Point-wise hierarchical reconstruction for discontinuous Galerkin and finite volume methods for solving conservation laws

We develop a new hierarchical reconstruction (HR) method  and  for limiting solutions of the discontinuous Galerkin and finite volume methods up to fourth order of accuracy without local characteristic decomposition for solving hyperbolic nonlinear conservation laws on triangular meshes. The new HR utilizes a set of point values when evaluating polynomials and remainders on neighboring cells, extending the technique introduced in Hu, Li and Tang [9]. The point-wise HR simplifies the implementation of the previous HR method which requires integration over neighboring cells and makes HR easier to extend to arbitrary meshes. We prove that the new point-wise HR method keeps the order of accuracy of the approximation polynomials. Numerical computations for scalar and system of nonlinear hyperbolic equations are performed on two-dimensional triangular meshes. We demonstrate that the new hierarchical reconstruction generates essentially non-oscillatory solutions for schemes up to fourth order on triangular meshes.     

The finite volume local evolution Galerkin method for solving the hyperbolic conservation laws

This paper presents a finite volume local evolution Galerkin (FVLEG) scheme for solving the hyperbolic conservation laws. The FVLEG scheme is the simplification of the finite volume evolution Galerkin method (FVEG). In FVEG, a necessary step is to compute the dependent variables at cell interfaces at t_n + τ (0 < τ ? Δt). The FVLEG scheme is constructed by taking τ → 0 in the evolution operators of FVEG. The FVLEG scheme greatly simplifies the evaluation of the numerical fluxes. It is also well suited with the semi-discrete finite volume method, making the flux evaluation being decoupled with the reconstruction procedure while maintaining the genuine multi-dimensional nature of the FVEG methods. The derivation of the FVLEG scheme is presented in detail. The performance of the proposed scheme is studied by solving several test cases. It is shown that FVLEG scheme can obtain very satisfactory numerical results in terms of accuracy and resolution.     

Beyond the mean-field: Quantum Monte Carlo approach for finite fermi systems

Large-amplitude fluctuations around the mean-field are important in hot finite Fermi systems. In the quantum Monte Carlo approach these fluctuations are taken into account exactly but often lead to a sign problem. A practical solution to the sign problem in the framework of the nuclear shell model allows realistic calculations in model spaces that are much larger than those that can be treated by conventional diagonalization methods. Recent applications of the Monte Carlo methods for microscopic calculations of collectivity and level densities in heavy nuclei are presented. Presented at the International Conference on “Atomic Nuclei and Metallic Clusters”, Prague, September 1–5, 1997. This work was supported in part by the Department of Energy grant No. DE-FG-0291-ER-40608.     

Duality for systems of conservation laws

Letters in Mathematical Physics - For one-dimensional systems of conservation laws admitting two additional conservation laws, we assign a ruled hypersurface of codimension two in projective space....     

Entropy viscosity method for nonlinear conservation laws

A new class of high-order numerical methods for approximating nonlinear conservation laws is described (entropy viscosity method). The novelty is that a nonlinear viscosity based on the local size of an entropy production is added to the numerical discretization at hand. This new approach does not use any flux or slope limiters, applies to equations or systems supplemented with one or more entropy inequalities and does not depend on the mesh type and polynomial approximation. Various benchmark problems are solved with finite elements, spectral elements and Fourier series to illustrate the capability of the proposed method.     

Relativistic calculations using Monte Carlo methods: one-electron systems

Variance minimization and Monte Carlo integration are used to evaluate the four-component Dirac equation for a number of one-electron atomic and diatomic systems. This combination produces accurate energies, is relatively simple to implement, and exhibits few of the problems associated with traditional techniques.     

Hyperbolic conservation laws on the sphere. A geometry-compatible finite volume scheme

We consider entropy solutions to the initial value problem associated with scalar nonlinear hyperbolic conservation laws posed on the two-dimensional sphere. We propose a finite volume scheme which relies on a web-like mesh made of segments of longitude and latitude lines. The structure of the mesh allows for a discrete version of a natural geometric compatibility condition, which arose earlier in the well-posedness theory established by Ben-Artzi and LeFloch. We study here several classes of flux vectors which define the conservation law under consideration. They are based on prescribing a suitable vector field in the Euclidean three-dimensional space and then suitably projecting it on the sphere’s tangent plane; even when the flux vector in the ambient space is constant, the corresponding flux vector is a non-trivial vector field on the sphere. In particular, we construct here “equatorial periodic solutions”, analogous to one-dimensional periodic solutions to one-dimensional conservation laws, as well as a wide variety of stationary (steady state) solutions. We also construct “confined solutions”, which are time-dependent solutions supported in an arbitrarily specified subdomain of the sphere. Finally, representative numerical examples and test cases are presented.     

