Non-Linear Error Ansatz Models in Space and Time for Solution Verification

Hemez, François M.; Brock, Jerry S.; Kamm, James R.

doi:10.2514/6.2006-1995

Cited by 10 publications

(11 citation statements)

References 5 publications

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“…Of course, if we cannot solve the primal equation (1), it is unlikely that we can solve the error equation (4). Thus, we will spatially discretize the error equation:…”

Section: Method-of-lines Formulationmentioning

confidence: 99%

“…We now specialize to the problem of discretization error. In general, we do not know how to solve either the original PDE (1) or the error equation (4). We can, however, discretize both and solve each approximately.…”

Section: Basic Conceptsmentioning

confidence: 99%

“…In this section, we briefly consider the one-dimensional ideal gas Euler equations of fluid dynamics, which represent a hyperbolic system of nonlinear equations. While treating a system with the error evolution approach is no different per se, we do note that the Burgers' equation possesses a particularly amenable nonlinearity; its quadratic form produces a particularly simple form of G(ẽ;ũ) in the nonlinear error equation (4).…”

Section: Application To the Euler Equationsmentioning

confidence: 99%

“…We can, however, discretize both and solve each approximately. There are thus three decisions to be made: the choice of discretization of the primal equation (1), the choice of discretization of the evolution operator in the error equation (4), and the choice of an evaluation technique for the residual in the error equation (4). Appropriate choices for the first two discretizations depend intimately on the nature of the continuous operator and initial conditions.…”

Section: Basic Conceptsmentioning

confidence: 99%

“…This includes cases where discontinuities are present and true weak solutions are sought. There have been attempts to consider richer ansätze with varying degrees of success [4].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Numerical error estimation for nonlinear hyperbolic PDEs via nonlinear error transport

Banks

Hittinger

Connors

et al. 2012

Computer Methods in Applied Mechanics and Engineering

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The estimation of discretization error in numerical simulations is a key component in the development of uncertainty quantification. In particular, there exists a need for reliable, robust estimators for finite volume and finite difference discretizations of hyperbolic partial differential equations. The approach espoused here, often called the error transport approach in the literature, is to solve an auxiliary error equation concurrently with the primal governing equation to obtain a point-wise (cell-wise) estimate of the discretization error. One-dimensional, nonlinear, time-dependent problems are considered. In contrast to previous work, fully nonlinear error equations are advanced, and potential benefits are identified. A systematic approach to approximate the local residual for both method-of-lines and space-time discretizations is developed. Behavior of the error estimates on problems that include weak solutions demonstrates that this nonlinear error evolution provides useful error estimates.

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“…Of course, if we cannot solve the primal equation (1), it is unlikely that we can solve the error equation (4). Thus, we will spatially discretize the error equation:…”

Section: Method-of-lines Formulationmentioning

confidence: 99%

Section: Basic Conceptsmentioning

confidence: 99%

Section: Application To the Euler Equationsmentioning

confidence: 99%

Section: Basic Conceptsmentioning

confidence: 99%

“…This includes cases where discontinuities are present and true weak solutions are sought. There have been attempts to consider richer ansätze with varying degrees of success [4].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Numerical error estimation for nonlinear hyperbolic PDEs via nonlinear error transport

Banks

Hittinger

Connors

et al. 2012

Computer Methods in Applied Mechanics and Engineering

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A Brief Overview of Code and Solution Verification in Numerical Simulation

Hemez

2017

Model Validation and Uncertainty Quantification, Volume 3

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Improving Time Step Convergence in an Atmosphere Model With Simplified Physics: The Impacts of Closure Assumption and Process Coupling

Wan

Woodward

Zhang

et al. 2020

J Adv Model Earth Syst

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Convergence testing is a common practice in the development of dynamical cores of atmospheric models but is not as often exercised for the parameterization of subgrid physics. An earlier study revealed that the stratiform cloud parameterizations in several predecessors of the Energy Exascale Earth System Model (E3SM) showed strong time step sensitivity and slower-than-expected convergence when the model's time step was systematically refined. In this work, a simplified atmosphere model is configured that consists of the spectral-element dynamical core of the E3SM atmosphere model coupled with a large-scale condensation parameterization based on commonly used assumptions. This simplified model also resembles E3SM and its predecessors in the numerical implementation of process coupling and shows poor time step convergence in short ensemble tests. We present a formal error analysis to reveal the expected time step convergence rate and the conditions for obtaining such convergence. Numerical experiments are conducted to investigate the root causes of convergence problems. We show that revisions in the process coupling and closure assumption help to improve convergence in short simulations using the simplified model; the same revisions applied to a full atmosphere model lead to significant changes in the simulated long-term climate. This work demonstrates that causes of convergence issues in atmospheric simulations can be understood by combining analyses from physical and mathematical perspectives. Addressing convergence issues can help to obtain a discrete model that is more consistent with the intended representation of the physical phenomena. Plain Language Summary Computer codes that simulate the time evolution of a physical system produce errors in the results due to the finite step sizes used to advance the calculations in time. These errors are expected to decrease as the time steps are shortened, at a rate determined by the characteristics of the equations and numerical methods. An earlier study revealed that the error reduction in several predecessors of the Energy Exascale Earth System Model (E3SM) was at a rate slower than expected. This study creates a simplified configuration of those models and investigates the causes of the unexpected behavior. We show that slow error reduction can be understood and improved by combining analyses from physical and mathematical perspectives. The required code modifications can lead to significant changes in the simulated long-term behavior of a full-fledged climate model. Furthermore, ensuring a proper rate of error reduction can help to obtain a computer code that is more consistent with the intended representation of the corresponding physical system.

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Non-Linear Error Ansatz Models in Space and Time for Solution Verification

Cited by 10 publications

References 5 publications

Numerical error estimation for nonlinear hyperbolic PDEs via nonlinear error transport

Numerical error estimation for nonlinear hyperbolic PDEs via nonlinear error transport

A Brief Overview of Code and Solution Verification in Numerical Simulation

Improving Time Step Convergence in an Atmosphere Model With Simplified Physics: The Impacts of Closure Assumption and Process Coupling

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