A Wavelet-Free Approach for Multiresolution-Based Grid Adaptation for Conservation Laws

doi:10.1007/s42967-020-00101-6

Abstract

Abstract: In recent years the concept of multiresolution-based adaptive discontinuous Galerkin (DG) schemes for hyperbolic conservation laws has been developed. The key idea is to perform a multiresolution analysis of the DG solution using multiwavelets defned on a hierarchy of nested grids. Typically this concept is applied to dyadic grid hierarchies where the explicit construction of the multiwavelets has to be performed only for one reference element. For non-uniform grid hierarchies multiwavelets have to be constructed for each element and, thus, becomes extremely expensive. To overcome this problem a multiresolution analysis is developed that avoids the explicit construction of multiwavelets.

Key words: Discontinuous Galerkin, Grid adaptivity, Multiwavelets, Multiresolution analysis, Conservation laws

CLC Number:

35L65
65M60

Nils Gerhard, Siegfried Müller, Aleksey Sikstel. A Wavelet-Free Approach for Multiresolution-Based Grid Adaptation for Conservation Laws[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 108-142.

TrendMD

References

1. Adjerid, S., Devine, K., Flaherty, J., Krivodonova, L.:A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Comput. Methods Appl. Mech. Eng. 191, 1097-1112 (2002)
2. Alpert, B.:A class of bases in L² for the sparse representation of integral operators. SIAM J. Math. Anal. 24, 246-262 (1993)
3. Alpert, B., Beylkin, G., Gines, D., Vozovoi, L.:Adaptive solution of partial diferential equation in multiwavelet bases. J. Comput. Phys. 182, 149-190 (2002)
4. Amat, S., Moncayo, M.:Non-uniform multiresolution analysis with supercompact multiwavelets. J. Comput. Appl. Math. 235, 334-340 (2010)
5. Archibald, R., Fann, G., Shelton, W.:Adaptive discontinuous Galerkin methods in multiwavelets bases. Appl. Numer. Math. 61, 879-890 (2011)
6. Arnold, D., Brezzi, F., Cockburn, B., Marini, L.:Unifed analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749-1779 (2002)
7. Arvanitis, C., Makridakis, C., Sfakianakis, N.:Entropy conservative schemes and adaptive mesh selection for hyperbolic conservation laws. J. Hyperbol. Difer. Eq. 7(3), 383-404 (2010)
8. Atkins, H., Pampell, A.:Robust and accurate shock capturing method for high-order discontinuous Galerkin methods. In:20th AIAA Computational Fluid Dynamics Conference, Fluid Dynamics and Co-located Conferences. Am. Inst. Aeronaut. Astron. (2011). https://doi.org/10.2514/6.2011-3058
9. Barter, G., Darmofal, D.:Shock capturing with higher-order, PDE-based artifcial viscosity. In:18th AIAA Computational Fluid Dynamics Conference, Fluid Dynamics and Co-located Conferences. Am. Inst. Aeronaut. Astron. (2007). https://doi.org/10.2514/6.2007-3823
10. Bassi, F., Crivellini, A., Rebay, S., Savini, M.:Discontinuous Galerkin solution of the Reynolds-averaged Navier-Stokes and k-ω turbulence model equations. Comput. Fluids 34, 507-540 (2005)
11. Bassi, F., Rebay, S.:A high order discontinuous Galerkin method for compressible turbulent fows. In:Cockburn, B., Karniadakis, G., Shu, C.-W. (eds.) Discontinuous Galerkin Methods, Volume 11 of Lecture Notes in Comput. Science and Engineering, Springer, pp. 77-88 (2000)
12. Bassi, F., Rebay, S., Mariotti, G., Pedinotti, S., Savini, M.:A high-order accurate discontinuous fnite element method for inviscid and viscous turbomachinery fows. In:Proceedings of 2nd European Conference on Turbomachinery, Fluid Dynamics and Thermodynamics, Antwerpen, Belgium, pp. 99-108 (1997)
13. Beam, R.M., Warming, R.:Multiresolution analysis and supercompact multiwavelets. SIAM J. Sci. Comput. 22(4), 1238-1268 (2000)
14. Bey, K., Oden, J.:hp-version discontinuous Galerkin methods for hyperbolic conservation laws. Comput. Methods Appl. Mech. Eng. 133(3/4), 259-286 (1996)
