On the Partial Difference Equations of Mathematical Physics

Courant, Richard; Friedrichs, K. O.; Lewy, Hans

doi:10.1147/rd.112.0215

Cited by 1,965 publications

(607 citation statements)

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“…The adaptive time-stepping functionality of TUFLOW, incorporating the Courant stability criterion (Courant et al, 1967), was employed to satisfy numerical stability criteria. To avoid the effects of initial instability, the model simulation period was selected to be larger than the actual analysis period and the first five days of each model run were used as model 'spin-up' and were not included in data analysis (Beven, 2008;Van Der Knijff et al, 2008).…”

Section: <Table 2 Here Please>mentioning

confidence: 99%

Where does all the water go? Partitioning water transmission losses in a data-sparse, multi-channel and low-gradient dryland river system using modelling and remote sensing

Jarihani

Larsen

Callow

et al. 2015

Journal of Hydrology

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Where does all the water go? Partitioning water transmission losses in a data-sparse, multi-channel and low-gradient dryland river system using modelling and remote sensing, Journal of Hydrology (2015), doi: http://dx.doi.org/10.1016/j.jhydrol. 2015.08.030 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. 1Where does all the water go? Partitioning water transmission losses in a data-sparse, multi-channel and low-gradient dryland river system using modelling and remote sensing and April 2012 along a 180 km reach of the Diamantina River in the Lake Eyre Basin, Australia.Transmission losses were found to be high, on average 46% of total inflow within 180 km reach segment or 7 GL/km (range: 4 to 10 GL/km). However, in 180 km reach, transmission losses vary non-linearly with flood discharge, with smaller flows resulting in higher losses (up to 68%), which diminish in higher flows (down to 24%) and in general there is a minor increase in losses with distance downstream. Partitioning these total losses into the major components shows that actual evaporation was the most significant component (21.6% of total inflow), followed by infiltration (13.2%) and terminal water storage (11.2%). Lateral inflow can be up to 200% of upstream inflow (mean = 86%) and is therefore a critical parameter in the water balance and 3 transmission loss calculations. This study shows that it is possible to constrain the water balance using hydrodynamic models in dryland river systems using remote sensing and simple field measurements to address the otherwise scarce availability of data. The results of this study also enable a better understanding of the water resources available for ecosystems in these unique multi-channel and large floodplain rivers. The combined modelling / remote sensing approach of this study can be applied elsewhere in the world to better understand the water balances and water transmission losses, important drivers of ecohydrological processes in dryland environments.

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Section: <Table 2 Here Please>mentioning

confidence: 99%

Where does all the water go? Partitioning water transmission losses in a data-sparse, multi-channel and low-gradient dryland river system using modelling and remote sensing

Jarihani

Larsen

Callow

et al. 2015

Journal of Hydrology

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“…The model was meshed to give 15 elements per wavelength at 3.5MHz, giving the total number of elements used in the model as just over 1.3 million. A time stability factor of 0.95 was implemented to set the time step to 95% of its maximum value, which satisfies the Courant-Friedrichs-Lewy condition [18].…”

Section: Figure 3 Pzflex Model Transducer Input Pulsementioning

confidence: 99%

“…Table 7 contains information on the number of elements in each simulation along with the individual simulation run time and time to generate the FMC date. The time stability factor was again set to 0.95 to satisfy the Courant-Friedrichs-Lewy condition [18]. …”

Section: -10mentioning

confidence: 99%

Finite element analysis simulations for ultrasonic array NDE inspections

Dobson¹,

Tweedie²,

Harvey³

et al. 2016

AIP Conference Proceedings

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Abstract. Advances in manufacturing techniques and materials have led to an increase in the demand for reliable and robust inspection techniques to maintain safety critical features. The application of modelling methods to develop and evaluate inspections is becoming an essential tool for the NDE community. Current analytical methods are inadequate for simulation of arbitrary components and heterogeneous materials, such as anisotropic welds or composite structures. Finite element analysis software (FEA), such as PZFlex, can provide the ability to simulate the inspection of these arrangements, providing the ability to economically prototype and evaluate improved NDE methods. FEA is often seen as computationally expensive for ultrasound problems however, advances in computing power have made it a more viable tool. This paper aims to illustrate the capability of appropriate FEA to produce accurate simulations of ultrasonic array inspectionsminimizing the requirement for expensive test-piece fabrication. Validation is afforded via corroboration of the FE derived and experimentally generated data sets for a test-block comprising 1D and 2D defects. The modelling approach is extended to consider the more troublesome aspects of heterogeneous materials where defect dimensions can be of the same length scale as the grain structure. The model is used to facilitate the implementation of new ultrasonic array inspection methods for such materials. This is exemplified by considering the simulation of ultrasonic NDE in a weld structure in order to assess new approaches to imaging such structures.

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“…To achieve this, we use a second-order Runge-Kutta time integration scheme to advance equation (16) in time. The time step is set locally within each computational cell according to the CFL condition (Courant et al 1967). The solution in each cell is then advanced in time according to its own local time step.…”

Section: Finite-volume Discretization and Time Stepping Gmentioning

confidence: 99%

A Multiscale Central Difference Scheme Applied to Magnetohydrodynamic Simulations of Cometary Atmospheres

Benna¹,

Mahaffy²,

MacNeice³

et al. 2004

ApJ

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The hyperbolic equations describing the interaction between a cometary atmosphere and the solar wind are solved through a central differencing scheme. This work will enable us to better predict conditions that robotic spacecraft will encounter on comet nucleus flyby or rendezvous missions. We are interested in studying the coma characteristics from large distances from the nucleus, where the solar wind just begins to be perturbed by the comet, to very near the nucleus. We employ a multifluid approach in which the equations governing the ions, neutrals, and electrons are coupled. We solve these equations by the use of a total variation diminishing LaxFriedrichs solver combined with an adaptive mesh refinement technique. Magnetic fields and plasma pressure, velocity, and density profiles are calculated and compared with other models. We show that this approach is well suited to provide an accurate solution over a large computational domain, and we present results for a Halley-like comet.

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On the Partial Difference Equations of Mathematical Physics

Cited by 1,965 publications

References 0 publications

Where does all the water go? Partitioning water transmission losses in a data-sparse, multi-channel and low-gradient dryland river system using modelling and remote sensing

Where does all the water go? Partitioning water transmission losses in a data-sparse, multi-channel and low-gradient dryland river system using modelling and remote sensing

Finite element analysis simulations for ultrasonic array NDE inspections

A Multiscale Central Difference Scheme Applied to Magnetohydrodynamic Simulations of Cometary Atmospheres

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