Time integration methods for finite element discretizations in weather forecasting
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Time integration methods for finite element discretizations in weather forecasting
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Time integration methods for finite element discretizations in weather forecasting

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CC Attribution 3.0 Germany:
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2019

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English

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Abstract 
There is a new interest in finite element methods for solving the equations in numerical weather forecasting. In contrast to finite difference and finite volume methods explicit time integration methods are hampered by nondiagonal mass matrices in front of the derivatives. We will compare different mixed finite and discontinuous Galerkin methods for the twodimensional linear Boussinesq approximation in the context of splitexplicit time integration schemes. Especially different lumping procedures are investigated which replaces nondiagonal mass matrices by simple diagonal blockdiagonal matrices. These methods are compared with energy conserving implicit RungeKutta methods for a nonhydrostatic gravity wave example.

00:00
Goodness of fit
Greatest element
Decision theory
Moment (mathematics)
Element (mathematics)
00:20
Prediction
Exponentialgleichung
Multiplication sign
Disintegration
Student's ttest
Collision
Infinity
Element (mathematics)
Finite element method
Mach's principle
01:01
Standard deviation
Presentation of a group
State of matter
Decision theory
Multiplication sign
Direction (geometry)
Numerical analysis
Mass
Icosahedron
Sphere
Infinity
Fraktalgeometrie
Duality (mathematics)
Goodness of fit
Mathematics
Calculus of variations
Equations of motion
Different (Kate Ryan album)
Gitterverfeinerung
Modulform
Glattheit <Mathematik>
Position operator
Stability theory
Physical system
Area
Predictability
Moment (mathematics)
Sampling (statistics)
Affine space
Element (mathematics)
Numerical analysis
Mathematics
Prediction
Volume
Finite element method
Spacetime
04:05
INTEGRAL
Multiplication sign
Outlier
Direction (geometry)
Mereology
Mathematics
Coefficient of determination
Tensor
Velocity
Pressure
Physical system
Area
Polynomial
Logical constant
Complex (psychology)
Element (mathematics)
Product (business)
Category of being
Velocity
Frequency
Spacetime
Metre
Functional (mathematics)
Connectivity (graph theory)
Student's ttest
Rule of inference
Continuous function
Tensorprodukt
Element (mathematics)
Independent set (graph theory)
Spacetime
Nichtlineares Gleichungssystem
Distribution (mathematics)
Scaling (geometry)
Linear equation
Military base
Projective plane
Basis <Mathematik>
Line (geometry)
Continuous function
Cartesian coordinate system
Affine space
Exponentialgleichung
Function (mathematics)
Autoregressive conditional heteroskedasticity
Einbettung <Mathematik>
Vertical direction
Pressure
Finite element method
11:04
Point (geometry)
Scaling (geometry)
Exponentialgleichung
Different (Kate Ryan album)
Multiplication sign
Model theory
Differential equation
Nichtlineares Gleichungssystem
Mereology
Pressure
Physical system
12:01
Point (geometry)
Functional (mathematics)
Momentum
Diagonal
Multiplication sign
Mass
Inverse element
Infinity
Sparse matrix
Mereology
Element (mathematics)
Mathematics
Independent set (graph theory)
Manysorted logic
Term (mathematics)
Operator (mathematics)
Exponentialgleichung
Matrix (mathematics)
Physical system
Stability theory
Differential (mechanical device)
Eigenvalues and eigenvectors
Moment (mathematics)
Physical law
Model theory
Computability
Mathematical analysis
Algebraic structure
Inverse element
Maxima and minima
Sparse matrix
Voting
Exponentialgleichung
Nichtlineares Gleichungssystem
Right angle
Iteration
Diagonal
Block (periodic table)
Metric system
Mathematician
Pressure
Matrix (mathematics)
Directed graph
Finite element method
18:27
Group action
Direction (geometry)
Multiplication sign
Decision theory
Price index
Time domain
Cartesian product
2 (number)
Time domain
Mach's principle
Wave
Velocity
Gravitation
Set theory
Spacetime
Stability theory
19:47
Point (geometry)
Wave
Term (mathematics)
Green's function
Matrix (mathematics)
Mach's principle
20:42
Supremum
Different (Kate Ryan album)
Forcing (mathematics)
Inverse element
Nichtlineares Gleichungssystem
Term (mathematics)
Sparse matrix
Finite element method
00:01
my philosophy there attempts to go from find bottoms to a planet element this good decision because it's at the moment and that the topic and
00:13
my last a at the at all research institute I want a moment a bit about that might
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be to do them have something also label and that the last vestige together with the bachelor student and chefs these it's a bachelor student worried that has only 1st steps and I want to explain a little bit what we have implemented and what are our experience though I want to give a short
00:43
