces, called Beppo Levi spaces. This in turn yields a very natural choice for the oscillation indicator, as required in the WENO reconstruction method. The key ingredients of the proposed polyharmonic splineWENO reconstruction algorithm are explained in detail, and one numerical example is given for illustration.2009-03-02T12:07:28ZAdaptivity and A Posteriori Error Estimation For DG Methods on Anisotropic Meshes
http://hdl.handle.net/2381/4272
Title: Adaptivity and A Posteriori Error Estimation For DG Methods on Anisotropic Meshes
Authors: Houston, Paul; Georgoulis, Emmanuil H.; Hall, Edward2009-02-26T16:08:54ZContinuous and Discontinuous Finite Element Methods for Convection-Diffusion Problems: A Comparison
http://hdl.handle.net/2381/4271
Title: Continuous and Discontinuous Finite Element Methods for Convection-Diffusion Problems: A Comparison
Authors: Cangiani, Andrea; Georgoulis, Emmanuil H.; Jensen, Max
Abstract: We compare numerically the performance of a new continuous-discontinuous finite element method (CDFEM) for linear convection-diffusion equations with three well-known upwind finite element formulations, namely with the streamline upwind Petrov-Galerkin finite element method, the residualfree bubble method and the discontinuous Galerkin finite element method. The defining feature of the CDFEM is that it uses discontinuous approximation spaces in the vicinity of layers while continuous FEM approximation are employed elsewhere.2009-02-26T16:04:23ZNorm Preconditioners for discontinuous Galerkin hp-Finite Element methods.
http://hdl.handle.net/2381/4270
Title: Norm Preconditioners for discontinuous Galerkin hp-Finite Element methods.
Authors: Georgoulis, Emmanuil H.; Loghin, Daniel
Abstract: We consider a norm-preconditioning approach for the solution of discontinuous Galerkin finite element discretizations of second order PDE with non-negative characteristic form. In particular, we perform an analysis for the general case of discontinuous hp-finite element discretizations. Our solution method is a norm-preconditioned three-term GMRES routine. We find that for symmetric positive-definite diffusivity tensors the convergence of our solver is independent of discretization, while for the semidefinite case both theory and experiment indicate dependence on both h and p. Numerical results are included to illustrate performance on several test cases.2009-02-26T15:53:35ZKrylov-Subspace Preconditioners for Discontinuous Galerkin Finite Element Methods
http://hdl.handle.net/2381/4269
Title: Krylov-Subspace Preconditioners for Discontinuous Galerkin Finite Element Methods
Authors: Georgoulis, Emmanuil H.; Loghin, Daniel
Abstract: Standard (conforming) finite element approximations of convection-dominated convection-diffusion problems often exhibit poor stability properties that manifest themselves as non-physical oscillations polluting the numerical solution. Various techniques have been proposed for the stabilisation of finite element methods (FEMs) for convection-diffusion problems, such as the popular streamline upwind Petrov-Galerkin (SUPG) method, and its variants. During the last decade, families of discontinuous Galerkin finite element methods (DGFEMs) have been proposed for the numerical solution of convection-diffusion problems, due to the many attractive properties they exhibit. In particular, DGFEMs admit good stability properties, they offer flexibility in the mesh design (irregular meshes are admissible) and in the imposition of boundary conditions (Dirichlet boundary conditions are weakly imposed), and they are increasingly popular in the context of hp-adaptive algorithms. The increase in popularity for DGFEMs has created a corresponding demand for developing corresponding linear solvers. This work aims to provide an overview of the current state of affairs in the solution of DGFEM-linear problems and present some recent results on the preconditioning of stiffness matrices arising from DGFEM discretisations of steady-state convection-diffusion boundary-value problems. More specifically, preconditioners are derived for which the2009-02-26T15:46:39ZStabilisation of the lattice-Boltzmann method using the Ehrenfests' coarse-graining
http://hdl.handle.net/2381/4268
Title: Stabilisation of the lattice-Boltzmann method using the Ehrenfests' coarse-graining
Authors: Brownlee, R.A.; Gorban, Alexander N.; Levesley, Jeremy
Abstract: The lattice-Boltzmann method (LBM) and its variants have emerged as promising, computationally efficient and increasingly popular numerical methods for modelling complex fuid flow. However, it is acknowledged that the method can demonstrate numerical instabilities, e.g., in the vicinity of shocks. We propose a simple and novel technique to stabilise the lattice-Boltzmann method by monitoring the difference between microscopic and macroscopic entropy. Populations are returned to their equilibrium states if a threshold value is exceeded. We coin the name Ehrenfests' steps for this procedure in homage to the vehicle that we use to introduce the procedure, namely, the Ehrenfests' idea of coarse-graining.2009-02-26T15:42:45ZA class of gradings of simple Lie algebras
http://hdl.handle.net/2381/4267
Title: A class of gradings of simple Lie algebras
Authors: Baur, Karin; Wallach, Nolan
Abstract: Abstract. In this paper we give a classification of parabolic subalgebras of simple Lie algebras over C that satisfy two properties. The first property is Lynch’s sufficient condition for the vanishing of certain Lie algebra cohomology spaces for generalized Whittaker modules associated with the parabolic subalgebra and the second is that the moment map of the cotangent bundle of the corresponding generalized flag variety be birational onto its image. We will call this condition the moment map condition.2009-02-26T15:05:27ZBasic Types of Coarse-Graining
http://hdl.handle.net/2381/4266
Title: Basic Types of Coarse-Graining
Authors: Gorban, Alexander N.
