A new wavelet-based method for the simulation of reacting flows on adaptive meshes is presented. The method is based on the removal of grid points whose wavelet coefficients are small with reference to some user-specified threshold. Unlike some other collocation methods, the scheme simulates flow behaviour in the physical (i.e. not transformed) domain, and the wavelets, thus, provide the method by which the derivatives appearing in the transport equations are calculated. The wavelet transformis based on a subtraction algorithm, and circumvents the hanging node problem associated with other adaptive strategies. Interpolating wavelets are applied to a compressible one-dimensional laminar flame problem with time dependent boundary conditions. We find that the resolution of the chemistry distribution is comparatively straightforward. The same is not true of the pressure field, which demonstrates sensitivity to the imposed threshold level. Conclusions and directions for future work are presented based on these findings.
WilliamsF.Combustion theory, second edition, 1985 (Addison Wesley, Menlo Park, California).
4.
HaarA.Zur theorie der orthogonalen funktionen-systeme. Math. Ann., 1910, 69, 331–371.
5.
MorletJ.ArensG.FourgeauI.GiardD.Wave propagation and sampling theory. Geophysics, 1982, 47, 203–236.
6.
MallatS.A theory for multiresolution signal decomposition: The wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell., 1989, 11(7), 674–693.
7.
MeyerY.Wavelets, algorithms and applications, 1993, (SIAM, Philadelphia).
8.
DaubechiesI.Ten lectures on wavelets. In CBMS lecture notes, No. 61, 1992 (SIAM, Philadelphia).
9.
FargeM.Wavelet transforms and their applications to turbulence. Annu. Rev. Fluid Mech., 1992, 24, 395–457.
10.
FargeM.PellegrinoG.SchneiderK.Coherent vortex extraction in 3D turbulent flows using orthogonal wavelets. Phys. Rev. Lett., 2001, 87, 1–4.
11.
FargeM.SchneiderK.KevlahanN.Non-gaussianity and coherent vortex simulation for two dimensional turbulence using an adaptive orthogonal wavelet basis. Phys. Fluids A, 1999, 11, 2187–2201.
LiandratJ.TchamitchianP.Resolution of the 1-D regularised burgers equation using a spatial wavelet approximation. Technical report 90–83, NASA (ICASE Report, December 1990.
14.
BacryE.MallatS.PapanicolaouG.A wavelet based space-time adaptive numerical method for partial differential equations. Math. Model. Numer. Anal., 1992, 26(7), 793–834.
15.
VasilyevO. V.PaolucciS.SenM.A multilevel wave let collocation method for solving partial differential equations in a finite domain. J. Comput. Phys., 1995, 120, 33–47.
16.
VasilyevO. V.PaolucciS.A dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain. J. Comput. Phys., 1996, 125, 498–512.
17.
VasilyevO. V.PaolucciS.A fast adaptive wavelet collocation algorithm for multidimensional PDEs. J. Comput. Phys., 1997, 138, 16–56.
18.
FrölichJ.SchneiderK.Numerical simulation of decaying turbulence in an adapted wavelet basis. Appl. Comput. Harmon. Anal., 1995, 2, 393–397.
19.
BihariB.Multiresolution schemes for conservation laws with viscosity. J. Comput. Phys., 1996, 123, 207–225.
20.
FrölichJ.SchneiderK.An adaptive wavelet-vaguelette algorithm for the solution of PDEs (preprint). J. Comput. Phys., 130, 174–190.
21.
FrölichJ.SchneiderK.A fast algorithm for lacunary wavelet bases related to the solution of PDE's. C. R. Math. Rep. Acad. Sci. Canada, 1996, 16, 83–86.
22.
BockhornH.FrölichJ.SchneiderK.An adaptive two-dimensional wavelet-vaguelette algorithm for the computation of flame balls(preprint). Technical report, Universität Kaiserslautern, 1995.
23.
SinghS.RastigejevY.PaolucciS.PowersJ.Viscous detonation in H2-O2-Ar using intrinsic low-dimensional manifolds and wavelet adaptive multilevel representation. Combust. Theory Model., 2001, 5, 163–184.
24.
ProsserR.Numerical methods for the computation of combustion. PhD Thesis, Cambridge University, 1997.
25.
