Engineering topics

Haut, Terry

Publications and source records attributed to Haut, Terry.

4 records

DOE OSTI2024

Multilevel Parareal Algorithm with Averaging for Oscillatory Problems

The present study is an extension of the work done by Peddle, Haut, and Wingate and Haut and Wingate, where a two-level Parareal method with mapping and averaging is examined. The method proposed in this paper is a multilevel Parareal method with arbitrarily many levels, which is not restricted to the two-level case. We give an asymptotic error estimate which reduces to the two-level estimate for the case when only two levels are considered. Introducing more than two levels has important consequences for the averaging procedure, as we choose separate averaging windows for each of the different levels, which is an additional new feature of the present study. The different averaging windows make the proposed method especially appropriate for nonlinear multiscale problems, because we can introduce a level for each intrinsic scale of the problem and adapt the averaging procedure such that we reproduce the behavior of the model on the particular scale resolved by the level. The method is applied to nonlinear differential equations. The nonlinearities can generate a range of frequencies in the problem. The computational cost of the new method is investigated and studied on several examples.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Mean Flow from Phase Averages in the 2D Boussinesq Equations

The atmosphere and ocean are described by highly oscillatory PDEs that challenge both our understanding of their dynamics and their numerical approximation. This paper presents a preliminary numerical study of one type of phase averaging applied to mean flows in the 2D Boussinesq equations that also has application to numerical methods. The phase averaging technique, well-known in dynamical systems theory, relies on a mapping using the exponential operator, and then an averaging over the phase. The exponential operator has connections to the Craya–Herring basis pioneered by Jack Herring to study the fluid dynamics of oscillatory, nonlinear fluid dynamics. In this paper, we perform numerical experiments to study the effect of this averaging technique on the time evolution of the solution. We explore its potential as a definition for mean flows. We also show that, as expected from theory, the phase-averaging method can reduce the magnitude of the time rate of change in the PDEs, making them potentially suitable for time stepping methods.

Wingate, Beth A.↗

DOE OSTI2021

A short note on the accuracy of the discontinuous Galerkin method with reentrant faces

In this work, we study the convergence of the discontinuous Galerkin (DG) method applied to the advection–reaction equation on meshes with reentrant faces. On such meshes, the upwind numerical flux is not smooth, and so the numerical integration of the resulting face terms can only be expected to be first-order accurate. Despite this inexact integration, we prove that the DG method converges with order $\mathscr{O}$(h p+1/2 ), which is the same rate as in the case of exact integration. Consequently, specialized quadrature rules that accurately integrate the non-smooth numerical fluxes are not required for high-order accuracy. These results are numerically corroborated on examples of linear advection and discrete ordinates transport equations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

Model Structural Inference Using Local Dynamic Operators

This paper focuses on the problem of quantifying the effects of model-structure uncertainty in the context of time-evolving dynamical systems. This is motivated by multi-model uncertainty in computer physics simulations: developers often make different modeling choices in numerical approximations and process simplifications, leading to different numerical codes that ostensibly represent the same underlying dynamics. We consider model-structure inference as a two-step methodology: the first step is to perform system identification on numerical codes for which it is possible to observe the full state; the second step is structural uncertainty quantification, in which the goal is to search candidate models "close" to the numerical code surrogates for those that best match a quantity of interest (QOI) from some empirical data sets. Here, we (1) define a discrete, local representation of the structure of a partial differential equation, which we refer to as the "local dynamical operator" (LDO); (2) identify model structure nonintrusively from numerical code output; (3) nonintrusively construct a reduced-order model (ROM) of the numerical model through POD-DEIM-Galerkin projection; (4) perturb the ROM dynamics to approximate the behavior of alternate model structures; and (5) apply Bayesian inference and energy conservation laws to calibrate a LDO to a given QOI. We demonstrate these techniques using the two-dimensional rotating shallow water equations as an example system.

97 MATHEMATICS AND COMPUTING↗