An analysis of arithmetic averaging in approximate Riemann solvers with an application to steady, supercritical flows

An analysis of various arithmetic averaging procedures for approximate Riemann solvers is made with a specific emphasis on efficiency and a jump capturing property. The various alternatives discussed are intended for future work, as well as the more immediate problem of steady, supercritical free-surface flows. Numerical results are shown for two test problems.

Riemann solvers with primitive parameter vectors for two-dimensional compressible flows of a real gas

An analysis of averaging procedures is presented for an approximate Riemann solver for the equations governing the compressible flow of a real gas. This study extends earlier work for the Euler equations with ideal gases.

High Order Multi-Moment Constrained Finite Volume Method. Part I: Basic Formulation

A new high-order finite volume method based on local reconstruction is presented in this paper. The method, so-called the multi-moment constrained finite volume (MCV) method, uses the point values defined within single cell at equally spaced points as the model variables (or unknowns). The time evolution equations used to update the unknowns are derived from a set of constraint conditions imposed on multi kinds of moments, i.e. the cell-averaged value and the point-wise value of the state variable and its derivatives. The finite volume constraint on the cell-average guarantees the numerical conservativeness of the method. Most constraint conditions are imposed on the cell boundaries, where the numerical flux and its derivatives are solved as general Riemann problems. A multi-moment constrained Lagrange interpolation reconstruction for the demanded order of accuracy is constructed over single cell and converts the evolution equations of the moments to those of the unknowns. The presented method provides a general framework to construct efficient schemes of high orders. The basic formulations for hyperbolic conservation laws in 1- and 2D structured grids are detailed with the numerical results of widely used benchmark tests. (C) 2009 Elsevier Inc. All rights reserved.

Stochastic Modeling of Reversible Biochemical Reaction-Diffusion Systems and High-Resolution Shock-Capturing Methods for Fluid Interfaces

Thesis (Ph.D.)--University of Washington, 2016-08

Effective Cell-Centred Time-Domain Maxwell's Equations Numerical Solvers

This research work analyses techniques for implementing a cell-centred finite-volume time-domain (ccFV-TD) computational methodology for the purpose of studying microwave heating. Various state-of-the-art spatial and temporal discretisation methods employed to solve Maxwell's equations on multidimensional structured grid networks are investigated, and the dispersive and dissipative errors inherent in those techniques examined. Both staggered and unstaggered grid approaches are considered. Upwind schemes using a Riemann solver and intensity vector splitting are studied and evaluated. Staggered and unstaggered Leapfrog and Runge-Kutta time integration methods are analysed in terms of phase and amplitude error to identify which method is the most accurate and efficient for simulating microwave heating processes. The implementation and migration of typical electromagnetic boundary conditions. from staggered in space to cell-centred approaches also is deliberated. In particular, an existing perfectly matched layer absorbing boundary methodology is adapted to formulate a new cell-centred boundary implementation for the ccFV-TD solvers. Finally for microwave heating purposes, a comparison of analytical and numerical results for standard case studies in rectangular waveguides allows the accuracy of the developed methods to be assessed.

Effective cell-centred time-domain Maxwell's equations numerical solvers

This research work analyses techniques for implementing a cell-centred finite-volume time-domain (ccFV-TD) computational methodology for the purpose of studying microwave heating. Various state-of-the-art spatial and temporal discretisation methods employed to solve Maxwell's equations on multidimensional structured grid networks are investigated, and the dispersive and dissipative errors inherent in those techniques examined. Both staggered and unstaggered grid approaches are considered. Upwind schemes using a Riemann solver and intensity vector splitting are studied and evaluated. Staggered and unstaggered Leapfrog and Runge-Kutta time integration methods are analysed in terms of phase and amplitude error to identify which method is the most accurate and efficient for simulating microwave heating processes. The implementation and migration of typical electromagnetic boundary conditions. from staggered in space to cell-centred approaches also is deliberated. In particular, an existing perfectly matched layer absorbing boundary methodology is adapted to formulate a new cell-centred boundary implementation for the ccFV-TD solvers. Finally for microwave heating purposes, a comparison of analytical and numerical results for standard case studies in rectangular waveguides allows the accuracy of the developed methods to be assessed. © 2004 Elsevier Inc. All rights reserved.

