IMEX finite volume methods for the shallow water equations

Die Flachwassergleichungen (SWE) sind ein hyperbolisches System von Bilanzgleichungen, die adäquate Approximationen an groß-skalige Strömungen der Ozeane, Flüsse und der Atmosphäre liefern. Dabei werden Masse und Impuls erhalten. Wir unterscheiden zwei charakteristische Geschwindigkeiten: die Advektionsgeschwindigkeit, d.h. die Geschwindigkeit des Massentransports, und die Geschwindigkeit von Schwerewellen, d.h. die Geschwindigkeit der Oberflächenwellen, die Energie und Impuls tragen. Die Froude-Zahl ist eine Kennzahl und ist durch das Verhältnis der Referenzadvektionsgeschwindigkeit zu der Referenzgeschwindigkeit der Schwerewellen gegeben. Für die oben genannten Anwendungen ist sie typischerweise sehr klein, z.B. 0.01. Zeit-explizite Finite-Volume-Verfahren werden am öftersten zur numerischen Berechnung hyperbolischer Bilanzgleichungen benutzt. Daher muss die CFL-Stabilitätsbedingung eingehalten werden und das Zeitinkrement ist ungefähr proportional zu der Froude-Zahl. Deswegen entsteht bei kleinen Froude-Zahlen, etwa kleiner als 0.2, ein hoher Rechenaufwand. Ferner sind die numerischen Lösungen dissipativ. Es ist allgemein bekannt, dass die Lösungen der SWE gegen die Lösungen der Seegleichungen/ Froude-Zahl Null SWE für Froude-Zahl gegen Null konvergieren, falls adäquate Bedingungen erfüllt sind. In diesem Grenzwertprozess ändern die Gleichungen ihren Typ von hyperbolisch zu hyperbolisch.-elliptisch. Ferner kann bei kleinen Froude-Zahlen die Konvergenzordnung sinken oder das numerische Verfahren zusammenbrechen. Insbesondere wurde bei zeit-expliziten Verfahren falsches asymptotisches Verhalten (bzgl. der Froude-Zahl) beobachtet, das diese Effekte verursachen könnte.Ozeanographische und atmosphärische Strömungen sind typischerweise kleine Störungen eines unterliegenden Equilibriumzustandes. Wir möchten, dass numerische Verfahren für Bilanzgleichungen gewisse Equilibriumzustände exakt erhalten, sonst können künstliche Strömungen vom Verfahren erzeugt werden. Daher ist die Quelltermapproximation essentiell. Numerische Verfahren die Equilibriumzustände erhalten heißen ausbalanciert.rnrnIn der vorliegenden Arbeit spalten wir die SWE in einen steifen, linearen und einen nicht-steifen Teil, um die starke Einschränkung der Zeitschritte durch die CFL-Bedingung zu umgehen. Der steife Teil wird implizit und der nicht-steife explizit approximiert. Dazu verwenden wir IMEX (implicit-explicit) Runge-Kutta und IMEX Mehrschritt-Zeitdiskretisierungen. Die Raumdiskretisierung erfolgt mittels der Finite-Volumen-Methode. Der steife Teil wird mit Hilfe von finiter Differenzen oder au eine acht mehrdimensional Art und Weise approximniert. Zur mehrdimensionalen Approximation verwenden wir approximative Evolutionsoperatoren, die alle unendlich viele Informationsausbreitungsrichtungen berücksichtigen. Die expliziten Terme werden mit gewöhnlichen numerischen Flüssen approximiert. Daher erhalten wir eine Stabilitätsbedingung analog zu einer rein advektiven Strömung, d.h. das Zeitinkrement vergrößert um den Faktor Kehrwert der Froude-Zahl. Die in dieser Arbeit hergeleiteten Verfahren sind asymptotisch erhaltend und ausbalanciert. Die asymptotischer Erhaltung stellt sicher, dass numerische Lösung das "korrekte" asymptotische Verhalten bezüglich kleiner Froude-Zahlen besitzt. Wir präsentieren Verfahren erster und zweiter Ordnung. Numerische Resultate bestätigen die Konvergenzordnung, so wie Stabilität, Ausbalanciertheit und die asymptotische Erhaltung. Insbesondere beobachten wir bei machen Verfahren, dass die Konvergenzordnung fast unabhängig von der Froude-Zahl ist.

