A fast adaptive diffusion wavelet method for Burger’s equation. (English) Zbl 1362.35230
Summary: A fast adaptive diffusion wavelet method is developed for solving the Burger’s equation. The diffusion wavelet is developed in [R. R. Coifman and M. Maggioni, Appl. Comput. Harmon. Anal. 21, No. 1, 53–94 (2006; Zbl 1095.94007)] and its most important feature is that it can be constructed on any kind of manifold. Classes of operators which can be used for construction of the diffusion wavelet include second order finite difference differentiation matrices. The efficiency of the method is that the same operator is used for the construction of the diffusion wavelet as well as for the discretization of the differential operator involved in the Burger’s equation. The diffusion wavelet is used for the construction of an adaptive grid as well as for the fast computation of the dyadic powers of the finite difference matrices involved in the numerical solution of Burger’s equation. In this paper, we have considered one dimensional and two dimensional Burger’s equation with Dirichlet and periodic boundary conditions. For each test problem the CPU time taken by fast adaptive diffusion wavelet method is compared with the CPU time taken by finite difference method and observed that the proposed method takes lesser CPU time. We have also verified the convergence of the given method.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
Citations:
Zbl 1095.94007References:
| [1] | Bateman, H., Some recent researches in motion of fluids, Mon. Weather Rev., 43, 163-170 (1915) |
| [2] | Burger, J. M., A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., 1, 177-199 (1948) |
| [3] | Whitham, G., Linear and Nonlinear Waves (1974), Wiley and Sons · Zbl 0373.76001 |
| [5] | Varoglu, E.; Finn, W., Space-time finite elements incorporating characterstics for burger’s equation, Int. J. Numer. Methods Eng., 16, 171-184 (1980) · Zbl 0449.76076 |
| [6] | Biringen, S.; Satti, A., Comaprison of several finite-difference methods, J. Aircr., 27, 90-92 (1990) |
| [7] | Bar-Yoseph, P.; Moses, E.; Zrahia, U.; Yarin, A. L., Space-time spectral element methods for one dimensional non linear advection diffusion problems, J. Comput. Phys., 119, 62-75 (1995) · Zbl 0827.65098 |
| [8] | Mansell, G.; Merryfield, W.; Shizgal, B.; Weinert, U., A comparison of differential quadrature methods for solution of partial differential equations, Comput. Methods Appl. Mech. Engrg., 104, 295-316 (1993) · Zbl 0781.65088 |
| [9] | Zhang, D. S.; Wei, G. W.; Kouri, D. J., Burger’s equation with high Reynolds number, Phys. Fluids, 9, 1853-1855 (1997) · Zbl 1185.76843 |
| [10] | Wei, G. W.; Zhang, D. S.; Kouri, D. J.; Hoffman, D. K., Distributed approximating functional approach to burger’s equation in one and two space dimensions, Comput. Phys. Commun., 111, 93-109 (1998) · Zbl 0933.65120 |
| [11] | Xia, K.; Zhan, M.; Wan, D.; Wei, G. W., Adaptively deformed mesh based interface method for elliptic equations with discontinuous coefficients, J. Comput. Phys., 231, 1440-1461 (2012) · Zbl 1242.65229 |
| [12] | Berger, M. J.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82, 64-84 (1989) · Zbl 0665.76070 |
| [13] | Jameson, L., A wavelet-optimized, very high order adaptive grid and order numerical method, SIAM J. Sci. Comput., 19, 1980-2013 (1998) · Zbl 0913.65090 |
| [14] | Garba, A., A wavelet collocation method for numerical solution of Burgers’ equation (1996), International Center for Theorytical Physics, (Tech. Rep.) · Zbl 0853.65122 |
| [15] | Wan, D. C.; Wei, G. W., The study of quasi wavelets based numerical method applied to Burger’s equation, Appl. Math. Mech., 21, 1099-1110 (2000) · Zbl 1003.76070 |
| [16] | Wei, G. W., Quasi wavelets and quasi interpolating wavelets, Chem. Phys. Lett., 296, 215-222 (1998) |
| [17] | Liandrat, J.; Tchamitchian, P., Resolution of the 1D regularized Burgers equation using a spatial wavelet approximation, Nasa contactor Report 187480: Icase Report no. 90-83 (1990) |
| [18] | Vasilyev, O. V.; Bowman, C., Second generation wavelet collocation method for the solution of partial differential equations, J. Comput. Phys., 165, 660-693 (2000) · Zbl 0984.65105 |
| [19] | Merriman, B.; Ruuth, S. J., Diffusion generated motion of curves on surfaces, J. Comput. Phys., 225, 2267-2282 (2007) · Zbl 1123.65011 |
| [20] | Macdonald, C. B.; Ruuth, S. J., The implicit closest point method for the numerical solutions of partial differential equations on the surfaces, SIAM J. Sci. Comput., 31, 4330-4350 (2009) · Zbl 1205.65238 |
| [21] | Bertalmio, M.; Cheng, L. T.; Osher, O.; Sapiro, G., Variational problems and partial differential equations on implicit surfaces, J. Comput. Phys., 174, 759-780 (2001) · Zbl 0991.65055 |
| [22] | Bergdorf, M.; Cottet, G.-H.; Koumoutsakos, P., Multilevel adaptive particle methods for convection-diffusion equations, Multiscale Model. Simul., 4, 328-357 (2005) · Zbl 1088.76055 |
| [23] | Mehra, M.; Kevlahan, N. K.-R., An adaptive wavelet collocation method for the solution of partial differential equations on the sphere, J. Comput. Phys., 227, 5610-5632 (2008) · Zbl 1147.65080 |
| [24] | Coifman, R. R.; Maggioni, M., Diffusion wavelets, Appl. Comput. Harmon. Anal., 21, 53-94 (2006) · Zbl 1095.94007 |
| [25] | Coifman, R. R.; Lafon, S., Diffusion maps, Appl. Comput. Harmon. Anal., 21, 5-30 (2006) · Zbl 1095.68094 |
| [26] | Donoho, D. L., Interpolating wavelet transforms (1992), Department of Statistics, Stanford University, Tech. Rep. 408 |
| [27] | Nilsen, O. M., Wavelets in scientific computing (1998), Technical University of Denamark, Lyngby, (Ph.D. thesis) |
| [28] | Bahadir, A. R., A fully implicit finite-difference scheme for two-dimensional Burgers’ equations, Appl. Math. Comput., 137, 131-137 (2003) · Zbl 1027.65111 |
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