Long Time Steps for Advection: MPDATA with implicit time stepping

2021 ◽  
Author(s):  
Hilary Weller ◽  
James Woodfield ◽  
Christian Kuehnlein
Keyword(s):  
Implicit Time ◽  
Courant Number ◽  
Time Stepping ◽  
Phase Errors ◽  
Advection Schemes ◽  
The Matrix ◽  
Long Time ◽  
Strong Winds ◽  
Lagrangian Advection ◽  
The Cost

<p>Semi-Lagrangian advection schemes are accurate and efficient and retain accuracy and stability even for large Courant numbers but are not conservative. Flux-form semi-Lagrangian is conservative and in principle can be used to achieve large Courant numbers. However this is complicated and would be prohibitively expensive on grids that are not logically rectangular. </p><p>Strong winds or updrafts can lead to localised violations of Courant number restrictions which can cause a model with explicit Eulerian advection to crash. Schemes are needed that remain stable in the presence of large Courant numbers. However accuracy in the presence of localised large Courant numbers may not be so crucial.</p><p>Implicit time stepping for advection is not popular in atmospheric science because of the cost of the global matrix solution and the phase errors for large Courant numbers. However implicit advection is simple to implement (once appropriate matrix solvers are available) and is conservative on any grid structure and can exploit improvements in solver efficiency and parallelisation. This talk will describe an implicit version of the MPDATA advection scheme and show results of linear advection test cases. To optimise accuracy and efficiency, implicit time stepping is only used locally where needed. This makes the matrix inversion problem local rather than global. With implicit time stepping MPDATA retains positivity, smooth solutions and accuracy in space and time.</p>

scholarly journals A Stability Analysis of Finite-Volume Advection Schemes Permitting Long Time Steps

2007 ◽  
Vol 135 (7) ◽  
pp. 2658-2673 ◽  
Author(s):  
Peter Hjort Lauritzen
Keyword(s):  
Stability Analysis ◽  
Finite Volume ◽  
Two Dimensional ◽  
Finite Volume Schemes ◽  
Courant Number ◽  
Von Neumann ◽  
Dispersion Properties ◽  
Advection Schemes ◽  
Long Time ◽  
The Stability

Abstract Finite-volume schemes developed in the meteorological community that permit long time steps are considered. These include Eulerian flux-form schemes as well as fully two-dimensional and cascade cell-integrated semi-Lagrangian (CISL) schemes. A one- and two-dimensional Von Neumann stability analysis of these finite-volume advection schemes is given. Contrary to previous analysis, no simplifications in terms of reducing the formal order of the schemes, which makes the analysis mathematically less complex, have been applied. An interscheme comparison of both dissipation and dispersion properties is given. The main finding is that the dissipation and dispersion properties of Eulerian flux-form schemes are sensitive to the choice of inner and outer operators applied in the scheme that can lead to increased numerical damping for large Courant numbers. This spurious dependence on the integer value of the Courant number disappears if the inner and outer operators are identical, in which case, under the assumptions used in the stability analysis, the Eulerian flux-form scheme becomes identical to the cascade scheme. To explain these properties a conceptual interpretation of the flux-based Eulerian schemes is provided. Of the two CISL schemes, the cascade scheme has superior stability properties.

scholarly journals A test of numerical instability and stiffness in the parametrizations of the ARPÉGE and ALADIN models

2013 ◽  
Vol 6 (4) ◽  
pp. 901-913 ◽  
Author(s):  
M. Tudor
Keyword(s):  
Differential Equations ◽  
Weather Prediction ◽  
Stiff Problem ◽  
Vertical Diffusion ◽  
Time Step ◽  
True Solution ◽  
Advection Schemes ◽  
Long Time ◽  
Lagrangian Advection ◽  
Nwp Model

Abstract. Meteorological numerical weather prediction (NWP) models solve a system of partial differential equations in time and space. Semi-lagrangian advection schemes allow for long time steps. These longer time steps can result in instabilities occurring in the model physics. A system of differential equations in which some solution components decay more rapidly than others is stiff. In this case it is stability rather than accuracy that restricts the time step. The vertical diffusion parametrization can cause fast non-meteorological oscillations around the slowly evolving true solution (fibrillations). These are treated with an anti-fibrillation scheme, but small oscillations remain in operational weather forecasts using ARPÉGE and ALADIN models. In this paper, a simple test is designed to reveal if the formulation of particular a physical parametrization is a stiff problem or potentially numerically unstable in combination with any other part of the model. When the test is applied to a stable scheme, the solution remains stable. However, applying the test to a potentially unstable scheme yields a solution with fibrillations of substantial amplitude. The parametrizations of the NWP model ARPÉGE were tested one by one to see which one may be the source of unstable model behaviour. The test identified the set of equations in the stratiform precipitation scheme (a diagnostic Kessler-type scheme) as a stiff problem, particularly the combination of terms arising due to the evaporation of snow.

