arXiv subjects

Clint Dawson

Publications and source records attributed to Clint Dawson.

42 records · Page 3Linked to original sources

arXiv2020

Discontinuous Galerkin methods for a dispersive wave hydro-morphodynamic model with bed-load transport

A dispersive wave hydro-morphodynamic model coupling the Green-Naghdi equations (the hydrodynamic part) with the sediment continuity Exner equation (the morphodynamic part) is presented. Numerical solution algorithms based on discontinuous Galerkin finite element discretizations of the model are proposed. The algorithms include both coupled and decoupled approaches for solving the hydrodynamic and morphodynamic parts simultaneously and separately from each other, respectively. The Strang operator splitting technique is employed to treat the dispersive terms separately, and it provides the ability to ignore the dispersive terms in specified regions, such as surf zones. Algorithms that can handle wetting-drying and detect wave breaking are presented. The numerical solution algorithms are validated with numerical experiments to demonstrate the ability of the algorithms to accurately resolve hydrodynamics of solitary and regular waves, and morphodynamic changes induced by such waves. The results indicate that the model has the potential to be used in studies of coastal morphodynamics driven by dispersive water waves, given that the hydrodynamic part resolves the water motion and dispersive wave effects with sufficient accuracy up to swash zones, and the morphodynamic model can capture the major features of bed erosion and deposition.

math.NA↗

arXiv2020

An Unconditionally Stable Space-Time FE Method for the Korteweg-de Vries Equation

We introduce an unconditionally stable finite element (FE) method, the automatic variationally stable FE (AVS-FE) method for the numerical analysis of the Korteweg-de Vries (KdV) equation. The AVS-FE method is a Petrov-Galerkin method which employs the concept of optimal discontinuous test functions of the discontinuous Petrov- Galerkin (DPG) method. However, since AVS-FE method is a minimum residual method, we establish a global saddle point system instead of computing optimal test functions element-by-element. This system allows us to seek both the approximate solution of the KdV initial boundary value problem (IBVP) and a Riesz representer of the approximation error. The AVS-FE method distinguishes itself from other minimum residual methods by using globally continuous Hilbert spaces, such as H1, while at the same time using broken Hilbert spaces for the test. Consequently, the AVS-FE approximations are classical C0 continuous FE solutions. The unconditional stability of this method allows us to solve the KdV equation space and time without having to satisfy a CFL condition. We present several numerical verifications for both linear and nonlinear versions of the KdV equation leading to optimal convergence behavior. Finally, we present a numerical verification of adaptive mesh refinements in both space and time for the nonlinear KdV equation.

math.NA↗

arXiv2020

Adaptive Total Variation Stable Local Timestepping for Conservation Laws

This paper proposes a first-order total variation diminishing (TVD) treatment for coarsening and refining of local timestep size in response to dynamic local variations in wave speeds for nonlinear conservation laws. The algorithm is accompanied with a proof of formal correctness showing that given a sufficiently small minimum timestep the algorithm will produce TVD solution for nonlinear scalar conservation laws. A key feature of the algorithm is its formulation as a discrete event simulation, which allows for easy and efficient parallelization using existing software. Numerical results demonstrate the stability and adaptivity of the method for the shallow water equations. We also introduce a performance model to load balance and explain the observed performance gains. Performance results are presented for a single node on Stampede2's Skylake partition using an optimistic parallel discrete event simulator. Results show the proposed algorithm recovering 59%-77% of the theoretically achievable speed-up with the discrepancies being attributed to the cost of computing the CFL condition and load imbalance.

math.NA↗

arXiv2014

Solving Stochastic Inverse Problems using Sigma-Algebras on Contour Maps

We compute approximate solutions to inverse problems for determining parameters in differential equation models with stochastic data on output quantities. The formulation of the problem and modeling framework define a solution as a probability measure on the parameter domain for a given $σ-$algebra. In the case where the number of output quantities is less than the number of parameters, the inverse of the map from parameters to data defines a type of generalized contour map. The approximate contour maps define a geometric structure on events in the $σ-$algebra for the parameter domain. We develop and analyze an inherently non-intrusive method of sampling the parameter domain and events in the given $σ-$algebra to approximate the probability measure. We use results from stochastic geometry for point processes to prove convergence of a random sample based approximation method. We define a numerical $σ-$algebra on which we compute probabilities and derive computable estimates for the error in the probability measure. We present numerical results to illustrate the various sources of error for a model of fluid flow past a cylinder.

math.NA↗

arXiv2014

Nearshore sticky waters

Wind- and current-driven flotsam, oil spills, pollutants, and nutrients, approaching the nearshore will frequently appear to park just beyond the break zone, where waves break. Moreover, the portion of these tracers that beach will do so only after a long time. Explaining why these tracers park and at what rate they reach the shore has important implications on a variety of different nearshore environmental issues, including the determination of what subscale processes are essential in computer models for the simulation of pollutant transport in the nearshore. Using a simple model we provide an explanation for the underlying mechanism responsible for the parking of tracers, the role played by the bottom topography, and the non-uniform dispersion which leads, in some circumstances, to the eventual landing of all or a portion of the tracers. We refer to the parking phenomenon in this environment as nearshore sticky waters.

physics.flu-dyn↗

arXiv2013

The regularizing Levenberg-Marquardt scheme for history matching of petroleum reservoirs,

In this paper we study a history matching approach that consists of finding stable approximations to the problem of minimizing the weighted least-squares functional that penalizes the misfit between the reservoir model predictions $G(u)$ and noisy observations $y^η$. In other words, we are interested in computing $u^η\equiv \arg\min_{u\in X}\frac{1}{2}\vert\vert Γ^{-1/2}(y-G(u))\vert\vert_{Y}^{2} $ where $Γ$ is the measurements error covariance, $Y$ is the observation space and $X$ is a set of admissible parameters. This is an ill-posed nonlinear inverse problem that we address by means of the regularizing Levenberg-Marquardt scheme developed in \cite{Hanke,Hanke2}. Under certain conditions on $G$, the theory of \cite{Hanke,Hanke2} ensures convergence of the scheme to stable approximations to the inverse problem. We propose an implementation of the regularizing Levenberg-Marquardt scheme that enforces prior knowledge of the geologic properties. In particular, the prior mean $\overline{u}$ is incorporated in the initial guess of the algorithm and the prior error covariance $C$ is enforced through the definition of the parameter space $X$. Our main goal is to numerically show that the proposed implementation of the regularizing Levenberg-Marquardt scheme of Hanke is a robust method capable of providing accurate estimates of the geologic properties for small noise measurements. The performance for recovering the true permeability with the regularizing Levenberg-Marquardt scheme is compared against the more standard techniques for history matching proposed in \cite{Li,Tavakoli,svdRML,Oliver}. Our numerical experiments suggest that the history matching approach based on iterative regularization is robust and could potentially be used to improve further on various methodologies already proposed as effective tools for history matching in petroleum reservoirs

math.OC↗