Semi-Lagrangian Subgrid Reconstruction for Advection-Dominant Multiscale Problems with Rough Data

Simon, Konrad; Behrens, Jörn

doi:10.1007/s10915-021-01451-w

Cited by 4 publications

(4 citation statements)

References 35 publications

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“…The essential idea of multi-scale methods is to resolve the fine-scale features such as strong gradients in the layers by locally precomputed generalized FE shape functions. Prime examples are variational multi-scale methods (VMS) [HFMQ98, LM09, JKL06, Må11], multi-scale FEMs [PH04,DLMN15], multi-scale hybrid-mixed methods [HPV15], multiscale discontinuous Galerkin methods [KW14,CL13], multi-scale virtual element methods [XWF21], multi-scale stabilization methods [CCEL16,CEL20], stabilization procedures by means of sub-grid scale [Cod00], energy minimizing generalized multi-scale methods [ZC22], or the multi-scale method for time-dependent convection-dominant problems recently proposed in [SB21].…”

Section: Introductionmentioning

confidence: 99%

Super-localized orthogonal decomposition for convection-dominated diffusion problems

Bonizzoni¹,

Freese²,

Peterseim³

2022

Preprint

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This paper presents a multi-scale method for convection-dominated diffusion problems in the regime of large Péclet numbers. The application of the solution operator to piecewise constant right-hand sides on some arbitrary coarse mesh defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the L 2 -norm, the Galerkin projection onto this generalized finite element space even yields ε-independent error bounds, ε being the singular perturbation parameter. By constructing an approximate local basis, the approach becomes a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization can be estimated in an a-posteriori way. In contrast to existing multi-scale methods, numerical experiments indicate ε-independent convergence without preasymptotic effects even in the under-resolved regime of large mesh Péclet numbers.

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Section: Introductionmentioning

confidence: 99%

Super-localized orthogonal decomposition for convection-dominated diffusion problems

Bonizzoni¹,

Freese²,

Peterseim³

2022

Preprint

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“…While it is common to construct a parameterization using assumptions on grid cell average quantities (Rasch & Kristjánsson, 1998; Sundqvist, 1978; Tiedtke, 1993; M. Zhang et al., 2003), an alternative approach is to first derive a formulation directly from a description of the physical phenomena that enforces the desired physical relationships and constraints, and then choose a numerical method to establish connections between this formulation and the grid cell average quantities the AGCM solves for. For example, both the reconstruct‐evolve‐average (LeVeque, 2002) and semi‐Lagrangian multiscale finite element method (Simon & Behrens, 2019) approaches reconstruct spatial profiles within a grid cell from discrete grid cell average values. Such profiles, referred to as subgrid profiles, can be evolved using equations independent of grid cell size and then be mapped back to discrete grid cell average values.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Deriving a parameterization for a given physical process is not a trivial task (McFarlane, 2011). As described by Simon and Behrens (2019), parameterizations derived in a heuristic or intuitive manner can lead to incorrect and unintended physical behaviors. Park et al (2014) provide an example for atmospheric parameterizations, discussing the inconsistency between diagnosed nonconvective cloud fraction and prognosed cloud condensate amount.…”

Section: Introductionmentioning

confidence: 99%

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Improving Time Step Convergence in an Atmosphere Model With Simplified Physics: Using Mathematical Rigor to Avoid Nonphysical Behavior in a Parameterization

Vogl

Wan

Zhang

et al. 2020

J Adv Model Earth Syst

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Global atmospheric models seek to capture physical phenomena across a wide range of time and length scales. For this to be a feasible task, the physical processes with time or length scales below that of a computational time step or grid cell size are simplified as one or more parameterizations. Inadvertent oversimplification can violate constraints or destroy relationships in the original physical system and consequently lead to unexpected and physically invalid behavior. An example of such a problem has been investigated in the work of Wan et al. (2020, https://doi.org/10.1029/2019MS001982). This work addresses the issues at a more fundamental level by revisiting the parameterization derivation. A derivation of an unaveraged condensation rate in the unaveraged equations, sometimes referred to as subgrid equations, provides a clear description and more accurate quantification of the condensation/evaporation processes associated with cloud growth/decay, while avoiding simplifications used in earlier studies. A subgrid reconstruction (SGR) methodology is used to connect the unaveraged condensation rate with the grid cell averaged equations solved by the global model. Analyses of the SGR method and the numerical results provide insights into root causes of inconsistent discrete formulations and nonphysical behavior. It is also shown that the SGR methodology provides a flexible framework for addressing such inconsistencies. This work serves as a demonstration that when nonphysical behavior in a parameterization of subgrid variability is avoided through rigorous mathematical derivation, the resulting formulation can exhibit both better numerical convergence properties and significant impact on long-term climate. Plain Language Summary Simulation of global climate is a difficult task even for the most powerful supercomputers available today. Instead of directly simulating every physical process in the atmosphere, average effects are typically considered to make global climate simulations feasible. As an example, it is currently not possible to track the entire lifespan of every cloud on the planet over tens or hundreds of years. Instead, the average cloud formation and evolution is modeled using a "parameterization." A parameterization typically makes assumptions about the underlying physical behaviors represented by the average values. This work highlights issues that arise when those assumptions are overly simple or inconsistent. It shows how a mathematically rigorous approach to parameterization development can help avoid, diagnose, and correct those issues. Specifically, an existing large-scale condensation (nonconvective cloud formation) parameterization is examined and improved upon, with the result both capturing more realistic physical behavior and being better positioned to make use of upcoming advances in computational power.

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Super-localized orthogonal decomposition for convection-dominated diffusion problems

Bonizzoni,

Freese,

Peterseim

2024

Bit Numer Math

View full text Add to dashboard Cite

This paper presents a novel multi-scale method for convection-dominated diffusion problems in the regime of large Péclet numbers. The method involves applying the solution operator to piecewise constant right-hand sides on an arbitrary coarse mesh, which defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the $$L^2$$ L 2 -norm, the Galerkin projection onto this generalized finite element space even yields $$\varepsilon $$ ε -independent error bounds, $$\varepsilon $$ ε being the singular perturbation parameter. By constructing an approximate local basis, the approach becomes a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization can be estimated in an a posteriori way. In contrast to existing multi-scale methods, numerical experiments indicate $$\varepsilon $$ ε -robust convergence without pre-asymptotic effects even in the under-resolved regime of large mesh Péclet numbers.

show abstract

Semi-Lagrangian Subgrid Reconstruction for Advection-Dominant Multiscale Problems with Rough Data

Cited by 4 publications

References 35 publications

Super-localized orthogonal decomposition for convection-dominated diffusion problems

Super-localized orthogonal decomposition for convection-dominated diffusion problems

Improving Time Step Convergence in an Atmosphere Model With Simplified Physics: Using Mathematical Rigor to Avoid Nonphysical Behavior in a Parameterization

Super-localized orthogonal decomposition for convection-dominated diffusion problems

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