15. Beylkin, G., Keiser, J.:On the adaptive numerical solution of nonlinear partial diferential equations in wavelet bases. J. Comput. Phys. 132(2), 233-259 (1997). http://www.sciencedirect.com/science/article/pii/S002199919695562X
16. Bramkamp, F., Lamby, P., Müller, S.:An adaptive multiscale fnite volume solver for unsteady and steady state fow computations. J. Comput. Phys. 197(2), 460-490 (2004)
17. Calle, J., Devloo, P., Gomes, S.:Wavelets and adaptive grids for the discontinuous Galerkin method. Numer. Algorithms 39(1/2/3), 143-154 (2005)
18. Carnicer, J., Dahmen, W., Peña, J.M.:Local decomposition of refnable spaces and wavelets. Appl. Comp. Harm. Anal. 3, 127-153 (1996)
19. Caviedes-Voullième, D., Kesserwani, G.:Benchmarking a multiresolution discontinuous Galerkin shallow water model:implications for computational hydraulics. Adv. Water Resour. 86, Part A, 14-31 (2015). http://www.sciencedirect.com/science/article/pii/S0309170815002237
20. Cockburn, B., Hou, S., Shu, C.-W.:The Runge-Kutta local projection discontinuous Galerkin fnite element method for conservation laws. IV:the multidimensional case. Math. Comp. 54(190), 545-581 (1990)
21. Cockburn, B., Lin, S.Y., Shu, C.-W.:TVB Runge-Kutta local projection discontinuous Galerkin fnite element method for conservation laws Ⅲ:one-dimensional systems. J. Comput. Phys. 84, 90-113 (1989)
22. Cockburn, B., Shu, C.-W.:TVB Runge-Kutta local projection discontinuous Galerkin fnite element method for conservation laws Ⅱ:general framework. Math. Comp. 52(186), 411-435 (1989)
23. Cockburn, B., Shu, C.-W.:The Runge-Kutta discontinuous Galerkin method for conservation laws V:multidimensional systems. J. Comput. Phys. 141, 199-244 (1998)
24. Cohen, A., Dahmen, W., DeVore, R.:Sparse evaluation of compositions of functions using multiscale expansions. SIAM J. Math. Anal. 35(2), 279-303 (2003). https://doi.org/10.1137/S0036141002412070
25. Cohen, A., Kaber, S., Müller, S., Postel, M.:Fully adaptive multiresolution fnite volume schemes for conservation laws. Math. Comput. 72(241), 183-225 (2003)
26. Coquel, F., Maday, Y., Müller, S., Postel, M., Tran, Q.:New trends in multiresolution and adaptive methods for convection-dominated problems. ESAIM Proc. 29, 1-7 (2009)
27. Dahmen, W.:Wavelet and multiscale methods for operator equations. Acta Numer. 6, 55-228 (1997)
28. Daru, V., Tenaud, C.:Evaluation of TVD high resolution schemes for unsteady viscous shocked fows. Comput. Fluids 30(1), 89-113 (2000). http://www.sciencedirect.com/science/article/pii/S0045793000000062
29. Daubechies, I.:Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, Philadelphia (1992)
30. Dedner, A., Makridakis, C., Ohlberger, M.:Error control for a class of Runge-Kutta discontinuous Galerkin methods for nonlinear conservation laws. SIAM J. Numer. Anal. 45, 514-538 (2007)
31. DeVore, R., Lorentz, C.:Constructive Approximation. Springer, New York (1993)
32. Di Pietro, D., Ern, A.:Mathematical Aspects of Discontinuous Galerkin Schemes. Mathèmatiques et Applications. Springer, New York (2012)
33. Domingues, M., Gomes, S., Roussel, O., Schneider, K.:Space-time adaptive multiresolution methods for hyperbolic conservation laws:applications to compressible Euler equations. Appl. Numer. Math. 59(9), 2303-2321 (2009). http://www.sciencedirect.com/science/article/pii/S0168927408002195
34. Gerhard, N.:An adaptive multiresolution discontinuous Galerkin scheme for conservation laws. PhD dissertation, RWTH Aachen (2017). https://doi.org/10.18154/RWTH-2017-06869
35. Gerhard, N., Caviedes-Voullième, D., Müller, S., Kesserwani, G.:Multiwavelet-based grid adaptation with discontinuous Galerkin schemes for shallow water equations. J. Comput. Phys. 301, 265-288 (2015)
36. Gerhard, N., Iacono, F., May, G., Müller, S., Schäfer, R.:A high-order discontinuous Galerkin discretization with multiwavelet-based grid adaptation for compressible fows. J. Sci. Comput. 62(1), 25-52 (2015)