Improved Monte Carlo methods for fermions

We describe an improved version of the Kuti-Von Neumann-Ulam algorithm useful for fermion contributions in lattice field theories. This is done by sampling the Neumann series for the propagator, which may be thought of as a sum over a set of weighted paths between two points on the lattice. Rather than selecting paths by a locally determined random walk, we average over sets of paths globally preselected for their importance in evaluating the few needed elements of the inverse. We also describe a method for the calculation of ratios of fermion determinants which is considerably less time consuming than the conventional one.     

Monte Carlo methods for yrast spectroscopy

We discuss the details of the recently proposed Monte Carlo method to evaluate the exact energies of yrast levels. Energy levels are evaluated up to J = 18 with small statistical errors using the Metropolis method for the case of ¹⁶⁶Er using the pairing plus quadrupole model within one major shell. We also discuss the evaluation of the probabilities of the Hartree-Fock-Bogoliubov wave functions in the corresponding yrast eigenstates and they are found to be large. The model displays a too strong backbending behaviour not seen experimentally.Received: 29 September 2003, Published online: 24 August 2004PACS: 21.60.-n Nuclear-structure models and methods - 02.70.Ss Quantum Monte Carlo methods - 21.60.Ka Monte Carlo models - 21.10.Re Collective levels     

Exact Monte Carlo for few-fermion systems

We have reconsidered the fundamental difficulties of fermion Monte Carlo as applied to few-body systems. We conclude that necessary ingredients of successful algorithms include the following: There must be equal populations of random walkers that carry positive and negative weights. The positions of positive walkers should be selected from a distribution that uses Green's functions to couple all walkers. The positions of negative walkers should be generated from those of positive walkers by means of odd permutations. The correct importance functions that take into account the global interactions of the populations are different for positive and negative walkers. Use of such importance functions breaks the symmetry that otherwise would exist between configurations (of the entire population) and configurations derived by interchanging positive and negative walkers. Based upon these observations, we have constructed a stable and accurate algorithm that solves a fully-polarized, three-dimensional, three-body model problem.     

Pseudo-spectral methods for the spatial symplectic reduction of open systems of conservation laws

A reduction method is presented for systems of conservation laws with boundary energy flow. It is stated as a generalized pseudo-spectral method which performs exact differentiation by using simultaneously several approximation spaces generated by polynomials bases and suitable choices of port-variables. The symplecticity of this spatial reduction method is proved when used for the reduction of both closed and open systems of conservation laws, for any choice of collocation points (i.e. for any polynomial bases). The symplecticity of some more usual collocation schemes is discussed and finally their accuracy on approximation of the spectrum, on the example of the ideal transmission line, is discussed in comparison with the suggested reduction scheme.     

A global formalism for nonlinear waves in conservation laws

We introduce a unifying framework for treating all of the fundamental waves occurring in general systems ofn conservation laws. Fundamental waves are represented as pairs of states statisfying the Rankine-Hugoniot conditions; after trivial solutions have been eliminated by means of a blow-up procedure, these pairs form an (n+1)-dimensional manifold, the fundamental wave manifold. There is a distinguishedn-dimensional submanifold of containing a single one-dimensonal foliation that represents the rarefaction curves for all families. Similarly, there is a foliation of itself that represent shock curves. We identify othern-dimensional submanifolds of that are naturally interpreted as boundaries of regions of admissible shock waves. These submanifolds also have one-dimensional foliations, which represent curves of composite waves. This geometric framework promises to simplify greatly the study of the stability and bifurcation propertiesThis work was supported in part by: the NSF/CNP_q U.S.-Latin America Cooperative Science Program under Grant INT-8612605; the Institute for Mathematics and its Applications with funds provided by the National Science Foundation; the Air Force Office of Scientific Research under Grant AFOSR 90-0075; the National Science Foundation under Grant 8901884; the U.S. Department of Energy under Grant DE-FG02-90ER25084; the U.S. Army Research Office under Grant DAAL03-89-K-0017; the Financiadora de Estudos e Projetos; the Conselho Nacional de Desenvolvimento Científico e Tecnológica (CNP_q); the Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ); the Coordenação de Aperfeiçamento de Pessoal de Ensino Superior (CAPES); and the Sociedade Brasileira de Matemática (SBM)     

