ENO adaptive method for solving one-dimensional conservation laws

Applied Numerical Mathematics

Volumn 59, Issue 9, 2009, Pages 2337-2355

  Kozakevicius, A J ^a   Santos, L C C ^b  

^a FEDERAL UNIVERSITY OF SANTA MARIA   (Brazil)

^b UNIVERSITY OF SÃO PAULO   (Brazil)

Author keywords

ENO scheme; Euler equations for gas dynamics; Multiresolution schemes; Thresholded wavelet transform

Indexed keywords

ADAPTIVE GRID GENERATION; ADAPTIVE METHODS; CLASSICAL EXAMPLE; CONSERVATION LAW; DIFFERENTIAL OPERATORS; ENO SCHEME; ESSENTIALLY NON-OSCILLATORY; FINITE DIFFERENCE SCHEME; FLUX SPLITTING; HYPERBOLIC SYSTEM; INTERPOLATING WAVELET; MULTIRESOLUTION SCHEMES; NON-LINEAR; NUMERICAL FLUX; SPARSE GRID; SPARSE REPRESENTATION; SPARSE SOLUTIONS; THIRD ORDER; THRESHOLD PARAMETERS; THRESHOLDED WAVELET TRANSFORM; TIME EVOLUTIONS; TIME STEP; TOTAL VARIATION DIMINISHING;

DIFFERENTIAL EQUATIONS; DYNAMICS; GAS DYNAMICS; INITIAL VALUE PROBLEMS; MATHEMATICAL OPERATORS; MESH GENERATION; MULTIRESOLUTION ANALYSIS; RUNGE KUTTA METHODS;

EULER EQUATIONS;

EID: 67349283525     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2008.12.020     Document Type: Article

References (40)