37. Gerhard, N., Müller, S.:Adaptive multiresolution discontinuous Galerkin schemes for conservation laws:multi-dimensional case. Computat. Appl. Math. 35(2), 312-349 (2016)
38. Giesselmann, J., Makridakis, C., Pryer, T.:A posteriori analysis of discontinuous Galerkin schemes for systems of hyperbolic conservation laws. SIAM J. Numer. Anal. 53(3), 1280-1303 (2015)
39. Gottlieb, S., Shu, C.-W., Tadmor, E.:Strong stability preserving high-order time discretization methods. SIAM Rev. 43(1), 89-112 (2001)
40. Gottschlich-Müller, B., Müller, S.:Adaptive fnite volume schemes for conservation laws based on local multiresolution techniques. In:Fey, M., Jeltsch, R. (eds.) Hyperbolic Problems:Theory, Numerics, Applications. Zürich, Switzerland, February 1998. Vol. I, pp. 385-394. Birkhäuser, Basel (1999)
41. Harten, A.:Discrete multi-resolution analysis and generalized wavelets. Appl. Numer. Math. 12, 153- 192 (1993)
42. Harten, A.:Adaptive multiresolution schemes for shock computations. J. Comput. Phys. 115, 319-338 (1994)
43. Harten, A.:Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Commun. Pure Appl. Math. 48, 1305-1342 (1995)
44. Harten, A.:Multiresolution representation of data:a general framework. SIAM J. Numer. Anal. 33(3), 1205-1256 (1996)
45. Hartmann, R., Houston, P.:Adaptive discontinuous Galerkin fnite element methods for nonlinear hyperbolic conservation laws. SIAM J. Sci. Comput. 24, 979-1004 (2002)
46. Hartmann, R., Houston, P.:Adaptive discontinuous Galerkin fnite element methods for the compressible Euler equations. J. Comput. Phys. 183, 508-532 (2002)
47. Houston, P., Senior, B., Süli, E.:hp-discontinuous Galerkin fnite element methods for hyperbolic problems:error analysis and adaptivity. Int. J. Numer. Methods Fluids 40(1/2), 153-169 (2002)
48. Hovhannisyan, N., Müller, S., Schäfer, R.:Adaptive multiresolution discontinuous Galerkin schemes for conservation laws. Math. Comput. 83, 113-151 (2014). https://doi.org/10.1090/S0025-5718-2013-02732-9
49. Hu, G.:An adaptive fnite volume method for 2d steady Euler equations with WENO reconstruction. J. Comput. Phys. 252, 591-605 (2013). http://www.sciencedirect.com/science/article/pii/S0021999113004786
50. Iacono, F., May, G., Müller, S., Schäfer, R.:A high-order discontinuous Galerkin discretization with multiwavelet-based grid adaptation for compressible fows. J. Sci. Comput. 62, 25-52 (2014). https://doi.org/10.1007/s10915-014-9846-9
51. Jaust, A., Schütz, J., Woopen, M.:An HDG Method for Unsteady Compressible Flows, Spectral and High Order Methods for Partial Diferential Equations ICOSAHOM 2014, pp. 267-274. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-19800-2_23
52. Kaber, S., Postel, M.:Finite volume schemes on triangles coupled with multiresolution analysis. Comptes Rendus de l'Académie des Sciences-Series I-Mathematics 328(9), 817-822 (1999). http://www.sciencedirect.com/science/article/pii/S076444429980278X
53. Kaibara, M., Gomes, S.:A fully adaptive multiresolution scheme for shock computations. In:Godunov Methods. Theory and Applications. International conference, Oxford, GB, October 1999, pp. 497-503. New York, NY:Kluwer Academic/Plenum Publishers (2001)
54. Lagha, M., Zhong, X., Eldredge, J., Kim, J.:A hybrid WENO scheme for simulation of shock wave-boundary layer interaction. In:47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Aerospace Sciences Meetings. American Institute of Aeronautics and Astronautics (2009). https://doi.org/10.2514/6.2009-1136
55. Mallat, S.:Multiresolution approximations and wavelet orthonormal bases for L²(R). Trans. Am. Math. Soc. 315(1), 69-87 (1989)
56. Müller, S.:Adaptive multiscale schemes for conservation laws. Lecture Notes in Computational Science and Engineering, vol. 27. Springer, Berlin (2003)
