Damrongsak Wirasaet

Adaptive hierarchic transformations for dynamically p-enriched slope-limiting over discontinuous Galerkin systems of generalized equations

We study a family of generalized slope limiters in two dimensions for Runge-Kutta discontinuous Galerkin (RKDG) solutions of advection-diffusion systems. We analyze the numerical behavior of these limiters applied to a pair of model problems, comparing the error of the approximate solutions, and discuss each limiters advantages and disadvantages. We then introduce a series of coupled p-enrichment schemes that may be used as standalone dynamic p-enrichment strategies, or may be augmented via any in the family of variable-in-p slope limiters presented.

WAMR: An adaptive wavelet method for the simulation of compressible reacting flow. Part I. Accuracy and efficiency of algorithm

Abstract The Wavelet Adaptive Multiresolution Representation (WAMR) method provides a robust method for controlling spatial grid adaptation — fine grid spacing in regions where a solution varies greatly (i.e., near steep gradients, or near-singularities) and a much coarser grid where the solution varies slowly. Subsequently, a wide range of spatial scales, often demanded in challenging continuum physics problems, can be efficiently captured. Furthermore, the wavelet transform provides a direct measure of local error at each collocation point, effectively producing automatically verified solutions. The method is applied to the solution of unsteady, compressible, reactive flow equations, and includes detailed diffusive transport and chemical kinetics models. Accuracy and performance of the method are examined on several test problems. The sparse grids produced by the WAMR method exhibit an impressive compression of the solution, reducing the number of collocation points used by factors of many orders of magnitude when compared to uniform grids of equivalent resolution.

Adaptive Wavelet Method for Incompressible Flows in Complex Domains

An adaptive wavelet-based method provides an alternative means to refine grids according to local demands of the physical solution. One of the prominent challenges of such a method is the application to problems defined on complex domains. In the case of incompressible flow, the application to problems with complicated domains is made possible by the use of the Navier-Stokes-Brinkman equations. These equations take into account solid obstacles by adding a penalized velocity term in the momentum equation. An adaptive wavelet collocation method, based on interpolating wavelets, is first applied to a benchmark problem defined on a simple domain to demonstrate the accuracy and efficiency of the method. Then the penalty technique is used to simulate flows over obstacles

A Comparison of Artificial Viscosity, Limiters, and Filters, for High Order Discontinuous Galerkin Solutions in Nonlinear Settings

Nonlinear systems of equations demonstrate complicated regularity features that are often obfuscated by overly diffuse numerical methods. Using a discontinuous Galerkin finite element method, we study a nonlinear system of advection–diffusion–reaction equations and aspects of its regularity. For numerical regularization, we present a family of solutions consisting of: (1) a sharp, computationally efficient slope limiter, known as the BDS limiter, (2) a standard spectral filter, and (3) a novel artificial diffusion algorithm with a solution-dependent entropy sensor. We analyze these three numerical regularization methods on a classical test in order to test the strengths and weaknesses of each, and then benchmark the methods against a large application model.

Dynamic p-enrichment schemes for multicomponent reactive flows

We present a family of p-enrichment schemes. These schemes may be separated into two basic classes: the first, called fixed tolerance schemes, rely on setting global scalar tolerances on the local regularity of the solution, and the second, called dioristic schemes, rely on time-evolving bounds on the local variation in the solution. Each class of p-enrichment scheme is further divided into two basic types. The first type (the Type I schemes) enrich along lines of maximal variation, striving to enhance stable solutions in ‘‘areas of highest interest.’’ The second type (the Type II schemes) enrich along lines of maximal regularity in order to maximize the stability of the enrichment process. Each of these schemes are tested on three model systems. The first is an academic exact system where basic analysis is easily performed. Then we discuss a pair of application model problems arising in coastal hydrology. The first being a contaminant transport model, which addresses a declinature problem for a contaminant plume with respect to a bay inlet setting. And the second, a multicomponent chemically reactive flow model of estuary eutrophication arising in the Gulf of Mexico.

The Application of an Adaptive Wavelet Method to the 3-D Natural-Convection Flow in a Differentially Heated Cavity

In this work, we describe a dynamically adaptive wavelet method for solving the natural-convection flow a differentially heated cavity in three spatial dimensions. The adaptive method takes advantage of an interpolating wavelet for the adaptive approximation in the design of a simple refinement strategy that reflects the local demand of the physical solution. The derivative approximation is computed via consistent finite-difference approximation on an adaptive grid. To demonstrate the versatility of the algorithm, we simulate the 3-D differentially heated cavity with various values of Rayleigh number. The results are compared with those obtained by other computational approaches.Copyright

Application of an Adaptive Wavelet Method to Natural-Convection Flow in a Differentially Heated Cavity

We describe an adaptive wavelet method based on interpolating wavelets applied to the solution of a natural-convection flow. The adaptive wavelet method, owing to the approximation capabilities and spatial localization of wavelet functions, enables the solution of problems with local grid resolution consistent with the local demand of the physical problem. The adaptive method is applied to simulate the flow in a differentially heated square cavity at large Rayleigh numbers. Numerical results, whenever possible, are compared with those previously published.Copyright

Artificial boundary layers in discontinuous Galerkin solutions to shallow water equations in channels

In this work, we consider the application of Discontinuous Galerkin (DG) solutions to open channel flow problems, governed by two-dimensional shallow water equations (SWE), with solid curved wall boundaries on which the no-normal flow boundary conditions are prescribed. A commonly used approach consists of straightforwardly imposing the no-normal flow condition on the linear approximation of curved walls. Numerical solutions indicate clearly that this approach could lead to unfavorable results and that a proper treatment of the no-normal flow condition on curved walls is crucial for an accurate DG solution to the SWE. In the test case used, errors introduced through the commonly used approach result in artificial boundary layers of one-grid-size thickness in the velocity field and a corresponding over-prediction of the surface elevation in the upstream direction. These significant inaccuracies, which render the coarse mesh solution unreliable, appear in all DG schemes employed including those using linear, quadratic, and cubic DG polynomials. The issue can be alleviated by either using an approach accounting for errors introduced by the geometric approximation or an approach that accurately represents the geometry.

An Adaptive Wavelet Method for the Incompressible Navier-Stokes Equations in Complex Domains

An adaptive wavelet-based method provides an alternative means to refine grids according to local demands of the physical solution. One of the prominent challenges of such a method is the application to problems defined on complex domains. In the case of incompressible flow, the application to problems with complicated domains is made possible by the use of the Navier-Stokes/Brinkman equations. These equations take into account solid obstacles by adding a penalized velocity term in the momentum equation. In this study, an adaptive wavelet collocation method, based on interpolating wavelets, is first applied to a benchmark problem defined on a simple domain to demonstrate the accuracy and efficiency of the method. Then the penalty technique is used to simulate flows over obstacles. The numerical results are compared with those obtained by other computational approaches as well as with experiments.© 2004 ASME