1. Arandiga F., Cohen A., Donat R., and Dyn N. Interpolation and approximation of piecewise smooth functions. SIAM J. Numer. Anal. 43 1 (2005) 41-57
2. Arandiga F., Donat R., and Harten A. Multiresolution based on weighted averages of the Hat function II: non-linear reconstruction techniques. SIAM J. Sci. Comput. 20 3 (1999) 1053
3. Berger M., and Oliger J. Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53 (1984) 484
4. Bihari B., and Harten A. Application of generalized wavelets: an adaptive multiresolution scheme. J. Comput. Appl. Math. 61 3 (1995) 275-321
5. Bürger R., and Kozakevicius A. Adaptive multiresolution WENO schemes for multi-species kinematic flow models. J. Comput. Phys. 224 2 (2007) 1190-1222
6. Bürger R., Kozakevicius A., and Sepúlveda M. Multiresolution schemes for strongly degenerate parabolic equations in one space dimension. Numer. Methods Partial Differential Equations 23 (2006) 706-730 10.1002/num.20206
7. Chiavassa G., and Donat R. Point value multiscale algorithms for 2D compressible flows. SIAM J. Sci. Comput. 23 3 (2001) 805-823
8. Cohen A. Wavelet methods in numerical analysis. In: Ciarlet P.G., and Lions J.L. (Eds). Handbook of Numerical Analysis vol. VII (2000), Elsevier, Amsterdam
9. Cohen A., Daubechies I., and Vial P. Multiresolution analysis, wavelets and fast algorithms on an interval. Appl. Comput. Harmon. Anal. 1 1 (1993) 100-115
10. Deslauriers G., and Dubuc S. Symmetric Iterative Interpolation Process, Constructive Approximation (1989), Springer-Verlag, New York/Berlin
11. Domingues M.O., Gomes S.M., and Diaz L.M.A. Adaptive wavelet representation and differentiation on block-structured grids. Appl. Numer. Math. 47 3-4 (2003) 421-437
12. Donat R. Studies on error propagation for certain nonlinear approximations to hyperbolic equations: discontinuities in derivatives. SIAM J. Numer. Anal. 31 3 (1994) 665
13. D.L. Donoho, Interpolating wavelet transform, Tech. report 408, Department of Statistics, Stanford University, Stanford, CA, 1992
14. Dubuc S. Interpolation through an iterative scheme. J. Math. Anal. Appl. 114 (1986) 185
15. Gottlieb S., and Shu C.-W. Total variation diminishing Runge-Kutta schemes. Math. Comp. 67 (1998) 73-85
16. Harten A. Adaptive multiresolution schemes for shock computations. J. Comput. Phys. 115 (1994) 319-338
17. Harten A. Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Comm. Pure Appl. Math. XLVIII (1995) 1305
18. A. Harten, Multiresolution representation of data, II. General Framework, CAM Report 94-10, UCLA, Los Angeles, CA, 1994
19. A. Harten, Multiresolution representation of cell average data, CAM Report 94-21, UCLA, Los Angeles, CA, 1994
20. Harten A., Engquist B., Osher S., and Chakravarthy S. Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71 (1987) 231
21. Holmström M. Solving hyperbolic PDE's using interpolating wavelets. SIAM J. Sci. Comput. 21 2 (1999) 405-420
22. Jameson L. A wavelet optimized, very high order adaptive grid and order numerical method. SIAM J. Sci. Comput. 19 (1998) 1980
23. L. Jameson, On the wavelet optimized finite difference method, Tech. report ICASE 94-9, NASA Langely Research Center, Hamptom, VA, 1994
24. Jansen M., and Bultheel A. Multiple wavelet threshold estimation by generalized cross validation for images with correlated noise. IEEE Trans. Image Processing 8 7 (1999) 947-953
25. Kadioglu S.Y., Sussman M., Osher S., Wright J.P., and Kang M. A second order primitive preconditioner for solving all speed multi-phase flows. J. Comput. Phys. 209 (2005) 477-503
26. M.K. Kaibara, Análise Multiresolução para Leis de Conservação em Malhas adaptativas, PhD Thesis, UNICAMP, Campinas, Brazil, 2000 (in Portuguese)
27. Kaibara M.K., and Gomes S.M. Fully adaptive multiresolution scheme for shock computations. In: Toro E.F. (Ed). Godunov Methods: Theory and Applications (2000), Kluwer Academic/Plenum Publishers, New York
28. Kevlahan N., and Vasilyev O. An adaptive wavelet collocation method for fluid-structure interaction at high Reynolds numbers. SIAM J. Sci. Comput. 26 6 (2005) 1894-1915
29. A. Kozakevicius, Interpolating wavelets as a tool for adaptive solving hyperbolic partial differential equations, PhD Thesis, USP, Brazil, 2002 (in Portuguese)
30. Liu X.-D., and Osher S. Convex ENO high order multi-dimensional schemes without field by field decomposition or staggered grids. J. Comput. Phys. 141 (1998) 1
31. O. Nielsen, Wavelet in scientific computing, PhD Thesis, Technical University of Denmark, 1998
32. Osher S.J., and Chakaravathy S. Very high order accurate TVD schemes. IMA Volumes in Mathematics and its Applications vol. 2 (1986), Springer-Verlag, New York/Berlin 229-274
33. Plantevin F. Wavelets on irregular meshes. Adv. Comput. Math. 4 (1995) 293-329
34. Roussel O., Schneider K., Tsigulin A., and Bockhorn H. A conservative fully adaptive multiresolution algorithm for parabolic PDE's. J. Comput. Phys. 188 (2003) 493-523
35. C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservations laws, ICASE Report, No. 97-65, NASA, 1997
36. Shu C.-W., and Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77 (1988) 439
37. B. Sjögreen, Numerical experiments with the multiresolution scheme for the compressible Euler equations, Report 159, Uppsala University, Dept. Scient. Computing, 1994
38. Stoer J., and Bulirsch R. Introduction to Numerical Analysis. second ed. (1993), Springer-Verlag, New York/Berlin
39. Strikwerda J. Finite Difference Schemes and Partial Differential Equations (1989), Chapmann & Hall, New York
40. Vasilyev O., Paolucci S., and Sen M. A multilevel wavelet collocation method for solving partial differential equations in a finite domain. J. Comput. Phys. 120 1 (1995) 33-47