Abstract
In this article, we conjugate time marching schemes with Finite Differences splittings into low and high modes in order to build fully explicit methods with enhanced temporal stability for the numerical solutions of PDEs. The main idea is to apply explicit schemes with less restrictive stability conditions to the linear term of the high modes equation, in order that the allowed time step for the temporal integration is only determined by the low modes. These conjugated schemes were developed in [10] for the spectral case and here we adapt them to the Finite Differences splittings provided by Incremental Unknowns, which steems from the Inertial Manifolds theory. We illustrate their improved capabilities with numerical solutions of Burgers equations, with uniform and nonuniform meshes, in dimensions one and two, when using modified Forward–Euler and Adams–Bashforth schemes. The resulting schemes use time steps of the same order of those used by semi-implicit schemes with comparable accuracy and reduced computational costs.
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REFERENCES
Ascher, U., and Petzold, L. (1998). Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM.
Calgaro, C., Laminie, J., and Temam, R. (1997). Dynamical multilevel schemes for the solution of evolution equations by hierarchical finite element discretization. Appl. Num. Math. 23, 403–442.
Chehab, J.-P., and Temam, R. (1995). Incremental unknowns for solving nonlinear eigenvalue problems: New multiresolution methods. Numer. Methods Partial Differential Equations 11, 199–228.
Chehab, J.-P. (1996). Solution of generalized Stokes problems using hierarchical methods and incremental unknowns. Appl. Numer. Math. 21, 9–42.
Chehab, J.-P. (1998). Incremental unknowns method and compact schemes. M2AN 32(1), 51–83.
Chehab, J.-P., and Miranville, A. (1998). Incremental unkowns method on nonuniform meshes. M2AN 32(5), 539–577.
Chen, M., and Temam, R. (1991). Incremental unknowns for solving partial differential equations. Numer. Math. 59, 255–271.
Chen, M., Miranville, A., and Temam, R. (1995). Incremental unknowns in finite difference in three space dimension. Comput. Appl. Math. 14(3), 1–15.
Costa, B, and Dettori, L. (1998). Fourier collocation splittings for partial differential equations, J. Comput. Phys. 20, 469–475.
Costa, B., Dettori, L., Gottlieb, D., and Temam, R. (2001). Time marching techniques for the nonlinear Galerkin method. SIAM. J. SC. Comp. 23(1), 46–65.
Debussche, A., Dubois, T., and Temam, R. (1995). The nonlinear Galerkin method: A multiscale method applied to the simulation of homogeneous turbulent flows, and theoretical and computational fluid dynamics 7(4), 279–315.
Dettori, L., Gottlieb, D., and Temam, R. (1995). A nonlinear Galerkin method: The two-level Fourier-collocation case. J. Sci. Comput. 10(4), 371–389.
Dettori, L., Gottlieb, D., and Temam, R. (1996). A nonlinear Galerkin method: The two-level Chebyshev-collocation case. In Ilin, A. V., and Scot, L. R. (eds.), Special Issue of Houston Journal of Mathematics, pp. 75–83.
Dubois, T., Jauberteau, F., and Temam, R. (1999). Dynamic, Multilevel Methods and the Numerical Simulation of Turbulence, Cambridge University Press.
Fornberg, B., and Whitham, G. B. (1978). A numerical and theoretical study of certain nonlinear wave phenomena. Philos. Trans. Roy. Soc. London Ser. A 289, 373–404.
Goubet, O. (1992). Construction of approximate inertial manifolds using wavelets, SIAM J. Math. Anal. 23(6), 1455–1481.
Lele, S. K. (1992). Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–42.
Jolly, M. S., Rosa, R., and Temam, R. Accurate computations on inertial manifolds. SIAM. J. Sci. Comput. 22(6), 2216–2238.
Marion, M., and Temam, R. (1989). Nonlinear Galerkin methods: SIAM J. Numer. Anal. 26, 1139–1157.
Marion, M., and Temam, R., (1990). Nonlinear Galerkin methods; The finite elements case. Numer. Math. 57, 205–226.
Pouit, F. (décembre 1998). Thèse, Université Paris-Sud, Orsay.
Simonnet, E. (décembre 1998). Thèse, Université Paris-Sud.
Tachim Medjo, T. (1997). Numerical solutions of the Navier–Stokes equations using wavelet-like incremental unknowns. RAIRO Modèl. Math. Anal. Numér. 31(7), 827–844.
Temam, R. (1997). Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer Verlag, Berlin, (2nd ed.).
Temam, R. (1990). Inertial manifolds and multigrid methods. SIAM J. Math. Anal. 21, 154–178.
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Chehab, JP., Costa, B. Time Explicit Schemes and Spatial Finite Differences Splittings. Journal of Scientific Computing 20, 159–189 (2004). https://doi.org/10.1023/B:JOMP.0000008719.48134.4f
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DOI: https://doi.org/10.1023/B:JOMP.0000008719.48134.4f