introduction the my what I loved about the finite elements will be done at the end that we only applied to to a linear problem and then my especially in the less time equation methods and I want to explain what other problems with this time the collision muscles if we all Kongfunded
01:04
finite elements of this to decisions some American sample and that's it so at the moment the
01:14
Hatfield listed some idea of what happens in the moment and I think the mathematics from the former about the prediction of reaches in some sense a really new a font us and then you can look in in different a key different direction in this in this research 1 as someone is that you are going from sound latitude and longitude words to unselected puts OK I'm smarter could see or the has made it was already 20 years ago or something like that or so years ago but I think in the last 10 years together with that we are changing from to fully composable or non had the systems that this unspotted goods also get the new the new interests and we got to do this quits in in different directions so we have different type of threats and we are at the same time we want we want put refinement and they want to have state also on disk with find things on handful different goods we want to have some type of unified mass arts and we want to have a stable methods and methods which a preserve as low as much as possible a form the continuous more and we also go at MIT this quits to more sophisticated space at the position methods so what is it we are looking on the top and for find the differences and on designs fractals which we are going to something which people in other areas called a method method of all combat that of affinity and and us all we are looking for another type of of presentation of the equations of motions like in and or number formulation and going then from this disk from this continuous formulation to a discrete land and from that that we get something about it this position and that of the pellet only go in space we do on the the same thing in time and there are now at moment also appeal not of sings about new retirement at this conversation methods which of follow from this Hamiltonian and formulation and that this year that for instance there is a special workshop about this so called socalled integrate in in the in that at the end of the year so it's also and you see that at all the time I think in in numerical weather prediction there are now people are sitting which to deep deep mathematics in some sense so what
04:06
now I them is no at the mathematics though what we've done with those of the bachelor student is that we want to solve this linearized equations there which is a the is a boson next type equation and you have this twodimensional you have tool to velocity components we have a pressure component and we have a buoyancy component and we have in the in the system as a sweet timescale so it's a advection timescale we have this is a time scale and we have this this Acosta timescale all the time scales which we also have to follow a constant cumbersome and that was a people I think the last of this year in quarterly Olmützer society we had people from the gungho project start presented sings about finite elements for this application and we following the bachelor worked through 2 we implement that and then to see how all tied equation as that's that can be useful so you already I a spirited equation in 2 parts so we have this no time scale the think about maybe 20 meter per 2nd and the thing about this fast timescale sound speed of 260 FIL segments and this 1 is somebody in in between and ice but already the equation here in this to rise version in this 2 parts which we also have later on I call this this rule part and this the 1st part the all finite elements so what we've done is we want those off this simple equation honor when the copies
06:03
input so we started simple with the student with simple onedimensional finite elements and all 210 support poked elements and implement that does 41 so and could not so good and key tells them out with a bachelor's then Bikel because you have not that knowledge about all that as I'm usually also that have appeared in 4 K we started with to aetioporphyrin had elements so we was on Monday on the 1 side we the was a continuous forget elements which also advise all all put in on the polymer found to be pleased the office mom and we denote it by policy on and we have this continuous elements which also was pollen functions which you also of we all plus 1 and been the by teach you all and then we take spaces we have to choose taking as basis for the for the good for the too about a component 3 have to take a space for for the buoyancy and they have to take 1 for the pressure and that what people 1st proposed a little bit is dude tested takes sweetie of this fund for the the affinity elements bases have to take which was disk this continuous and area feel about topic elements never take a space which is which is continuous and you have and and on dementia case you have linear elements and then that you have to take for the buoyancy that for the for the velocity and for the pressure you have to take at this continuous space and this 3 spaces then the therefore if it has this property that you can