57. Müller, S.:Multiresolution schemes for conservation laws. In:DeVore, R., Kunoth, A. (eds.) Multiscale, Nonlinear and Adaptive Approximation, pp. 379-408, Springer, Berlin (2009)
58. Pan, L., Xu, K., Li, Q., Li, J.:An efcient and accurate two-stage fourth-order gas-kinetic scheme for the Euler and Navier-Stokes equations. J. Comput. Phys. 326, 197-221 (2016). http://www.sciencedirect.com/science/article/pii/S0021999116304119
59. Persson, P.O., Peraire, J.:Sub-cell shock capturing for discontinuous Galerkin methods. In:44th AIAA Aerospace Sciences Meeting and Exhibit, Aerospace Sciences Meetings. American Institute of Aeronautics and Astronautics (2006). http://dx.doi.org/10.2514/6.2006-112
60. Pongsanguansin, T., Mekchay, K., Maleewong, M.:Adaptive TVD-RK discontinuous Galerkin algorithms for shallow water equations. Int. J. Math. Comput. Simul. 6(2), 257-273 (2012)
61. Remacle, J.F., Flaherty, J., Shephard, M.:An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible fow problems. SIAM Rev. 45(1), 53-72 (2003)
62. Remacle, J.F., Frazão, S., Li, X., Shephard, M.:An adaptive discretization of shallow-water equations based on discontinuous Galerkin methods. Int. J. Numer. Meth. Fl. 52(8), 903-923 (2006). https://doi.org/10.1002/fd.1204
63. Roussel, O., Schneider, K.:A fully adaptive multiresolution scheme for 3D reaction-difusion equations. In:Finite Volumes for Complex Applications Ⅲ. Problems and Perspectives. Papers from the 3rd Symposium of Finite Volumes for Complex Applications, Porquerolles, France, June 24-28, 2002, pp. 833-840. Hermes Penton Science, London (2002)
64. Roussel, O., Schneider, K., Tsigulin, A., Bockhorn, H.:A conservative fully adaptive multiresolution algorithm for parabolic PDEs. J. Comput. Phys. 188(2), 493-523 (2003). http://www.sciencedirect.com/science/article/pii/S002199910300189X
65. Schäfer, R.:Adaptive multiresolution discontinuous Galerkin schemes for conservation laws. Ph.D. thesis, RWTH Aachen University (2011)
66. Shelton, A.:A multi-resolution discontinuous Galerkin method for unsteady compressible fows. Ph.D. thesis, Georgia Institute of Technology (2008)
67. Shu, C.-W., Osher, S.:Efcient implementation of essentially non-oscillatory shock-capturing schemes Ⅱ. J. Comput. Phys. 83, 32-78 (1989)
68. Sjögreen, B., Yee, H.C.:Grid convergence of high order methods for multiscale complex unsteady viscous compressible fows. J. Comput. Phys. 185(1), 1-26 (2003). http://www.sciencedirect.com/science/article/pii/S002199910200044X
69. Sod, G.A.:A survey of several fnite diference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27, 1-31 (1978)
70. Vuik, M., Ryan, J.:Multiwavelet troubled-cell indicator for discontinuity detection of discontinuous Galerkin schemes. J. Comput. Phys. 270, 138-160 (2014)
71. Wang, L., Mavriplis, D.:Adjoint-based hp adaptive discontinuous Galerkin methods for the 2D compressible Euler equations. J. Comput. Phys. 228(20), 7643-7661 (2009)
72. Weber, Y., Oran, E., Boris, J., Anderson, J.:The numerical simulation of shock bifurcation near the end wall of a shock tube. Phys. Fluids 7(10), 2475-2488 (1995). http://scitation.aip.org/content/aip/journal/pof2/7/10/10.1063/1.868691
73. Woodward, P.:Trade-ofs in designing explicit hydrodynamical schemes for vector computers. In:Rodrigue G. (ed.) Parallel Computations, pp. 153-172, Academic Press (1982)
74. Woodward, P., Colella, P.:The numerical simulation of two-dimensional fuid fow with strong shocks. J. Comput. Phys. 54(1), 115-173 (1984). http://www.sciencedirect.com/science/article/pii/0021999184901426
75. Yu, T., Kolarov, K., Lynch, W.:Barysymmetric multiwavelets on triangles. IRC Report 1997-006, Standford University (1997)
76. Zingan, V., Guermond, J.L., Popov, B.:Implementation of the entropy viscosity method with the discontinuous Galerkin method. Comput. Methods Appl. Mech. Eng. 253, 479-490 (2013)