A tale of two solvers: Eager and lazy approaches to bit-vectors

The standard method for deciding bit-vector constraints is via eager reduction to propositional logic. This is usually done after first applying powerful rewrite techniques. While often efficient in practice, this method does not scale on problems for which top-level rewrites cannot reduce the problem size sufficiently. A lazy solver can target such problems by doing many satisfiability checks, each of which only reasons about a small subset of the problem. In addition, the lazy approach enables a wide range of optimization techniques that are not available to the eager approach. In this paper we describe the architecture and features of our lazy solver (LBV). We provide a comparative analysis of the eager and lazy approaches, and show how they are complementary in terms of the types of problems they can efficiently solve. For this reason, we propose a portfolio approach that runs a lazy and eager solver in parallel. Our empirical evaluation shows that the lazy solver can solve problems none of the eager solvers can and that the portfolio solver outperforms other solvers both in terms of total number of problems solved and the time taken to solve them.

Implementation of parallel tridiagonal solvers for a heterogeneous computing environment

Tridiagonal diagonally dominant linear systems arise in many scientific and engineering applications. The standard Thomas algorithm for solving such systems is inherently serial forming a bottleneck in computation. Algorithms such as cyclic reduction and SPIKE reduce a single large tridiagonal system into multiple small independent systems which can be solved in parallel. We have developed portable cyclic reduction and SPIKE algorithm OpenCL implementations with the intent to target a range of co-processors in a heterogeneous computing environment including Field Programmable Gate Arrays (FPGAs), Graphics Processing Units (GPUs) and other multi-core processors. In this paper, we evaluate these designs in the context of solver performance, resource efficiency and numerical accuracy.

R-parameter: A local truncation error based adaptive framework for finite volume compressible flow solvers

A residual-based strategy to estimate the local truncation error in a finite volume framework for steady compressible flows is proposed. This estimator, referred to as the -parameter, is derived from the imbalance arising from the use of an exact operator on the numerical solution for conservation laws. The behaviour of the residual estimator for linear and non-linear hyperbolic problems is systematically analysed. The relationship of the residual to the global error is also studied. The -parameter is used to derive a target length scale and consequently devise a suitable criterion for refinement/derefinement. This strategy, devoid of any user-defined parameters, is validated using two standard test cases involving smooth flows. A hybrid adaptive strategy based on both the error indicators and the -parameter, for flows involving shocks is also developed. Numerical studies on several compressible flow cases show that the adaptive algorithm performs excellently well in both two and three dimensions.

Flat connections, geometric invariants and energy of harmonic functions on compact Riemann surfaces

A geometric invariant is associated to the space of fiat connections on a G-bundle over a compact Riemann surface and is related to the energy of harmonic functions.

Focussed Crawling with large scale Ordinal Regression Solvers

In this paper we propose a novel, scalable, clustering based Ordinal Regression formulation, which is an instance of a Second Order Cone Program (SOCP) with one Second Order Cone (SOC) constraint. The main contribution of the paper is a fast algorithm, CB-OR, which solves the proposed formulation more eficiently than general purpose solvers. Another main contribution of the paper is to pose the problem of focused crawling as a large scale Ordinal Regression problem and solve using the proposed CB-OR. Focused crawling is an efficient mechanism for discovering resources of interest on the web. Posing the problem of focused crawling as an Ordinal Regression problem avoids the need for a negative class and topic hierarchy, which are the main drawbacks of the existing focused crawling methods. Experiments on large synthetic and benchmark datasets show the scalability of CB-OR. Experiments also show that the proposed focused crawler outperforms the state-of-the-art.

Directional Diffusion Regulator (DDR) for some numerical solvers of hyperbolic conservation laws

A computational tool called ``Directional Diffusion Regulator (DDR)'' is proposed to bring forth real multidimensional physics into the upwind discretization in some numerical schemes of hyperbolic conservation laws. The direction based regulator when used with dimension splitting solvers, is set to moderate the excess multidimensional diffusion and hence cause genuine multidimensional upwinding like effect. The basic idea of this regulator driven method is to retain a full upwind scheme across local discontinuities, with the upwind bias decreasing smoothly to a minimum in the farthest direction. The discontinuous solutions are quantified as gradients and the regulator parameter across a typical finite volume interface or a finite difference interpolation point is formulated based on fractional local maximum gradient in any of the weak solution flow variables (say density, pressure, temperature, Mach number or even wave velocity etc.). DDR is applied to both the non-convective as well as whole unsplit dissipative flux terms of some numerical schemes, mainly of Local Lax-Friedrichs, to solve some benchmark problems describing inviscid compressible flow, shallow water dynamics and magneto-hydrodynamics. The first order solutions consistently improved depending on the extent of grid non-alignment to discontinuities, with the major influence due to regulation of non-convective diffusion. The application is also experimented on schemes such as Roe, Jameson-Schmidt-Turkel and some second order accurate methods. The consistent improvement in accuracy either at moderate or marked levels, for a variety of problems and with increasing grid size, reasonably indicate a scope for DDR as a regular tool to impart genuine multidimensional upwinding effect in a simpler framework. (C) 2012 Elsevier Inc. All rights reserved.