Uncertainty quantification for systems of conservation laws

Uncertainty quantification through stochastic spectral methods has been recently applied to several kinds of non-linear stochastic PDEs. In this paper, we introduce a formalism based on kinetic theory to tackle uncertain hyperbolic systems of conservation laws with Polynomial Chaos (PC) methods. The idea is to introduce a new variable, the entropic variable, in bijection with our vector of unknowns, which we develop on the polynomial basis: by performing a Galerkin projection, we obtain a deterministic system of conservation laws. We state several properties of this deterministic system in the case of a general uncertain system of conservation laws. We then apply the method to the case of the inviscid Burgers’ equation with random initial conditions and we present some preliminary results for the Euler system. We systematically compare results from our new approach to results from the stochastic Galerkin method. In the vicinity of discontinuities, the new method bounds the oscillations due to Gibbs phenomenon to a certain range through the entropy of the system without the use of any adaptative random space discretizations. It is found to be more precise than the stochastic Galerkin method for smooth cases but above all for discontinuous cases.     

Multi-moment ADER-Taylor methods for systems of conservation laws with source terms in one dimension

A new integration method combining the ADER time discretization with a multi-moment finite-volume framework is introduced. ADER runtime is reduced by performing only one Cauchy–Kowalewski (C–K) procedure per cell per time step and by using the Differential Transform Method for high-order derivatives. Three methods are implemented: (1) single-moment WENO (WENO), (2) two-moment Hermite WENO (HWENO), and (3) entirely local multi-moment (MM-Loc). MM-Loc evolves all moments, sharing the locality of Galerkin methods yet with a constant time step during p-refinement.Five 1-D experiments validate the methods: (1) linear advection, (2) Burger’s equation shock, (3) transient shallow-water (SW), (4) steady-state SW simulation, and (5) SW shock. WENO and HWENO methods showed expected polynomial h-refinement convergence and successfully limited oscillations for shock experiments. MM-Loc showed expected polynomial h-refinement and exponential p-refinement convergence for linear advection and showed sub-exponential (yet super-polynomial) convergence with p-refinement in the SW case.HWENO accuracy was generally equal to or better than a five-moment MM-Loc scheme. MM-Loc was less accurate than RKDG at lower refinements, but with greater h- and p-convergence, RKDG accuracy is eventually surpassed. The ADER time integrator of MM-Loc also proved more accurate with p-refinement at a CFL of unity than a semi-discrete RK analog of MM-Loc. Being faster in serial and requiring less frequent inter-node communication than Galerkin methods, the ADER-based MM-Loc and HWENO schemes can be spatially refined and have the same runtime, making them a competitive option for further investigation.     

Monte Carlo simulations for two-dimensional quantum systems

The Trotter-Suzuki transformation has been used to obtain the classical representation ford-dimensional lattice systems with boson and fermion degrees of freedom. A Monte Carlo algorithm for the equivalent (d+1)-dimensional classical system is presented. Numerical results are shown for the Heisenberg-spin-glass, the XY model and the spinless fermion lattice gas in two dimensions.     

Positivity-preserving high-resolution schemes for systems of conservation laws

A new class of flux-limited schemes for systems of conservation laws is presented that is both high-resolution and positivity-preserving. The schemes are obtained by extending the Steger–Warming method to second-order accuracy through the use of component-wise TVD flux limiters while ensuring that the coefficients of the discretization equation are positive. A coefficient is considered positive if it has all-positive eigenvalues and has the same eigenvectors as those of the convective flux Jacobian evaluated at the corresponding node. For certain systems of conservation laws, such as the Euler equations for instance, this condition is sufficient to guarantee positivity-preservation. The method proposed is advantaged over previous positivity-preserving flux-limited schemes by being capable to capture with high resolution all wave types (including contact discontinuities, shocks, and expansion fans). Several test cases are considered in which the Euler equations in generalized curvilinear coordinates are solved in 1D, 2D, and 3D. The test cases confirm that the proposed schemes are positivity-preserving while not being significantly more dissipative than the conventional TVD methods. The schemes are written in general matrix form and can be used to solve other systems of conservation laws, as long as they are homogeneous of degree one.