take the great enough that you could coming here and take the dog built and on Monday the wage and again the end up some they're here but now we have to go to Judea and 2 would be we have to choose the buoyancy the well a city and pressure and now we take for the buoyancy be take this this space he as a tensor product but for the for the for the velocity components we have a a tensor product which is linear a full for you in this direction the tagger the take we take continuous space and on the other line we take this continuous space which 1 although less for the it's it's wise for us are inappropriate for P again we take a tensor product of 2 this continuous spaces and and this raises if you take it again we will have this type of embedding bubble so it can starting with the buoyancy in you go compute this type of a call you end up in in this in this neck buffeted amend space and then you make that elections and end up in this space so now that that's a small more no OK good so what we have we have a we make only experiments with the low although I'm that's functions we have piecewise constant and that's function piecewiselinear and that's functions and they have to work was of it the world is german yeah and peace last but not think functions you have to implement at all and so I had we have love admit how to do that in some efficient and that's all too but easy if you go to find other elements and here you see how is like this fine and elements and look like and and and if you know if you want to go to find out and amends to have to go from this to to a weak formulation and we have to choose all the I'll trial functions under the velocity components and the pressure and the B from this and that's French so that I have showed and again be take test functions which are often from from the same and that's functions the multiplied with this test functions make here for the pressure to term some of the Pasha integrations and offer that we are looking for outliers functions infinite element space which of fuel for all test functions this this equation and if you again compute all so I want not to explain in detail how we compute the advection tome especially for the for the pressure or he'll for the it's it's this continuous audio phobia so we have to do something special at the end we then we end
11:06
up but the modeling our system so you have a system of all the different differential equations and I have have written again here old got lecture part if you if you cut that this part the new see seawall do equations all each equation you can solve by itself and their feel the pressure Tamiflu the boy tell you it's the acoustic Germany have been African a boy and 2 2 so and we have to solve best that equation in time so now
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comes in we my favorite methods which I use in in the final point context that all it experts that matter so for this simple models I want to solve this fully explicit models but with different time steps that because of this different time scales so it's
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it's not true so we are going back to
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this this system here so I have and have so part and the fastball and we have we have developed is
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sometimes of a combined of of a special type of work with them as art which we use for the slow parts of seemed that this fast part this year then what we do is we in each stage we solve again as the French iteration in all in all cases it's all or iterations of now but he this part this is frozen so it's fixed and only this part which is the FastPath is this this really the right hand side function which we can compute in each point and we can solve this in a system with this fixed source film and then at the end the we get a new speech right we can compute again this slow pop can form again this type of work all right inside soft this we know I creation and read feel maybe 3 stages with a fixed time stamp and then we take smaller prime steps to solve this system but the the special muscle and if you are looking in our for this for this fast part here then you you you will see a avid a little bit simplified if you take only momentum and and pressure that this fast part has this symplectic structure so you see if you would it depends on p and p depends on on you and then you can of what's what mathematicians call and symplectic all I think you as a small arrow or so we solve fills the 1st creation with with a full what all along and then use that you should be and plus 1 then dissolved a 2nd iteration but at the Beck model this is this is the stable mass art and there's also something which is this is a centralized symplectic are law which is also called in the you met little the literature something like of of time much as a dumping so we make here again before what all of that and the place you at this this for what us that also with some with some lecture on the peak the the bits the head and to stabilize the whole procedure that is also not fully understand it at the moment so now we are going to this to apply this