IMPROVED ACCURACY FOR LOCALLY ONE-DIMENSIONAL METHODS FOR PARABOLIC EQUATIONS

2001 ◽  
Vol 11 (09) ◽  
pp. 1563-1579 ◽  
Author(s):  
JIM DOUGLAS ◽  
SEONGJAI KIM
Keyword(s):  
Parabolic Equations ◽  
Wave Equations ◽  
Computational Cost ◽  
Implicit Time ◽  
Differential Problem ◽  
Time Stepping ◽  
Backward Euler ◽  
The Cost ◽  
Improved Accuracy ◽  
Crank Nicolson

Classical alternating direction (AD) and fractional step (FS) methods for parabolic equations, based on some standard implicit time-stepping procedure such as Crank–Nicolson, can have errors associated with the AD or FS perturbations that are much larger than the errors associated with the underlying time-stepping procedure. We show that minor modifications in the AD and FS procedures can virtually eliminate the perturbation errors at an additional computational cost that is less than 10% of the cost of the original AD or FS method. Moreover, after these modifications, the AD and FS procedures produce identical approximations of the solution of the differential problem. It is also shown that the same perturbation of the Crank–Nicolson procedure can be obtained with AD and FS methods associated with the backward Euler time-stepping scheme. An application of the same concept is presented for second-order wave equations.

On Some Properties of Semi-rings

2018 ◽  
pp. 35-50
Author(s):  
A. I. Belousov
Keyword(s):  
Linear Equations ◽  
Oriented Graph ◽  
Theoretical Computer Science ◽  
Alternative Methods ◽  
Main Role ◽  
Cost Matrix ◽  
Sequential Elimination ◽  
The Matrix ◽  
Systems Of Linear Equations ◽  
The Cost

The main objective of this paper is to prove a theorem according to which a method of successive elimination of unknowns in the solution of systems of linear equations in the semi-rings with iteration gives the really smallest solution of the system. The proof is based on the graph interpretation of the system and establishes a relationship between the method of sequential elimination of unknowns and the method for calculating a cost matrix of a labeled oriented graph using the method of sequential calculation of cost matrices following the paths of increasing ranks. Along with that, and in terms of preparing for the proof of the main theorem, we consider the following important properties of the closed semi-rings and semi-rings with iteration.We prove the properties of an infinite sum (a supremum of the sequence in natural ordering of an idempotent semi-ring). In particular, the proof of the continuity of the addition operation is much simpler than in the known issues, which is the basis for the well-known algorithm for solving a linear equation in a semi-ring with iteration.Next, we prove a theorem on the closeness of semi-rings with iteration with respect to solutions of the systems of linear equations. We also give a detailed proof of the theorem of the cost matrix of an oriented graph labeled above a semi-ring as an iteration of the matrix of arc labels.The concept of an automaton over a semi-ring is introduced, which, unlike the usual labeled oriented graph, has a distinguished "final" vertex with a zero out-degree.All of the foregoing provides a basis for the proof of the main theorem, in which the concept of an automaton over a semi-ring plays the main role.The article's results are scientifically and methodologically valuable. The proposed proof of the main theorem allows us to relate two alternative methods for calculating the cost matrix of a labeled oriented graph, and the proposed proofs of already known statements can be useful in presenting the elements of the theory of semi-rings that plays an important role in mathematical studies of students majoring in software technologies and theoretical computer science.

scholarly journals Seawater Absorption and Adhesion Properties of Hydrophobic and Superhydrophobic Thermoset Epoxy Nanocomposite Coatings

Nanomaterials ◽  
2021 ◽  
Vol 11 (2) ◽  
pp. 272
Author(s):  
Ayman M. Atta ◽  
Mohamed H. El-Newehy ◽  
Meera Moydeen Abdulhameed ◽  
Mohamed H. Wahby ◽  
Ahmed I. Hashem
Keyword(s):  
Thermal Stability ◽  
Oleic Acid ◽  
Salt Spray ◽  
Adhesion Properties ◽  
In Situ Modification ◽  
Long Time ◽  
Seawater Environment ◽  
Thermoset Epoxy ◽  
The Cost

The enhancement of both thermal and mechanical properties of epoxy materials using nanomaterials becomes a target in coating of the steel to protect it from aggressive environmental conditions for a long time, with reducing the cost. In this respect, the adhesion properties of the epoxy with the steel surfaces, and its proper superhyrophobicity to repel the seawater humidity, can be optimized via addition of green nanoparticles (NPs). In-situ modification of silver (Ag) and calcium carbonate (CaCO3) NPs with oleic acid (OA) was carried out during the formation of Ag−OA and CaCO3−OA, respectively. The epoxide oleic acid (EOA) was also used as capping for Ca−O3 NPs by in-situ method and epoxidation of Ag−OA NPs, too. The morphology, thermal stability, and the diameters of NPs, as well as their dispersion in organic solvent, were investigated. The effects of the prepared NPs on the exothermic curing of the epoxy resins in the presence of polyamines, flexibility or rigidity of epoxy coatings, wettability, and coatings durability in aggressive seawater environment were studied. The obtained results confirmed that the proper superhyrophobicity, coating adhesion, and thermal stability of the epoxy were improved after exposure to salt spray fog for 2000 h at 36 °C.