MonasseP.PerrierV.Orthonormal wavelet bases adapted for partial differential equations with boundary conditions. SIAM J. Math. Anal., 1998, 28, 1040–1065.
26.
CohenA.DaubechiesI.VialP.Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal., 1993, 1, 54–81.
27.
JamesonL.On the differentiation matrix for daubechies-based wavelets on an interval. SIAM J. Sci. Comput., 1996, 17, 498–516.
28.
CohenA.DaubechiesI.FeauveauJ.Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math., 1992, 45, 485–560.
29.
SweldensW.The lifting scheme: A custom design construction of biorthogonal wavelets. Appl. Comput. Harmon. Anal., 1996, 3, 186–200.
30.
SweldensW.The lifting scheme: A construction of second generation wavelets. SIAM J. Math. Anal., 1997, 29, 511–546.
31.
SchröderP.SweldensW.Building your own wavelets at home. In ACM SIGGRAPH course notes, 1996.
DonohoD.Interpolating wavelet transforms. Presented at The NATO Advanced Study Institute Conference on Wavelets and Applications, Il Ciocco, Italy, August 1992.
34.
ProsserR.CantR.A wavelet-based method for the efficient simulation of combustion. J. Comput. Phys., 1998, 147, 337–361.
35.
ProsserR.CantR.The application of wavelets to reacting flows. J. Mech. Eng. Sci., 2000, 24, 1355–1362.
36.
ProsserR.CantR.The theoretical development of wavelets in reacting flows. J. Mech. Eng. Sci., 2000, 24, 1363–1373.
37.
VasilyevO. V.BowmanC.Second-generation wavelet collocation method for the solution of partial differential equations. J. Comput. Phys., 2000, 165, 660–693.
38.
VasilyevO. V.KevlahanN.An adaptive multilevel wavelet collocation method for elliptic problems. J. Comput. Phys., 2005, 206, 412–431.
39.
RestrepoJ.LeafG.SchlossnagleG.Periodized daubechies wavelets. Technical report MCS-P423–0394, Mathematics and Computer Science Division, Argonne National Lab, Illinois, 1994, (preprint), available at http://www.mathsoft.com/wavelet.html.
40.
PressW.TeukolskyS.VetterlingW.FlanneryB.Numerical recipes in FORTRAN, 2nd edition, 1992 (Cambridge University Press, Cambridge, UK).
41.
CanutoC.HussainiM.QuarteroniA.ZangT.Spectral methods in fluid dynamics, 1st edition, 1998 (Springer-Verlag, Berlin).
42.
BeylkinG.CoifmanR.RokhlinV.Fast wavelet transforms and numerical algorithms I. Commun. Pure Appl. Math., 1991, 44, 141–183.
43.
AràndigaF.CandelaV.Multiresolution standard form of a matrix. SIAM J. Num. Anal., 1998, 33(2), 417–434.
44.
WilliamsF.A review of some theoretical considerations of turbulent flame structure. In Analytical and numerical methods for investigation of flow fields with chemical reactions, especially related to combustion. In AGARD Conference Proceedings No. 164, NATO, 1975, pp. 1–25.
45.
MaasU.PopeS.Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space. Combust. Flame, 1992, 88, 239–264.
46.
PoinsotT.LeleS.Boundary conditions for direct simulations of compressible viscous flows. J. Comput. Phys.1992101, 104–129.
47.
ThompsonK.Time dependent boundary conditions for hyperbolic systems. J. Comput. Phys., 1987, 68, 1–24.
48.
ThevéninD.BaumM.PoinsotT.(Eds) Direct numerical simulation for turbulent reacting flows, 1996 (Editions Technip, Paris).
49.
ProsserR.Improved boundary conditions for the direct numerical simulation of turbulent subsonic flows I: Inviscid flows. J. Comput. Phys., 2005, 207, 736–768.
50.
WrayA.Minimal storage time-advancement schemes for spectral methods. NASA Ames Research Center, 1990.
51.
RoacheP.Quantification of uncertainty in CFD. Annu. Rev. Fluid Mech., 1997, 29, 123–160.
52.
RoacheP.Verification of codes and calculations. AIAA J., 1998, 36, 696–702.
53.
ProsserR.On the effects of lifting on interpolating wavelets for combustion simulations. Proc. IMechE Part C: J. Mechanical Engineering Science, 2007, 221, in press.