Abstract: Summary. We consider two basic types of coarse-graining: the Ehrenfests’ coarse-graining and its extension to a general principle of non-equilibrium thermodynamics, and the coarse-graining based on uncertainty of dynamical models and ε-motions (orbits). Non-technical discussion of basic notions and main coarse-graining theorems are presented: the theorem about entropy overproduction for the Ehrenfests’ coarse-graining and its generalizations, both for conservative and for dissipative systems, and the theorems about stable properties and the Smale order for ε-motions of general dynamical systems including structurally unstable systems. Computational kinetic models of macroscopic dynamics are considered. We construct a theoretical basis for these kinetic models using generalizations of the Ehrenfests’ coarse-graining. General theory of reversible regularization and filtering semi-groups in kinetics is presented, both for linear and non-linear filters. We obtain explicit expressions and entropic stability conditions for filtered equations. A brief discussion of coarse-graining by rounding and by small noise is also presented.2009-02-26T14:58:19ZSecant Dimensions of Minimal Orbits
http://hdl.handle.net/2381/4265
Title: Secant Dimensions of Minimal Orbits
Authors: Baur, Karin; Draisma, Jan; de Graff, Willem
Abstract: We present an algorithm for computing the dimensions of higher secant varieties of minimal orbits. Experiments with this algorithm lead to many conjectures on secant dimensions, especially of Grassmannians and Segre products. For these two classes of minimal orbits, we also point out a relation between the existence of certain codes and non-defectiveness of certain higher secant varieties.2009-02-26T14:43:49ZExtending the range of error estimates for radial approximation in Euclidean space and on spheres
http://hdl.handle.net/2381/4264
Title: Extending the range of error estimates for radial approximation in Euclidean space and on spheres
Authors: Brownlee, R.A.; Georgoulis, Emmanuil H.; Levesley, Jeremy
Abstract: We adapt Schaback's error doubling trick [13] to give error estimates for radial interpolation of functions with smoothness lying (in some sense) between that of the usual native space and the subspace with double the smoothness. We do this for both bounded subsets of IRd and spheres. As a step on the way to our ultimate goal we also show convergence of pseudo-derivatives of the interpolation error.2009-02-26T14:40:11ZInverse-Type Estimates on hp-Finite Element Spaces and Applications.
http://hdl.handle.net/2381/4263
Title: Inverse-Type Estimates on hp-Finite Element Spaces and Applications.
Authors: Georgoulis, Emmanuil H.
Abstract: Abstract. This work is concerned with the development of inverse-type inequalities for piecewise polynomial functions and, in particular, functions belonging to hp-finite element spaces. The cases of positive and negative Sobolev norms are considered for both continuous and discontinuous finite element functions. The inequalities are explicit both in the local polynomial degree and the local mesh-size. The assumptions on the hp-finite element spaces are very weak, allowing anisotropic (shape-irregular) elements and varying polynomial degree across elements. Finally, the new inverse-type inequalities are used to derive bounds for the condition number of symmetric stiffness matrices of hp-boundary element method discretisations of integral equations, with element-wise discontinuous basis functions constructed via scaled tensor products of Legendre polynomials.2009-02-26T14:12:18ZEnhancing SPH using Moving Least-Squares and Radial Basis Functions.
http://hdl.handle.net/2381/4262
Title: Enhancing SPH using Moving Least-Squares and Radial Basis Functions.
Authors: Brownlee, R.A.; Houston, Paul; Levesley, Jeremy; Rosswog, S.
