DOI QR코드

DOI QR Code

ADAPTIVE GRID SIMULATION OF HYPERBOLIC EQUATIONS

  • Li, Haojun (Department of Mathematical Sciences, Seoul National University) ;
  • Kang, Myungjoo (Department of Mathematical Sciences, Seoul National University)
  • Received : 2013.07.09
  • Accepted : 2013.10.23
  • Published : 2013.12.25

Abstract

We are interested in an adaptive grid method for hyperbolic equations. A multiresolution analysis, based on a biorthogonal family of interpolating scaling functions and lifted interpolating wavelets, is used to dynamically adapt grid points according to the physical field profile in each time step. Traditional finite-difference schemes with fixed stencils produce high oscillations around sharp discontinuities. In this paper, we hybridize high-resolution schemes, which are suitable for capturing singularities, and apply a finite-difference approach to the scaling functions at non-singular points. We use a total variation diminishing Runge-Kutta method for the time integration. The computational cost is proportional to the number of points present after compression. We provide several numerical examples to verify our approach.

References

  1. M. A. Alves, P. Cruz, A. Mendes, F. D. Magalhaes, F. T. Pinho, P. J. Oliveira, Adaptive multiresolution approach for solution of hyperbolic PDEs, Comput. Methods Appl. Mech. Engrg. 191 (2002) 3909-3928. https://doi.org/10.1016/S0045-7825(02)00334-1
  2. B. L. Bihari, A. Harten, Application of generalized wavelets: An adaptive multiresolution scheme, J. Comput. Appl. Math. 61 (1995), 275-321. https://doi.org/10.1016/0377-0427(94)00070-1
  3. M. S. Darwish, F. Moukalled, Normalized variable and space formulation methodology for high-resolution schemes, Numer. Heat Transfer Part B 30 (1994) 217-237.
  4. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Series in Appl. Math. 61, SIAM, 1992.
  5. M. O. Domingues, S. M. Gomes, O. Roussel, K. Schneider, An adaptive multiresolution scheme with local time stepping for evolutionary PDEs, J. Comput. Phys. 227 (2008), 3758-3780. https://doi.org/10.1016/j.jcp.2007.11.046
  6. M. O. Domingues, S. M. Gomes, O. Roussel, K. Schneider, Space-time adaptive multiresolution methods for hyperbolic conservation laws: Applications to compressible Euler equations, Appl. Nume. Math. 59 (2009), 2303-2321. https://doi.org/10.1016/j.apnum.2008.12.018
  7. D. L. Donoho, Interpolating wavelet transforms, Tech. rep., Department of Statistics, Stanford University (1992).
  8. S. Dubuc, Interpolation through an iterative scheme, J. Math. Anal. Appl. 114 (1986) 185-204. https://doi.org/10.1016/0022-247X(86)90077-6
  9. S. Goedecker, Wavelets and their applications for the solution of partial differential equations in physics, Vol. 4, Presses Polytechniques et Universitaires Romandes, 1998.
  10. G. Deslauriers, S. Dubuc, Symmetric iterative interpolation processes, Constr. Approx. 5 (1989) 49-68. https://doi.org/10.1007/BF01889598
  11. P. H. Gaskell and A. K. C. Lau, Curvature-Compensated Convective Transport: SMART, A New Boundednesspreserving Transport Algorithm, Inter. J. Nume. Methods in Fluids 8 (1988), 617-641. https://doi.org/10.1002/fld.1650080602
  12. S. Gottlieb, C.-W. Shu, Total Variation Diminishing Runge-Kutta Schemes, Math. of Computation 67(2231) (1998) 73-85. https://doi.org/10.1090/S0025-5718-98-00913-2
  13. A. Harten, High Resolution Schemes for Hyperbolic Conservation Laws, J. Comput. Phys. 49 (1983), 357-393. https://doi.org/10.1016/0021-9991(83)90136-5
  14. A. Harten, B. Engquist, S. Osher and S. Chakravarthy, Uniformly high order essentially non-oscillatory schemes, III, J. Comput. Phys. 71 (1987), 231-303. https://doi.org/10.1016/0021-9991(87)90031-3
  15. J. Hesthaven, T.Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer Texts in Applied Mathematics, Springer Verlag, 2008.
  16. L. Jameson, A Wavelet-optimized Very High Order Adaptive Grid And Order Numerical Method, SIAM J. Sci. Comput. 19(6) (1998) 1980-2013. https://doi.org/10.1137/S1064827596301534
  17. A. J. Kozakevicius, L. C. C. Santos, ENO adaptive method for solving one-dimensional conservation laws, Appl. Nume. Math. 59 (2009), 2337-2355. https://doi.org/10.1016/j.apnum.2008.12.020
  18. A. Kurganov, E. Tadmor, New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection-Diffusion Equations, J. Comput. Phys. 160 (2000), 241-282. https://doi.org/10.1006/jcph.2000.6459
