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Fully adaptive multiresolution finite volume schemes for conservation laws. (English) Zbl 1010.65035
This paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes for the numerical solution of the Cauchy problem for hyperbolic systems of conservation laws of the form \( \partial_t u+\text{Div}_x(f(u(t,x)))=0\), \(u\in\mathbb R^m\), \(x\in\mathbb R^d\), \(t>0\), with initial value \(u(t=0,x)=u_0(x)\). The authors present an alternative strategy for adaptive discretizations, based on coupling of multiresolution representation and finite volume schemes, which allows a relatively simple implementation and a rigorous error analysis.
The solution is represented and computed in a dynamically evolved adaptive grid. A crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages. Several solutions of this problem are proposed, analyzed, and compared in terms of accuracy and complexity. Numerical tests for 1D and 2D problems are presented together with a discussion concerning the practical relevance of the refinement strategy and error estimate.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65T60 Numerical methods for wavelets
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
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[1] Abgrall, R. (1997) Multiresolution analysis on unstructured meshes: applications to CFD, in Chetverushkin and al. eds. Experimentation, modeling and computation in flow, turbulence and combustion, vol.2, John Wiley & Sons.
[2] Arandiga, F. and R. Donat (1999) A class of nonlinear multiscale decomposition, to appear in Numerical Algorithms.
[3] Berger, M. J. and P. Collela (1989) Local adaptive mesh refinement for shock hydrodynamics, Journal of Computational Physics, 82, pp 64-84. · Zbl 0665.76070
[4] I. Babuška and B. Q. Guo, The \?-\? version of the finite element method for domains with curved boundaries, SIAM J. Numer. Anal. 25 (1988), no. 4, 837 – 861. · Zbl 0655.65124 · doi:10.1137/0725048 · doi.org
[5] Barna L. Bihari and Ami Harten, Multiresolution schemes for the numerical solution of 2-D conservation laws. I, SIAM J. Sci. Comput. 18 (1997), no. 2, 315 – 354. · Zbl 0878.35007 · doi:10.1137/S1064827594278848 · doi.org
[6] J. M. Carnicer, W. Dahmen, and J. M. Peña, Local decomposition of refinable spaces and wavelets, Appl. Comput. Harmon. Anal. 3 (1996), no. 2, 127 – 153. · Zbl 0859.42025 · doi:10.1006/acha.1996.0012 · doi.org
[7] Alfred S. Cavaretta, Wolfgang Dahmen, and Charles A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (1991), no. 453, vi+186. · Zbl 0741.41009 · doi:10.1090/memo/0453 · doi.org
[8] Chiavassa, G. and R. Donat (1999) Numerical experiments with point value multiresolution for 2d compressible flows, Technical Report GrAN-99-4, University of Valencia. · Zbl 1043.76046
[9] Bernardo Cockburn, Frédéric Coquel, and Philippe LeFloch, An error estimate for finite volume methods for multidimensional conservation laws, Math. Comp. 63 (1994), no. 207, 77 – 103. · Zbl 0855.65103
[10] Cohen, A. (2000) Wavelet methods in numerical analysis, Handbook of Numerical Analysis, vol. VII, P.G. Ciarlet and J.L. Lions, eds., North-Holland, Amsterdam, pp. 417-711. CMP 2001:08
[11] Cohen, A., W. Dahmen and R. DeVore (2001) Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. Comp. 70, 27-75. CMP 2001:06 · Zbl 0980.65130
[12] A. Cohen, Ingrid Daubechies, and J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992), no. 5, 485 – 560. · Zbl 0776.42020 · doi:10.1002/cpa.3160450502 · doi.org
[13] Albert Cohen, Nira Dyn, Sidi Mahmoud Kaber, and Marie Postel, Multiresolution schemes on triangles for scalar conservation laws, J. Comput. Phys. 161 (2000), no. 1, 264 – 286. · Zbl 0959.65105 · doi:10.1006/jcph.2000.6503 · doi.org
[14] Cohen, A., S.M. Kaber and M. Postel (1999) Multiresolution analysis on triangles: application to conservation laws, in Finite volumes for complex applications II, R. Vielsmeier, F. Benkhaldoun and D. Hänel eds., Hermes, Paris.
