An efficient numerical scheme for the Euler equations

A finite difference scheme based on flux difference splitting is presented for the solution of the Euler equations for the compressible flow of an ideal gas. A linearised Riemann problem is defined, and a scheme based on numerical characteristic decomposition is presented for obtaining approximate solutions to the linearised problem. An average of the flow variables across the interface between cells is required, and this average is chosen to be the arithmetic mean for computational efficiency, leading to arithmetic averaging. This is in contrast to the usual ‘square root’ averages found in this type of Riemann solver, where the computational expense can be prohibitive. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second order scheme which avoids nonphysical, spurious oscillations. The scheme is applied to a shock tube problem and a blast wave problem. Each approximate solution compares well with those given by other schemes, and for the shock tube problem is in agreement with the exact solution.

Euler's theorem under the microscope

In this article Geoff Tennant summarises the first half of Imre Lakatos's seminal 1976 book, "Proofs and refutations: the logic of mathematical discovery". Implications are drawn for the classroom treatment of proof.

On the existence of two-dimensional Euler flows satisfying energy-Casimir stability criteria

The energy-Casimir stability method, also known as the Arnold stability method, has been widely used in fluid dynamical applications to derive sufficient conditions for nonlinear stability. The most commonly studied system is two-dimensional Euler flow. It is shown that the set of two-dimensional Euler flows satisfying the energy-Casimir stability criteria is empty for two important cases: (i) domains having the topology of the sphere, and (ii) simply-connected bounded domains with zero net vorticity. The results apply to both the first and the second of Arnold’s stability theorems. In the spirit of Andrews’ theorem, this puts a further limitation on the applicability of the method. © 2000 American Institute of Physics.

Nonlinear stability of Euler flows in two-dimensional periodic domains

The non-quadratic conservation laws of the two-dimensional Euler equations are used to show that the gravest modes in a doubly-periodic domain with aspect ratio L = 1 are stable up to translations (or structurally stable) for finite-amplitude disturbances. This extends a previous result based on conservation of energy and enstrophy alone. When L 1, a saturation bound is established for the mode with wavenumber |k| = L −1 (the next-gravest mode), which is linearly unstable. The method is applied to prove nonlinear structural stability of planetary wave two on a rotating sphere.

Curl free pressure gradients over orography in a solution of the fully compressible Euler equations with implicit treatment of acoustic and gravity waves

Steep orography can cause noisy solutions and instability in models of the atmosphere. A new technique for modelling flow over orography is introduced which guarantees curl free gradients on arbitrary grids, implying that the pressure gradient term is not a spurious source of vorticity. This mimetic property leads to better hydrostatic balance and better energy conservation on test cases using terrain following grids. Curl-free gradients are achieved by using the co-variant components of velocity over orography rather than the usual horizontal and vertical components. In addition, gravity and acoustic waves are treated implicitly without the need for mean and perturbation variables or a hydrostatic reference profile. This enables a straightforward description of the implicit treatment of gravity waves. Results are presented of a resting atmosphere over orography and the curl-free pressure gradient formulation is advantageous. Results of gravity waves over orography are insensitive to the placement of terrain-following layers. The model with implicit gravity waves is stable in strongly stratified conditions, with N∆t up to at least 10 (where N is the Brunt-V ̈ais ̈al ̈a frequency). A warm bubble rising over orography is simulated and the curl free pressure gradient formulation gives much more accurate results for this test case than a model without this mimetic property.

A DISCRETIZATION SCHEME FOR AN ONE-DIMENSIONAL REACTION-DIFFUSION EQUATION WITH DELAY AND ITS DYNAMICS

The goal of this paper is to present an approximation scheme for a reaction-diffusion equation with finite delay, which has been used as a model to study the evolution of a population with density distribution u, in such a way that the resulting finite dimensional ordinary differential system contains the same asymptotic dynamics as the reaction-diffusion equation.

Interface de cálculo modal e visualização para estruturas flexíveis tipo Euler-Bernoulli

Neste trabalho, foram analisadas, implementadas e mostradas as características de uma interface com estrutura de programação genérica para o ambiente Windows, criando facilidades de rapidez, confiabilidade e a apresentação de resultados aos usuários do sistema matemático Maple na criação e uso de aplicações matemáticas. A interface utilizou como modelo de implementação cálculos modais de vigas clássicas de Euler-Bernoulli. O usuário encontra, em um único sistema, cálculo para vigas clássicas, terá uma entrada de dados facilitada de variáveis que serão substituídas automaticamente no programa fonte da viga e a geração de resultados em um ambiente amigável com dados e gráficos mostrados de forma organizados.

On the statistical estimation of diffusion processes: a survey

Data available on continuos-time diffusions are always sampled discretely in time. In most cases, the likelihood function of the observations is not directly computable. This survey covers a sample of the statistical methods that have been developed to solve this problem. We concentrate on some recent contributions to the literature based on three di§erent approaches to the problem: an improvement of the Euler-Maruyama discretization scheme, the use of Martingale Estimating Functions and the application of Generalized Method of Moments (GMM).