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Title: An arbitrary-order Runge–Kutta discontinuous Galerkin approach to reinitialization for banded conservative level sets

Here, we present a Runge-Kutta discontinuous Galerkin method for solving conservative reinitialization in the context of the conservative level set method. This represents an extension of the method recently proposed by Owkes and Desjardins [21], by solving the level set equations on the refined level set grid and projecting all spatially-dependent variables into the full basis used by the discontinuous Galerkin discretization. By doing so, we achieve the full k+1 order convergence rate in the L1 norm of the level set field predicted for RKDG methods given kth degree basis functions when the level set profile thickness is held constant with grid refinement. Shape and volume errors for the 0.5-contour of the level set, on the other hand, are found to converge between first and second order. We show a variety of test results, including the method of manufactured solutions, reinitialization of a circle and sphere, Zalesak's disk, and deforming columns and spheres, all showing substantial improvements over the high-order finite difference traditional level set method studied for example by Herrmann. We also demonstrate the need for kth order accurate normal vectors, as lower order normals are found to degrade the convergence rate of the method.
Authors:
ORCiD logo [1] ;  [2]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  2. Arizona State Univ., Tempe, AZ (United States). Dept. of Mechanical and Aerospace Engineering
Publication Date:
Report Number(s):
LA-UR-17-27700
Journal ID: ISSN 0021-9991; TRN: US1800405
Grant/Contract Number:
AC52-06NA25396; CBET-1054272
Type:
Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 349; Journal Issue: C; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Research Org:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org:
USDOE; National Science Foundation (NSF)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Discontinuous Galerkin; Arbitrary order; Conservative level set; Multiphase flow
OSTI Identifier:
1412902

Jibben, Zechariah Joel, and Herrmann, Marcus. An arbitrary-order Runge–Kutta discontinuous Galerkin approach to reinitialization for banded conservative level sets. United States: N. p., Web. doi:10.1016/j.jcp.2017.08.035.
Jibben, Zechariah Joel, & Herrmann, Marcus. An arbitrary-order Runge–Kutta discontinuous Galerkin approach to reinitialization for banded conservative level sets. United States. doi:10.1016/j.jcp.2017.08.035.
Jibben, Zechariah Joel, and Herrmann, Marcus. 2017. "An arbitrary-order Runge–Kutta discontinuous Galerkin approach to reinitialization for banded conservative level sets". United States. doi:10.1016/j.jcp.2017.08.035. https://www.osti.gov/servlets/purl/1412902.
@article{osti_1412902,
title = {An arbitrary-order Runge–Kutta discontinuous Galerkin approach to reinitialization for banded conservative level sets},
author = {Jibben, Zechariah Joel and Herrmann, Marcus},
abstractNote = {Here, we present a Runge-Kutta discontinuous Galerkin method for solving conservative reinitialization in the context of the conservative level set method. This represents an extension of the method recently proposed by Owkes and Desjardins [21], by solving the level set equations on the refined level set grid and projecting all spatially-dependent variables into the full basis used by the discontinuous Galerkin discretization. By doing so, we achieve the full k+1 order convergence rate in the L1 norm of the level set field predicted for RKDG methods given kth degree basis functions when the level set profile thickness is held constant with grid refinement. Shape and volume errors for the 0.5-contour of the level set, on the other hand, are found to converge between first and second order. We show a variety of test results, including the method of manufactured solutions, reinitialization of a circle and sphere, Zalesak's disk, and deforming columns and spheres, all showing substantial improvements over the high-order finite difference traditional level set method studied for example by Herrmann. We also demonstrate the need for kth order accurate normal vectors, as lower order normals are found to degrade the convergence rate of the method.},
doi = {10.1016/j.jcp.2017.08.035},
journal = {Journal of Computational Physics},
number = C,
volume = 349,
place = {United States},
year = {2017},
month = {8}
}