[1]	Xiaofeng Cai, Wei Guo, Jing-Mei Qiu. Comparison of Semi-Lagrangian Discontinuous Galerkin Schemes for Linear and Nonlinear Transport Simulations [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 3-33.
[2]	Francis Filbet, Tao Xiong. Conservative Discontinuous Galerkin/Hermite Spectral Method for the Vlasov-Poisson System [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 34-59.
[3]	Zhanjing Tao, Juntao Huang, Yuan Liu, Wei Guo, Yingda Cheng. An Adaptive Multiresolution Ultra-weak Discontinuous Galerkin Method for Nonlinear Schrödinger Equations [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 60-83.
[4]	Hongjuan Zhang, Boying Wu, Xiong Meng. A Local Discontinuous Galerkin Method with Generalized Alternating Fluxes for 2D Nonlinear Schrödinger Equations [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 84-107.
[5]	Vít Dolejší, Filip Roskovec. Goal-Oriented Anisotropic hp-Adaptive Discontinuous Galerkin Method for the Euler Equations [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 143-179.
[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]	Qi Tao, Yan Xu, Xiaozhou Li. Negative Norm Estimates for Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for Nonlinear Hyperbolic Equations [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 250-270.
[9]	Haijin Wang, Qiang Zhang. The Direct Discontinuous Galerkin Methods with Implicit-Explicit Runge-Kutta Time Marching for Linear Convection-Difusion Problems [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 271-292.
[10]	Aycil Cesmelioglu, Sander Rhebergen. A Compatible Embedded-Hybridized Discontinuous Galerkin Method for the Stokes-Darcy-Transport Problem [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 293-318.
[11]	Yuan Xu, Qiang Zhang. Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 319-352.
[12]	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.
[13]	F. L. Romeo, M. Dumbser, M. Tavelli. A Novel Staggered Semi-implicit Space-Time Discontinuous Galerkin Method for the Incompressible Navier-Stokes Equations [J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 607-647.
[14]	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.
[15]	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.