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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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 non-diagonal mass matrices in front of the derivatives. We will compare different mixed finite and discontinuous Galerkin methods for the two-dimensional linear Boussinesq approximation in the context of split-explicit time integration schemes. Especially different lumping procedures are investigated which replaces non-diagonal mass matrices by simple diagonal block-diagonal matrices. These methods are compared with energy conserving implicit Runge-Kutta methods for a non-hydrostatic gravity wave example.
Goodness of fit Greatest element Decision theory Moment (mathematics) Element (mathematics)
Prediction Exponentialgleichung Multiplication sign Disintegration Student's t-test Collision Infinity Element (mathematics) Finite element method Mach's principle
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
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 t-test 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
Point (geometry) Scaling (geometry) Exponentialgleichung Different (Kate Ryan album) Multiplication sign Model theory Differential equation Nichtlineares Gleichungssystem Mereology Pressure Physical system
Point (geometry) Functional (mathematics) Momentum Diagonal Multiplication sign Mass Inverse element Infinity Sparse matrix Mereology Element (mathematics) Mathematics Independent set (graph theory) Many-sorted 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
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
Point (geometry) Wave Term (mathematics) Green's function Matrix (mathematics) Mach's principle
Supremum Different (Kate Ryan album) Forcing (mathematics) Inverse element Nichtlineares Gleichungssystem Term (mathematics) Sparse matrix Finite element method
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it's not true so we are going back to
this this system here so I have and have so part and the fastball and we have we have developed is
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 May-Britt 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 rule-based 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
copy in domain so 150-kilometre 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
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
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
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
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