[15] Wolfgang Dahmen, Wavelet and multiscale methods for operator equations, Acta numerica, 1997, Acta Numer., vol. 6, Cambridge Univ. Press, Cambridge, 1997, pp. 55 – 228. · Zbl 0884.65106 · doi:10.1017/S0962492900002713 · doi.org
[16] Dahmen, W., B. Gottschlich-Müller and S. Müller (2000) Multiresolution Schemes for Conservation Laws, Numerische Mathematik DOI 10.1007/s00210000222
[17] Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. · Zbl 0776.42018
[18] Ronald A. DeVore, Nonlinear approximation, Acta numerica, 1998, Acta Numer., vol. 7, Cambridge Univ. Press, Cambridge, 1998, pp. 51 – 150. · Zbl 0931.65007 · doi:10.1017/S0962492900002816 · doi.org
[19] Dyn, N. (1992) Subdivision algorithms in computer-aided geometric design, in: Advances in Numerical Analysis II, W.A. Light ed., Clarendon Press, Oxford. · Zbl 0760.65012
[20] Birgit Gottschlich-Müller and Siegfried Müller, Adaptive finite volume schemes for conservation laws based on local multiresolution techniques, Hyperbolic problems: theory, numerics, applications, Vol. I (Zürich, 1998) Internat. Ser. Numer. Math., vol. 129, Birkhäuser, Basel, 1999, pp. 385 – 394. · Zbl 0929.65081
[21] Ami Harten, Adaptive multiresolution schemes for shock computations, J. Comput. Phys. 115 (1994), no. 2, 319 – 338. · Zbl 0925.65151 · doi:10.1006/jcph.1994.1199 · doi.org
[22] Ami Harten, Multiresolution algorithms for the numerical solution of hyperbolic conservation laws, Comm. Pure Appl. Math. 48 (1995), no. 12, 1305 – 1342. · Zbl 0860.65078 · doi:10.1002/cpa.3160481201 · doi.org
[23] Ami Harten, Björn Engquist, Stanley Osher, and Sukumar R. Chakravarthy, Uniformly high-order accurate essentially nonoscillatory schemes. III, J. Comput. Phys. 71 (1987), no. 2, 231 – 303. · Zbl 0652.65067 · doi:10.1016/0021-9991(87)90031-3 · doi.org
[24] S. Jaffard, Pointwise smoothness, two-microlocalization and wavelet coefficients, Publ. Mat. 35 (1991), no. 1, 155 – 168. Conference on Mathematical Analysis (El Escorial, 1989). · Zbl 0760.42016 · doi:10.5565/PUBLMAT_35191_06 · doi.org
[25] Dietmar Kröner, Numerical schemes for conservation laws, Wiley-Teubner Series Advances in Numerical Mathematics, John Wiley & Sons, Ltd., Chichester; B. G. Teubner, Stuttgart, 1997. · Zbl 0872.76001
[26] Randall J. LeVeque, Numerical methods for conservation laws, 2nd ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. · Zbl 0847.65053
[27] Bradley J. Lucier, A moving mesh numerical method for hyperbolic conservation laws, Math. Comp. 46 (1986), no. 173, 59 – 69. · Zbl 0592.65062
[28] Yves Meyer, Ondelettes et opérateurs. I, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990 (French). Ondelettes. [Wavelets]. Yves Meyer, Ondelettes et opérateurs. II, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990 (French). Opérateurs de Calderón-Zygmund. [Calderón-Zygmund operators]. Yves Meyer and R. R. Coifman, Ondelettes et opérateurs. III, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1991 (French). Opérateurs multilinéaires. [Multilinear operators].
[29] Richard Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983), no. 161, 91 – 106. · Zbl 0533.65061
[30] Sjögreen, B. (1995) Numerical experiments with the multiresolution scheme for the compressible Euler equations, J. Comp. Phys., 117, 251-261. · Zbl 0821.76054
[31] Schröder-Pander, F. and T. Sonar (1995) Preliminary investigations on multiresolution analysis on unstructured grids, DLR Report IB 223-95 A 36, 1995, DLR Göttingen. · Zbl 0987.65083
[32] L.J. Durlofsky, B. Engquist, and S. Osher (1992) Triangle based adaptive stencils for the solution of hyperbolic conservation laws. J. Comp. Phys., 98, 64-73. · Zbl 0747.65072
[33] Edwige Godlewski and Pierre-Arnaud Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, vol. 118, Springer-Verlag, New York, 1996. · Zbl 0860.65075
[34] Sidi Mahmoud Kaber and Marie Postel, Finite volume schemes on triangles coupled with multiresolution analysis, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 9, 817 – 822 (English, with English and French summaries). · Zbl 0933.65114 · doi:10.1016/S0764-4442(99)80278-X · doi.org
[35] A. Voss and S. Müller (1999) A manual for the template class library igpm_t_lib. Technical Report 197, IGPM, RWTH Aachen.
[36] J. Ballmann, F. Bramkamp, and S. Müller (2000) Development of a flow solver employing local adaptation based on multiscale analysis on B-spline grids. In Proceedings of 8th Annual Conf. of the CFD Society of Canada, June, 11 to 13, 2000 Montreal.
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