Examples of PDEs all whose points are characteristic

Let π : FM ! M be the bundle of linear frames of a manifold M. A basis Lijk , j < k, of diffeomorphism invariant Lagrangians on J1 (FM) was determined in [J. Muñoz Masqué, M. E. Rosado, Invariant variational problems on linear frame bundles, J. Phys. A35 (2002) 2013-2036]. The notion of a characteristic hypersurface for an arbitrary first-order PDE system on an ar- bitrary bred manifold π : P → M, is introduced and for the systems dened by the Euler-Lagrange equations of Lijk every hypersurface is shown to be characteristic. The Euler-Lagrange equations of the natural basis of Lagrangian densities Lijk on the bundle of linear frames of a manifold M which are invariant under diffeomorphisms, are shown to be an underdetermined PDEs systems such that every hypersurface of M is characteristic for such equations. This explains why these systems cannot be written in the Cauchy-Kowaleska form, although they are known to be formally integrable by using the tools of geometric theory of partial differential equations, see [J. Muñoz Masqué, M. E. Rosado, Integrability of the eld equations of invariant variational problems on linear frame bundles, J. Geom. Phys. 49 (2004), 119-155]

A second order in time modified Lagrange-Galerkin finite element method for the incompressible Navier-Stokes equations

We introduce a second order in time modified Lagrange--Galerkin (MLG) method for the time dependent incompressible Navier--Stokes equations. The main ingredient of the new method is the scheme proposed to calculate in a more efficient manner the Galerkin projection of the functions transported along the characteristic curves of the transport operator. We present error estimates for velocity and pressure in the framework of mixed finite elements when either the mini-element or the $P2/P1$ Taylor--Hood element are used.

Predictive validity of three ActiGraph energy expenditure equations for children

Purpose: This Study evaluated the predictive validity of three previously published ActiGraph energy expenditure (EE) prediction equations developed for children and adolescents. Methods: A total of 45 healthy children and adolescents (mean age: 13.7 +/- 2.6 yr) completed four 5-min activity trials (normal walking. brisk walking, easy running, and fast running) in ail indoor exercise facility. During each trial, participants were all ActiGraph accelerometer oil the right hip. EE was monitored breath by breath using the Cosmed K4b(2) portable indirect calorimetry system. Differences and associations between measured and predicted EE were assessed using dependent t-tests and Pearson correlations, respectively. Classification accuracy was assessed using percent agreement, sensitivity, specificity, and area under the receiver operating characteristic (ROC) curve, Results: None of the equations accurately predicted mean energy expenditure during each of the four activity trials. Each equation, however, accurately predicted mean EE in at least one activity trial. The Puyau equation accurately predicted EE during slow walking. The Trost equation accurately predicted EE during slow running. The Freedson equation accurately predicted EE during fast running. None of the three equations accurately predicted EE during brisk walking. The equations exhibited fair to excellent classification accuracy with respect to activity intensity. with the Trost equation exhibiting the highest classification accuracy and the Puyau equation exhibiting the lowest. Conclusions: These data suggest that the three accelerometer prediction equations do not accurately predict EE on a minute-by-minute basis in children and adolescents during overground walking and running. The equations maybe, however, for estimating participation in moderate and vigorous activity.

Singular Solutions of Protter’s Problem for a Class of 3-D Hyperbolic Equations

Недю Иванов Попиванов, Алексей Йорданов Николов - През 1952 г. М. Протър формулира нови гранични задачи за вълновото уравнение, които са тримерни аналози на задачите на Дарбу в равнината. Задачите са разгледани в тримерна област, ограничена от две характеристични конуса и равнина. Сега, след като са минали повече от 50 години, е добре известно, че за безброй гладки функции в дясната страна на уравнението тези задачи нямат класически решения, а обобщеното решение има силна степенна особеност във върха на характеристичния конус, която е изолирана и не се разпространява по конуса. Тук ние разглеждаме трета гранична задача за вълновото уравнение с младши членове и дясна страна във формата на тригонометричен полином. Дадена е по-нова от досега известната априорна оценка за максимално възможната особеност на решенията на тази задача. Оказва се, че при по-общото уравнение с младши членове възможната сингулярност е от същия ред като при чисто вълновото уравнение.

On the Zeros of the Solutions to Nonlinear Hyperbolic Equations with Delays

2002 Mathematics Subject Classification: Primary 35В05; Secondary 35L15

Estimating glomerular filtration rate in acute coronary syndromes: Different equations, different mortality risk prediction

AIMS: Renal dysfunction is a powerful predictor of adverse outcomes in patients hospitalized for acute coronary syndrome. Three new glomerular filtration rate (GFR) estimating equations recently emerged, based on serum creatinine (CKD-EPIcreat), serum cystatin C (CKD-EPIcyst) or a combination of both (CKD-EPIcreat/cyst), and they are currently recommended to confirm the presence of renal dysfunction. Our aim was to analyse the predictive value of these new estimated GFR (eGFR) equations regarding mid-term mortality in patients with acute coronary syndrome, and compare them with the traditional Modification of Diet in Renal Disease (MDRD-4) formula. METHODS AND RESULTS: 801 patients admitted for acute coronary syndrome (age 67.3±13.3 years, 68.5% male) and followed for 23.6±9.8 months were included. For each equation, patient risk stratification was performed based on eGFR values: high-risk group (eGFR<60ml/min per 1.73m2) and low-risk group (eGFR⩾60ml/min per 1.73m2). The predictive performances of these equations were compared using area under each receiver operating characteristic curves (AUCs). Overall risk stratification improvement was assessed by the net reclassification improvement index. The incidence of the primary endpoint was 18.1%. The CKD-EPIcyst equation had the highest overall discriminate performance regarding mid-term mortality (AUC 0.782±0.20) and outperformed all other equations (ρ<0.001 in all comparisons). When compared with the MDRD-4 formula, the CKD-EPIcyst equation accurately reclassified a significant percentage of patients into more appropriate risk categories (net reclassification improvement index of 11.9% (p=0.003)). The CKD-EPIcyst equation added prognostic power to the Global Registry of Acute Coronary Events (GRACE) score in the prediction of mid-term mortality. CONCLUSION: The CKD-EPIcyst equation provides a novel and improved method for assessing the mid-term mortality risk in patients admitted for acute coronary syndrome, outperforming the most widely used formula (MDRD-4), and improving the predictive value of the GRACE score. These results reinforce the added value of cystatin C as a risk marker in these patients.