with explicit methods and we to slide and we have to bring the the year the mess MayBritt switches on front what you have seen it if you had a welcome at the time that elements have to bring this to the right hand side so you'll for each computation of HBO's of terms you have to soften some sense an assistant who can make before that it wouldn't good they composition but if you go to Sweetie'' it's it's a major problem now what people do in the finite element of literature they did do some some slumping the the the data the rulebased this mate by diner got all blocked by the bricks which are called though that you have now the filled options you'll you replace pose of them but this the lump matrix or the other 1 you only use a because that I compute on the on the large times that he I take the exact messmate matrix that I do on a small that so I take you didn't metrics rare have sort of a simple linear system so it's diagonal Proc no problem and SSL a cell the idea so what I do know I sup upright shot something taken in the media I sup like does he did that he'll from a for the from the wonderful for Malaysian and I added again so now this is something with which I called my slow term I have to compute it only when this is my fast left CompuAdd often and we tried that fills and we see OK this was of 10 people OK did a lot with us but experts at methods by this idea doesn't work so that was unstable the watermarked then I make some eigen value analysis and see that that at least if you see that the that this operator has some eigenvalues which are sitting on the not on the left hand side and and this makes the problems and now the idea walls and not to take it this simple last lump breaks but to go to a matrix made say beyond looking for something which isn't in those metrics of of of the mass matrix quite has a has a sparse structure something like an approximate inverse and that we work here that the at this this term here and and again though you don't change the equation the only change what is the slope or the voters the fast and for this involves matrix we tested 2 ideas the first one is we taken and be taken and in in those metrics which is the the minimizer of this of this will bring a small and in Ingalls has cut sparsity peplum so you could put in for the food is is so the so the minimum this Matsuo or the idea is that we compute the inverse and then it came to take only a sparse pattern of this in the matrix A and the sparsity deep pet is you should be the same as as the sparsity pattern of of 10 or something like that so and then this idea we made some experiments so we take the a
18:33
copy in domain so 150kilometre to do to the east and the west and thank can move us in the IoT and sweet thousand segments and we start with the atmosphere requests that ran out the the velocity and this this budget but the business well and buoyancy has has this type of a special spot show and this 1 you put the patient that you have a give us some of some some of some movement that so you're all EU Parliament us and we take a space of this decision so It's index I action and also in this set direction it's it's like the the metal so in the developed a direction it's my course and we take a slow times that of 20 seconds and they have to take something like a 90 and small time steps below the large times that to fulfill this this stability to Terence for this for the and this is the this
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is the this is the buoyancy at the at at th the the stopping pardon and then you after the 3 thousand seconds if you also
19:49
include some some of the some horizontal movement here you get this type of picture and we are now looking somewhere here in the middle of this point see you in the case where we have no movement to see what happens with this different must lumping
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procedures so I have blocked it here now sweet tired of of of lumping procedure so I have 1st the 1 of the the this to this in this blue is we make no lumping what we call the exact solution some sense then be made a full lumping so we have 4 boast terms we have landing matrix and then we make we in there we have only the slumping on on the fast and there said all things will work so and then I'll there should be a little bit
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of and now we of going the maybe that's the last 1 now what we now do is that go to this the the spores this this idea that these th sup thanked foster and in the case that we take the indoors which is off boss then you see for us if we take we do it with our this this station we at his some problems and then if you make also this this small southern decision in fast for the Caucasus that belies the sprit experts and then you see that the exact solution and that what we get here all on this user you see no no differences will this force experiments and that was something like outcome of this bechamel rock and what I want to send the and now I am also have already a finite element code into these useful for the fully compostable equations but notice that that so that's what I want to say sentence
21:51
and