Dual-Wind Discontinuous Galerkin Methods for Stationary Hamilton-Jacobi Equations and Regularized Hamilton-Jacobi Equations

doi:10.1007/s42967-021-00130-9

Communications on Applied Mathematics and Computation ›› 2022, Vol. 4 ›› Issue (2): 563-596.doi: 10.1007/s42967-021-00130-9

• ORIGINAL PAPERS • 上一篇下一篇

Dual-Wind Discontinuous Galerkin Methods for Stationary Hamilton-Jacobi Equations and Regularized Hamilton-Jacobi Equations

Xiaobing Feng¹, Thomas Lewis², Aaron Rapp³

1 Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA;
2 Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, NC 27402, USA;
3 Department of Mathematical Sciences, University of the Virgin Islands, Kingshill, USVI 00850-9781, US Virgin Islands

收稿日期:2020-09-03 修回日期:2021-01-30 出版日期:2022-06-20 发布日期:2022-04-29
通讯作者: Xiaobing Feng, Thomas Lewis, Aaron Rapp E-mail:xfeng@utk.edu;tllewis3@uncg.edu;aaron.rapp@uvi.edu

Dual-Wind Discontinuous Galerkin Methods for Stationary Hamilton-Jacobi Equations and Regularized Hamilton-Jacobi Equations

Xiaobing Feng¹, Thomas Lewis², Aaron Rapp³

1 Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA;
2 Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, NC 27402, USA;
3 Department of Mathematical Sciences, University of the Virgin Islands, Kingshill, USVI 00850-9781, US Virgin Islands

Received:2020-09-03 Revised:2021-01-30 Online:2022-06-20 Published:2022-04-29
Contact: Xiaobing Feng, Thomas Lewis, Aaron Rapp E-mail:xfeng@utk.edu;tllewis3@uncg.edu;aaron.rapp@uvi.edu

摘要/Abstract

摘要： This paper develops and analyzes a new family of dual-wind discontinuous Galerkin (DG) methods for stationary Hamilton-Jacobi equations and their vanishing viscosity regularizations. The new DG methods are designed using the DG fnite element discrete calculus framework of[17] that defnes discrete diferential operators to replace continuous differential operators when discretizing a partial diferential equation (PDE). The proposed methods, which are non-monotone, utilize a dual-winding methodology and a new skewsymmetric DG derivative operator that, when combined, eliminate the need for choosing indeterminable penalty constants. The relationship between these new methods and the local DG methods proposed in[38] for Hamilton-Jacobi equations as well as the generalized-monotone fnite diference methods proposed in[13] and corresponding DG methods proposed in[12] for fully nonlinear second order PDEs is also examined. Admissibility and stability are established for the proposed dual-wind DG methods. The stability results are shown to hold independent of the scaling of the stabilizer allowing for choices that go beyond the Godunov barrier for monotone schemes. Numerical experiments are provided to gauge the performance of the new methods.

关键词: Hamilton-Jacobi equations, Discontinuous Galerkin methods, Vanishing viscosity method

Abstract: This paper develops and analyzes a new family of dual-wind discontinuous Galerkin (DG) methods for stationary Hamilton-Jacobi equations and their vanishing viscosity regularizations. The new DG methods are designed using the DG fnite element discrete calculus framework of[17] that defnes discrete diferential operators to replace continuous differential operators when discretizing a partial diferential equation (PDE). The proposed methods, which are non-monotone, utilize a dual-winding methodology and a new skewsymmetric DG derivative operator that, when combined, eliminate the need for choosing indeterminable penalty constants. The relationship between these new methods and the local DG methods proposed in[38] for Hamilton-Jacobi equations as well as the generalized-monotone fnite diference methods proposed in[13] and corresponding DG methods proposed in[12] for fully nonlinear second order PDEs is also examined. Admissibility and stability are established for the proposed dual-wind DG methods. The stability results are shown to hold independent of the scaling of the stabilizer allowing for choices that go beyond the Godunov barrier for monotone schemes. Numerical experiments are provided to gauge the performance of the new methods.

Key words: Hamilton-Jacobi equations, Discontinuous Galerkin methods, Vanishing viscosity method

中图分类号:

65N30

Xiaobing Feng, Thomas Lewis, Aaron Rapp. Dual-Wind Discontinuous Galerkin Methods for Stationary Hamilton-Jacobi Equations and Regularized Hamilton-Jacobi Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 563-596.