  19. P. D. Lax, B. Wendroff, Systems of conservation laws, Commun. Pure Appl. Math. 13 (1960), 217-237. https://doi.org/10.1002/cpa.3160130205
  20. B. van. Leer, Towards the ultimate conservative difference scheme I. The quest of monotonicity, Springer Lecture Notes Phys. 18 (1973), 163-168.
  21. B. van. Leer, Towards the ultimate conservative difference scheme II. Monotonicity and conservation combined in a second order scheme, J. Comput. Phys. 14 (1974), 361-370. https://doi.org/10.1016/0021-9991(74)90019-9
  22. B. van. Leer, Towards the ultimate conservative difference scheme III. Upstream-centered finite difference schemes for ideal compressible flow, J. Comput. Phys. 23 (1977), 263-275. https://doi.org/10.1016/0021-9991(77)90094-8
  23. B. van. Leer, Towards the ultimate conservative difference scheme IV. A new approach to numerical convection, J. Comput. Phys. 23 (1977), 276-299. https://doi.org/10.1016/0021-9991(77)90095-X
  24. B. van. Leer, Towards the ultimate conservative difference scheme V. A second order sequel to Godunov's method, J. Comput. Phys. 32 (1979), 101-136. https://doi.org/10.1016/0021-9991(79)90145-1
  25. B. P. Leonard, A stable and accurate convective modelling procedure based on quadratic upstream interpolation, Comput. Methods Appl. Mech. Eng, 19 (1979) 59-98. https://doi.org/10.1016/0045-7825(79)90034-3
  26. B. P. Leonard, Locally Modified Quick Scheme for Highly Convective 2-D and 3-D Flows, Num. Methods in Lam. and Turb. Flows, eds. Taylor C., Morgan K., Pineridge Press, Swansea, UK 5 (1987) 35-47.
  27. B. P. Leonard, Simple High-accuracy Resolution Program For Convective Modelling Of Discontinuities, Inter. J. Numer. Meth. Fluids 8 (1988) 1291-1318. https://doi.org/10.1002/fld.1650081013
  28. R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.
  29. X.-D. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys. 115 (1994) 200-212. https://doi.org/10.1006/jcph.1994.1187
  30. S. Mallat, A Theory for Multiresolution Signal Decomposition: The Wavelet Representation, IEEE Trans. Patte. Machi. Intel. 11(7) (1989).
  31. S. Mallat, Multiresolution approximations and wavelet orthonormal bases of $L^2$(R), Trans. Amer. Math. Soc. 315 (1989) 69-87.
  32. S. Mallat, A Wavelet Tour of Signal Processing, 2nd Edition, Academic Press, 1998.
  33. P. L. Roe, Approximate Riemann solvers, parameter vectors and difference scheme, J. Comput. Phys. 43 (1981) 357-372. https://doi.org/10.1016/0021-9991(81)90128-5
  34. C.-W. Shu, S. Osher, Efficient Implementation of Essentially Non-oscillatory Shock-Capturing Schemes, J. Comput. Phys. 77 (1988) 439-471. https://doi.org/10.1016/0021-9991(88)90177-5
  35. G. A. Sod, A Survey of Several Finite Difference Methods for Systems of Nonlinear Hyperbolic Conservation Laws, J. Comput. Phys. 27 (1978) 1-31. https://doi.org/10.1016/0021-9991(78)90023-2
  36. W. Sweldens, The lifting scheme: A custom-design construction of biorthogonal wavelets, Appl. Comput. Harmon. Anal. 3 (1996) 186-200. https://doi.org/10.1006/acha.1996.0015
  37. W. Sweldens, The lifting scheme: A construction of second generation wavelets, SIAM, J. Math. Anal 29 (2) (1998) 511-546. https://doi.org/10.1137/S0036141095289051
  38. E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, third edition, Springer, 2009.
  39. O. V. Vasilyev, S. Paolucci, A dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain, J. Comput. Phys. 125 (1996) 498-512. https://doi.org/10.1006/jcph.1996.0111
  40. O. V. Vasilyev, S. Paolucci, A fast adaptive wavelet collocation algorithm for multidimensional PDEs, J. Comput. Phys. 138 (1997) 16-56. https://doi.org/10.1006/jcph.1997.5814
  41. O. V. Vasilyev, C. Bowman, Second-generation wavelet collocation method for the solution of partial differential equations, J. Comput. Phys. 165 (2000) 660-693. https://doi.org/10.1006/jcph.2000.6638
  42. O. V. Vasilyev, Solving multi-dimensional evolution problems with localized structures using second generation wavelets, Int. J. Comput. Fluid Dynam. 17 (2) (2003) 151-168. https://doi.org/10.1080/1061856021000011152
  43. J. D. Regele, O. V. Vasilyev, An adaptive wavelet-collocation method for shock computations, Inter. J. Comput. Fluid Dynam. 23(7) (2009) 503-518. https://doi.org/10.1080/10618560903117105
  44. R. F. Warming, R. M. Beam, Upwind second-order difference schemes and applications in unsteady aerodynamic flows, in Proc. AIAA 2nd Computational Fluid Dynamics Conf., Hartford, Conn, 1975.