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.

Some Remarks on the Approximation of Transitional Density in Stochastic Differential Equations

Aijt-Sahalia (2002) introduced a method to estimate transitional probability densities of di®usion processes by means of Hermite expansions with coe±cients determined by means of Taylor series. This note describes a numerical procedure to ¯nd these coe±cients based on the calculation of moments. One advantage of this procedure is that it can be used e®ectively when the mathematical operations required to ¯nd closed-form expressions for these coe±cients are otherwise infeasible.

Development of a biaxial compression device for biological samples : preliminary experimental results for a closed cell foam

Biological tissues are subjected to complex loading states in vivo and in order to define constitutive equations that effectively simulate their mechanical behaviour under these loads, it is necessary to obtain data on the tissue's response to multiaxial loading. Single axis and shear testing of biological tissues is often carried out, but biaxial testing is less common. We sought to design and commission a biaxial compression testing device, capable of obtaining repeatable data for biological samples. The apparatus comprised a sealed stainless steel pressure vessel specifically designed such that a state of hydrostatic compression could be created on the test specimen while simultaneously unloading the sample along one axis with an equilibrating tensile pressure. Thus a state of equibiaxial compression was created perpendicular to the long axis of a rectangular sample. For the purpose of calibration and commissioning of the vessel, rectangular samples of closed cell ethylene vinyl acetate (EVA) foam were tested. Each sample was subjected to repeated loading, and nine separate biaxial experiments were carried out to a maximum pressure of 204 kPa (30 psi), with a relaxation time of two hours between them. Calibration testing demonstrated the force applied to the samples had a maximum error of 0.026 N (0.423% of maximum applied force). Under repeated loading, the foam sample demonstrated lower stiffness during the first load cycle. Following this cycle, an increased stiffness, repeatable response was observed with successive loading. While the experimental protocol was developed for EVA foam, preliminary results on this material suggest that this device may be capable of providing test data for biological tissue samples. The load response of the foam was characteristic of closed cell foams, with consolidation during the early loading cycles, then a repeatable load-displacement response upon repeated loading. The repeatability of the test results demonstrated the ability of the test device to provide reproducible test data and the low experimental error in the force demonstrated the reliability of the test data.

Constructive development of the solutions of linear equations in introductory ordinary differential equations

The solution of linear ordinary differential equations (ODEs) is commonly taught in first year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognising what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to tables of solutions, is an important skill for students to carry with them to advanced studies in mathematics. In this study we describe a teaching and learning strategy that replaces the traditional algorithmic, transmission presentation style for solving ODEs with a constructive, discovery based approach where students employ their existing skills as a framework for constructing the solutions of first and second order linear ODEs. We elaborate on how the strategy was implemented and discuss the resulting impact on a first year undergraduate class. Finally we propose further improvements to the strategy as well as suggesting other topics which could be taught in a similar manner.

Characteristic analysis for multisampled digital implementation of fixed-switching-frequency closed-loop modulation of voltage-source inverter

In this paper, a fixed-switching-frequency closed-loop modulation of a voltage-source inverter (VSI), upon the digital implementation of the modulation process, is analyzed and characterized. The sampling frequency of the digital processor is considered as an integer multiple of the modulation switching frequency. An expression for the determination of the modulation design parameter is developed for smooth modulation at a fixed switching frequency. The variation of the sampling frequency, switching frequency, and modulation index has been analyzed for the determination of the switching condition under closed loop. It is shown that the switching condition determined based on the continuous-time analysis of the closed-loop modulation will ensure smooth modulation upon the digital implementation of the modulation process. However, the stability properties need to be tested prior to digital implementation as they get deteriorated at smaller sampling frequencies. The closed-loop modulation index needs to be considered maximum while determining the design parameters for smooth modulation. In particular, a detailed analysis has been carried out by varying the control gain in the sliding-mode control of a two-level VSI. The proposed analysis of the closed-loop modulation of the VSI has been verified for the operation of a distribution static compensator. The theoretical results are validated experimentally on both single- and three-phase systems.