scholarly journals Data-Driven Model Reduction for Stochastic Burgers Equations

Entropy ◽  
2020 ◽  
Vol 22 (12) ◽  
pp. 1360
Author(s):  
Fei Lu
Keyword(s):  
Energy Spectrum ◽  
Space Time ◽  
Time Step ◽  
Burgers Equations ◽  
Time Reduction ◽  
Flow Map ◽  
Invariant Densities ◽  
Stability Limits ◽  
Maximal Space ◽  
Maximal Time

We present a class of efficient parametric closure models for 1D stochastic Burgers equations. Casting it as statistical learning of the flow map, we derive the parametric form by representing the unresolved high wavenumber Fourier modes as functionals of the resolved variable’s trajectory. The reduced models are nonlinear autoregression (NAR) time series models, with coefficients estimated from data by least squares. The NAR models can accurately reproduce the energy spectrum, the invariant densities, and the autocorrelations. Taking advantage of the simplicity of the NAR models, we investigate maximal space-time reduction. Reduction in space dimension is unlimited, and NAR models with two Fourier modes can perform well. The NAR model’s stability limits time reduction, with a maximal time step smaller than that of the K-mode Galerkin system. We report a potential criterion for optimal space-time reduction: the NAR models achieve minimal relative error in the energy spectrum at the time step, where the K-mode Galerkin system’s mean Courant–Friedrichs–Lewy (CFL) number agrees with that of the full model.

scholarly journals Data-driven Model Reduction for Stochastic Burgers Equations

2020 ◽  
Author(s):  
Fei Lu
Keyword(s):  
Energy Spectrum ◽  
Space Time ◽  
Time Step ◽  
Burgers Equations ◽  
Time Reduction ◽  
Flow Map ◽  
Invariant Densities ◽  
Stability Limits ◽  
Maximal Time ◽  
Optimal Space

We present a class of efficient parametric closure models for 1D stochastic Burgers equations. Casting it as statistical learning of the flow map, we derive the parametric form by representing the unresolved high wavenumber Fourier modes as functionals of the resolved variables’ trajectory. The reduced models are nonlinear autoregression (NAR) time series models, with coefficients estimated from data by least squares. The NAR models can accurately reproduce the energy spectrum, the invariant densities, and the autocorrelations. Taking advantage of the simplicity of the NAR models, we investigate maximal and optimal space-time reduction. Reduction in space dimension is unlimited, and NAR models with two Fourier modes can perform well. The NAR model’s stability limits time reduction, with a maximal time step smaller than that of the K-mode Galerkin system. We report a potential criterion for optimal space-time reduction: the NAR models achieve minimal relative error in the energy spectrum at the time step where the K-mode Galerkin system’s mean CFL number agrees with the full model’s.

scholarly journals Accuracy Analysis of a Spectral Element Atmospheric Model Using a Fully Implicit Solution Framework

2010 ◽  
Vol 138 (8) ◽  
pp. 3333-3341 ◽  
Author(s):  
Katherine J. Evans ◽  
Mark A. Taylor ◽  
John B. Drake
Keyword(s):  
Shallow Water Equation ◽  
Time Integration ◽  
Atmospheric Model ◽  
Test Cases ◽  
Implicit Methods ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Fully Implicit ◽  
Stability Limits

Abstract A fully implicit (FI) time integration method has been implemented into a spectral finite-element shallow-water equation model on a sphere, and it is compared to existing fully explicit leapfrog and semi-implicit methods for a suite of test cases. This experiment is designed to determine the time step sizes that minimize simulation time while maintaining sufficient accuracy for these problems. For test cases without an analytical solution from which to compare, it is demonstrated that time step sizes 30–60 times larger than the gravity wave stability limits and 6–20 times larger than the advective-scale stability limits are possible using the FI method without a loss in accuracy, depending on the problem being solved. For a steady-state test case, the FI method produces error within machine accuracy limits as with existing methods, but using an arbitrarily large time step size.

Local Space-Time Adaptive Discontinuous Galerkin Finite Element Methods for Time-Dependent Waves

2003 ◽  
Author(s):  
Lonny L. Thompson ◽  
Dantong He
Keyword(s):  
Discontinuous Galerkin ◽  
Acoustic Radiation ◽  
Adaptive Strategy ◽  
Adaptive Methods ◽  
Space Time ◽  
Time Dependent ◽  
Time Interval ◽  
Time Step ◽  
Continuous Space ◽  
Local Space

Local space-time adaptive methods are developed including high-order accurate nonreflecting boundary conditions (NRBC) for time-dependent waves. The time-discontinuous Galerkin (TDG) variational method is used to divide the time-interval into space-time slabs, the solution advanced from one slab to the next. Within each slab, a continuous space-time mesh is used which enables local sub-time steps. By maintaining orthogonality of the space-time mesh and pre-integrating analytically through the time-slab, we obtain an efficient yet robust local space-time adaptive method. Any standard spatial element may be used together with standard spatial mesh generation and visualization methods. Recovery based error estimates are used in both space and time dimensions to determine the number and size of local space-time elements within a global time step such that both the spatial and temporal estimated error is equally distributed throughout the space-time approximation. The result is an efficient and reliable adaptive strategy which distributes local space-time elements where needed to accurately track time-dependent waves over large distances and time. Numerical examples of time-dependent acoustic radiation are given which demonstrate the accuracy, reliability and efficiency gained from this new technology.

scholarly journals THE ANYON MODEL: AN EXAMPLE INSPIRED BY STRING THEORY

2011 ◽  
Vol 26 (16) ◽  
pp. 2757-2772 ◽  
Author(s):  
S. V. TALALOV
Keyword(s):  
Dynamical System ◽  
Energy Spectrum ◽  
String Theory ◽  
Arbitrary Number ◽  
Simple Formula ◽  
Fock Space ◽  
Space Time ◽  
Fermionic Field ◽  
Spectral Problems

We investigate the enlarged class of open finite strings in (2+1)D space–time. The new dynamical system related to this class is constructed and quantized here. As the result, the energy spectrum of the model is defined by a simple formula [Formula: see text]; the spin [Formula: see text] is an arbitrary number here but the constants αn and cn are eigenvalues for certain spectral problems in fermionic Fock space Hψ constructed for the free 2D fermionic field.

Unified Space–Time Finite Element Methods for Dissipative Continua Dynamics

2017 ◽  
Vol 09 (02) ◽  
pp. 1750019 ◽  
Author(s):  
Jinkyu Kim ◽  
Gary F. Dargush ◽  
Hwasung Roh ◽  
Jaeho Ryu ◽  
Dongkeon Kim
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Analytical Investigation ◽  
Space Time ◽  
Time Step ◽  
Average Acceleration ◽  
Acceleration Method ◽  
Step Algorithm ◽  
Displacement Based ◽  
Continuum Dynamics

Based upon the extended framework of Hamilton’s principle, unified space–time finite element methods for viscoelastic and viscoplastic continuum dynamics are presented, respectively. For numerical efficiency, mixed time-step algorithm in time- and displacement-based algorithm in space are adopted. Through analytical investigation, we demonstrate that the Newmark’s constant average acceleration method and the present method are the same for viscoelasticity. With spatial eight-node brick elements, some numerical simulations are undertaken to validate and investigate the performance of the present non-iterative space–time finite element method for viscoplasticity.

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