参考文献

1.Ayuso, B., Marini, L.D.:Discontinuous Galerkin methods for advection-difusion-reaction problems.SIAM J.Numer.Anal.47, 1391-1420 (2009)
2.Barles, G., Souganidis, P.E.:Convergence of approximation schemes for fully nonlinear second order equations.Asympt.Anal.4, 271-283 (1991)
3.Barth, T.J., Sethian, J.A.:Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains.J.Comp.Phys.145, 1-40 (1998)
4.Brenner, S.C., Scott, L.R.:The Mathematical Theory of Finite Element Methods, 3rd edn.Springer, Berlin (2008)
5.Brooks, A.N., Huges, T.J.:Streamline upwind/Petrov-Galerkin formulations for convection dominated fows with particular emphasis on the incompressible Navier-Stokes equations.Comput.Methods Appl.Mech.Engrg.32, 199-259 (1982)
6.Ciarlet, P.G.:The Finite Element Method for Elliptic Problems.North-Holland, Amsterdam (1978)
7.Cockburn, B., Dong, B.:An analysis of the minimal dissipation local discontinuous Galerkin method for convection-difusion problems.J.Sci.Comput.32, 233-262 (2007)
8.Crandall, M.G., Lions, P.-L.:Viscosity solutions of Hamilton-Jacobi equations.Trans.Am.Math.Soc.277, 1-42 (1983)
9.Crandall, M.G., Lions, P.-L.:Two approximations of solutions of Hamilton-Jacobi equations.Math.Comp.43, 1-19 (1984)
10.Crandall, M.G., Ishii, H., Lions, P.-L.:User's guide to viscosity solutions of second order partial differential equations.Bull.Am.Math.Soc.27, 1-67 (1992)
11.Feng, X., Lewis, T.:Local discontinuous Galerkin methods for one-dimensional second order fully nonlinear elliptic and parabolic equations.J.Sci.Comput.59, 129-157 (2014)
12.Feng, X., Lewis, T.:Nonstandard local discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic equations in high dimensions.J.Sci.Comput.77, 1534-1565 (2018)
13.Feng, X., Lewis, T.:A narrow-stencil fnite diference method for approximating viscosity solutions of Hamilton-Jacobi-Bellman equations.SIAM J.Numer.Anal.59(2), 886-924 (2021)
14.Feng, X., Neilan, M.:The vanishing moment method for fully nonlinear second order partial diferential equations:formulation, theory, and numerical analysis, arXiv:1109.1183[math.NA]
15.Feng, X., Kao, C., Lewis, T.:Convergent fnite diference methods for one-dimensional fully nonlinear second order partial diferential equations.J.Comput.Appl.Math.254, 81-98 (2013)
16.Feng, W., Lewis, T.L., Wise, S.M.:Discontinuous Galerkin derivative operators with applications to second-order elliptic problems and stability.Math.Meth.Appl.Sci.38, 5160-5182 (2015)
17.Feng, X., Lewis, T., Neilan, M.:Discontinuous Galerkin fnite element diferential calculus and applications to numerical solutions of linear and nonlinear partial diferential equations.J.Comput.Appl.Math.299, 68-91 (2016)
18.Hu, C., Shu, C.-W.:A discontinuous Galerkin fnite element method for Hamilton-Jacobi equations.SIAM J.Sci.Comp.21, 666-690 (1999)
19.Huang, S.-C., Wang, S., Teo, K.L.:Solving Hamilton-Jacobi-Bellman equations by a modifed method of characteristics.Nonlinear Anal.40, 279-293 (2000)
20.Hughes, T.J.R., Franca, L.P., Hulbert, G.M.:A new fnite element formulation for computational fuid dynamics:VIII.The Galerkin/least-squares method for advective-difusive equations.Comput.Methods Appl.Mech.Engrg.73, 173-189 (1989)
21.Kao, C.Y., Osher, S., Qian, J.:Lax-Friedrichs sweeping scheme for static Hamilton-Jacobi equations.J.Comput.Phys.196, 367-391 (2004)
22.Katzourakis, N.:An Introduction to Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L∞, SpringerBriefs in Mathematics.Springer, Cham (2015)
23.Lewis, T., Neilan, M.:Convergence analysis of a symmetric dual-wind discontinuous Galerkin method.J.Sci.Comput.59, 602-625 (2014)
24.Lewis, T., Rapp, A., Zhang, Y.:Convergence analysis of symmetric dual-wind discontinuous Galerkin approximation methods for the obstacle problem.J.Math.Anal.Appl.(2020).https://doi.org/10.1016/j.jmaa.2020.123840
25.Li, F., Shu, C.-W.:Reinterpretation and simplifed implementation of a discontinuous Galerkin method for Hamilton-Jacobi equations.Appl.Math.Lett.18, 1204-1209 (2005)
26.Li, F., Yakovlev, S.:A central discontinuous Galerkin method for Hamilton-Jacobi equations.J.Sci.Comput.45, 404-428 (2010)
27.Lions, P.-L.:Generalized Solutions of Hamilton-Jacobi Equations.Pitman, London (1982)
28.Osher, S., Fedkiw, R.:Level Set Methods and Dynamic Implicit Surfaces.Applied Mathematical Sciences, vol.153.Springer-Verlag, New York (2003)
29.Osher, S., Shu, C.-W.:High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations.SIAM J.Numer.Anal.28, 907-922 (1991)
30.Rudin, W.:Functional Analysis.McGraw-Hill, New York (1991)
31.Sethian, J.A.:Level Set Methods and Fast Marching Methods:Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, volume 3 of Cambridge Monographs on Applied and Computational Mathematics.2nd ed.Cambridge University Press, Cambridge (1999)
32.Sethian, J.A., Vladimirsky, A.:Ordered upwind methods for static Hamilton-Jacobi equations:theory and algorithms.SIAM J.Numer.Anal.41, 325-363 (2003)
33.Shu, C.-W.:High order numerical methods for time dependent Hamilton-Jacobi equations.In:Mathematics and Computation in Imaging Science and Information Processing, Vol.11 of Lect.Notes Ser.Inst.Math.Sci.Natl.Univ.Singap., World Sci.Publ., Hackensack, NJ, 2007, pp.47-91 (2007)
34.Tadmor, E.:Approximate solutions of nonlinear conservation laws and related equations.Available at https://home.cscamm.umd.edu/people/faculty/tadmor/pub/PDEs/Tadmor.Lax+Nirenberg-97.pdf (1997)
35.Tadmor, E.:The numerical viscosity of entropy stable schemes for systems of conservation laws, I.Math.Comp.49, 91-103 (1987)
36.Tsai, Y.R., Cheng, L.-T., Osher, S., Zhao, H.K.:Fast sweeping algorithms for a class of HamiltonJacobi equations.SIAM J.Numer.Anal.41, 673-694 (2003)
37.Van der Vegt, J.J.W., Xia, Y., Xu, Y.:Positivity preserving limiters for time-implicit higher order accurate discontinuous Galerkin discretizations.SIAM J.Sci.Comput.41, 2037-2063 (2019)
38.Yan, J., Osher, S.:Direct discontinuous local Galerkin methods for Hamilton-Jacobi equations.J.Comp.Phys.230, 232-244 (2011)