A Numerical Model of In-Channel Pebble Bed Thermal Storage for a Solar Chimney

2012 ◽  
Author(s):  
Brandon Schulte ◽  
O. A. Plumb
Keyword(s):  
Numerical Model ◽  
Present Model ◽  
Thermal Storage ◽  
Implicit Time ◽  
Heat Transfer Characteristics ◽  
Solar Chimney ◽  
Pebble Bed ◽  
One Dimensional ◽  
Time Stepping ◽  
Daily Energy

In this study, solar chimney performance is numerically modeled. Previously published models have considered water bags and natural earth as means to store daytime thermal energy for nighttime operation of the system. The present model considers in-channel pebble bed thermal storage. A one-dimensional, implicit time stepping numerical model is developed to predict solar chimney performance throughout a 24 hour period. The model is partially verified with available experimental data. The daily energy, daily efficiency and heat transfer characteristics of the solar chimney with pebble bed thermal storage are summarized. The numerical simulation showed that by introducing a pebble bed, nightly exit velocities reach 40% of the peak daytime velocity. However, the daily kinetic energy delivered by a solar chimney with pebble bed thermal storage is much less than a traditional solar chimney, suggesting pebble bed thermal storage is more practicable in building heating applications as opposed to power generation.

Wartime activities of the Vavilov Institute

2021 ◽  
Vol 182 (2) ◽  
pp. 151-162
Author(s):  
I. G. Loskutov
Keyword(s):  
Plant Genetic Resources ◽  
Cultivated Plants ◽  
Left Behind ◽  
Long Time ◽  
Potato Collection ◽  
The Government ◽  
The Cost ◽  
The City ◽  
Harsh Reality ◽  
Difficult Part

Among the chronicles relating the heroism of the besieged Leningrad, there are pages dedicated to the deeds performed by the staff the world-famous All-Union Research Institute of Plant Industry (VIR, now the N.I. Vavilov All-Russian Institute of Plant Genetic Resources). With the beginning of the war, even before the city was surrounded by the Nazi troops, the government decided to evacuate a number of factories and institutes from Leningrad, including VIR, but the plan failed. Only in winter did the Institute start partial evacuation, although preparations had been going on for a long time. The largest and most important part of the collection was left behind in the besieged city. The remaining employees were forced to work under the hardest conditions of the siege, in unheated premises. In the harsh reality of the winter in 1941–1942, the daily bread rationing was cut down, and hunger raged in the city, killing tens of thousands of city residents, including VIR employees who kept the stored seeds and tubers untouched. The most difficult part was preserving the potato collection. In the spring of 1942, preparations were made for sowing to restore the viability of seeds and tubers in the fields of Leningrad’s suburban area under the fire from the enemy artillery. Only the heroic efforts of VIR’s staff helped to save the collection from destruction and loss of germination. This heroism cost more than 20 experts and scientists their lives. So, the most dangerous period for the Institute was overcome at such price. Immediately after the siege was lifted, a group of experts was sent to Leningrad from Krasnoufimsk to help with selecting seed accessions for urgent reproduction. Working under extreme physical exhaustion in frozen premises, without water or electricity, under continuous shelling, they saved, many at the cost of their own lives, the collection of cultivated plants and their wild relatives, the herbarium, and the scientific library for future generations. 

scholarly journals Complex Event Processing for Sensor Stream Data

Sensors ◽  
2018 ◽  
Vol 18 (9) ◽  
pp. 3084 ◽  
Author(s):  
Kyoungsoo Bok ◽  
Daeyun Kim ◽  
Jaesoo Yoo
Keyword(s):  
Real Time ◽  
Complex Event Processing ◽  
Superior Performance ◽  
Event Processing ◽  
Stream Data ◽  
Internet Of Things Environment ◽  
Processing Cost ◽  
Single Operator ◽  
Long Time ◽  
The Cost

As a large amount of stream data are generated through sensors over the Internet of Things environment, studies on complex event processing have been conducted to detect information required by users or specific applications in real time. A complex event is made by combining primitive events through a number of operators. However, the existing complex event-processing methods take a long time because they do not consider similarity and redundancy of operators. In this paper, we propose a new complex event-processing method considering similar and redundant operations for stream data from sensors in real time. In the proposed method, a similar operation in common events is converted into a virtual operator, and redundant operations on the same events are converted into a single operator. The event query tree for complex event detection is reconstructed using the converted operators. Through this method, the cost of comparison and inspection of similar and redundant operations is reduced, thereby decreasing the overall processing cost. To prove the superior performance of the proposed method, its performance is evaluated in comparison with existing methods.

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