TY - JOUR
T1 - An arbitrary-order Runge–Kutta discontinuous Galerkin approach to reinitialization for banded conservative level sets
AU - Jibben, Z.
AU - Herrmann, Marcus
N1 - Funding Information:
This work was supported by the National Science Foundation grant CBET-1054272 and the Department of Energy at Los Alamos National Laboratory under contract DE-AC52-06NA25396 . Calculations were performed using the ASU Advanced Computing Center Saguaro and Ocotillo clusters.
Publisher Copyright:
© 2017 Elsevier Inc.
PY - 2017/11/15
Y1 - 2017/11/15
N2 - We present a Runge–Kutta discontinuous Galerkin method for solving conservative reinitialization in the context of the conservative level set method [18,19]. 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 [9] 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 [9]. 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.
AB - We present a Runge–Kutta discontinuous Galerkin method for solving conservative reinitialization in the context of the conservative level set method [18,19]. 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 [9] 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 [9]. 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.
KW - Arbitrary order
KW - Conservative level set
KW - Discontinuous Galerkin
KW - Multiphase flow
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U2 - 10.1016/j.jcp.2017.08.035
DO - 10.1016/j.jcp.2017.08.035
M3 - Article
AN - SCOPUS:85028561312
VL - 349
SP - 453
EP - 473
JO - Journal of Computational Physics
JF - Journal of Computational Physics
SN - 0021-9991
ER -