[1]	Ohannes A. Karakashian, Michael M. Wise. A Posteriori Error Estimates for Finite Element Methods for Systems of Nonlinear, Dispersive Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 823-854.
[2]	Poorvi Shukla, J. J. W. van der Vegt. A Space-Time Interior Penalty Discontinuous Galerkin Method for the Wave Equation[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 904-944.
[3]	Jiawei Sun, Shusen Xie, Yulong Xing. Local Discontinuous Galerkin Methods for the abcd Nonlinear Boussinesq System[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 381-416.
[4]	Gang Chen, Bernardo Cockburn, John R. Singler, Yangwen Zhang. Superconvergent Interpolatory HDG Methods for Reaction Difusion Equations II: HHO-Inspired Methods[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 477-499.
[5]	Xue Hong, Yinhua Xia. Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Methods for KdV Type Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 530-562.
[6]	Yuqing Miao, Jue Yan, Xinghui Zhong. Superconvergence Study of the Direct Discontinuous Galerkin Method and Its Variations for Difusion Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 180-204.
[7]	Liyao Lyu, Zheng Chen. Local Discontinuous Galerkin Methods with Novel Basis for Fractional Difusion Equations with Non-smooth Solutions[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 227-249.
[8]	Jie Du, Eric Chung, Yang Yang. Maximum-Principle-Preserving Local Discontinuous Galerkin Methods for Allen-Cahn Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 353-379.
[9]	Zheng Sun, Chi-Wang Shu. Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity[J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 671-700.
[10]	Oleksii Beznosov, Daniel Appel?. Hermite-Discontinuous Galerkin Overset Grid Methods for the Scalar Wave Equation[J]. Communications on Applied Mathematics and Computation, 2021, 3(3): 391-418.
[11]	Rathan Samala, Biswarup Biswas. Arc Length-Based WENO Scheme for Hamilton-Jacobi Equations[J]. Communications on Applied Mathematics and Computation, 2021, 3(3): 481-496.
[12]	Yong Liu, Chi-Wang Shu, Mengping Zhang. Superconvergence of Energy-Conserving Discontinuous Galerkin Methods for Linear Hyperbolic Equations[J]. Communications on Applied Mathematics and Computation, 2019, 1(1): 101-116.

Dual-Wind Discontinuous Galerkin Methods for Stationary Hamilton-Jacobi Equations and Regularized Hamilton-Jacobi Equations

Dual-Wind Discontinuous Galerkin Methods for Stationary Hamilton-Jacobi Equations and Regularized Hamilton-Jacobi Equations

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 12

编辑推荐

Metrics

本文评价