Abstract: In this paper we consider two sources of enhancement for the meshfree Lagrangian particle method smoothed particle hydrodynamics (SPH) by improving the accuracy of the particle approximation. Namely, we will consider shape functions constructed using: moving least-squares approximation (MLS); radial basis functions (RBF). Using MLS approximation is appealing because polynomial consistency of the particle approximation can be enforced. RBFs further appeal as they allow one to dispense with the smoothing-length { the parameter in the SPH method which governs the number of particles within the support of the shape function. Currently, only ad hoc methods for choosing the smoothing-length exist. We ensure that any enhancement retains the conservative and meshfree nature of SPH. In doing so, we derive a new set of variationally-consistent hydrodynamic equations. Finally, we demonstrate the performance of the new equations on the Sod shock tube problem.2009-02-26T14:04:28ZSecant Dimensions of Low-Dimensional Homogeneous Varieties
http://hdl.handle.net/2381/4261
Title: Secant Dimensions of Low-Dimensional Homogeneous Varieties
Authors: Baur, Karin; Draisma, Jan
Abstract: We completely describe the higher secant dimensions of all connected homogeneous projective varieties of dimension at most 3, in all possible equivariant embeddings. In particular, we calculate these dimensions for all Segre-Veronese embeddings of P1 × P1, P1 × P1 × P1, and P2 × P1, as well as for the variety F of incident point-line pairs in P2. For P2 × P1 and F the results are new, while the proofs for the other two varieties are more compact than existing proofs. Our main tool is the second author’s tropical approach to secant dimensions.2009-02-26T13:52:23ZA note on the design of hp-version interior penalty discontinuous Galerkin finite element methods for degenerate problems
http://hdl.handle.net/2381/4260
Title: A note on the design of hp-version interior penalty discontinuous Galerkin finite element methods for degenerate problems
Authors: Georgoulis, Emmanuil H.; Lasis, Andris
Abstract: We construct cellular homotopy theories for categories of simplicial presheaves on small Grothendieck sites and discuss applications to the motivic homotopy category of Morel and Voevodsky.2009-02-26T13:41:56ZSecondary Theories for Simplicial Manifolds and Classifying Spaces
http://hdl.handle.net/2381/4259
Title: Secondary Theories for Simplicial Manifolds and Classifying Spaces
Authors: Felisatti, Marcello; Neumann, Frank
Abstract: Abstract. We define secondary theories and characteristic classes for simplicial smooth manifolds generalizing Karoubi’s multiplicative Ktheory and multiplicative cohomology groups for smooth manifolds. As a special case we get versions of the groups of differential characters of Cheeger and Simons for simplicial smooth manifolds. Special examples include classifying spaces of Lie groups and Lie groupoids.2009-02-26T13:34:39Zhp–Version Interior Penalty Discontinuous Galerkin Finite
http://hdl.handle.net/2381/4258
Title: hp–Version Interior Penalty Discontinuous Galerkin Finite
Authors: Georgoulis, Emmanuil H.
Abstract: We consider the hp-version interior penalty discontinuous Galerkin finite element method (hp-DGFEM) for linear second-order elliptic reaction-diffusion-advection equations with mixed Dirichlet and Neumann boundary conditions. Our main concern is the extension of the error analysis of the hp-DGFEM to the case when anisotropic (shape-irregular) elements and anisotropic polynomial degrees are used. For this purpose, extensions of well known approximation theory results are derived. In particular, new error bounds for the approximation error of the L2- and H1-projection operators are presented, as well as generalizations of existing inverse inequalities to the anisotropic setting. Equipped with these theoretical developments, we derive general error bounds for the hp-DGFEM on anisotropic meshes, and anisotropic polynomial degrees. Moreover, an improved choice for the (user-defined) discontinuity-penalisation parameter of the method is proposed, which takes into account the anisotropy of the mesh. These results collapse to previously known ones when applied to problems on shape-regular elements. The theoretical findings are justified by numerical experiments, indicating that the use of anisotropic elements, together with our newly suggested choice of the discontinuity-penalisation parameter, improves the stability, the accuracy and the efficiency of the method.2009-02-26T13:26:17ZOn cellularization for simplicial presheaves and motivic homotopy theory.
http://hdl.handle.net/2381/4256
Title: On cellularization for simplicial presheaves and motivic homotopy theory.
Authors: Neumann, Frank; Rodriguez, J.L.
Abstract: We construct cellular homotopy theories for categories of simplicial presheaves on small Grothendieck sites and discuss applications to the motivic homotopy category of Morel and Voevodsky.2009-02-26T13:01:15Z