zbMATH — the first resource for mathematics

A multilevel characteristics method for periodic convection-dominated diffusion problems. (English) Zbl 0953.65065
The authors consider the linear unsteady convection-diffusion problem that involves a function \(u(x,t)\) from \(]0,2\pi[^d\times \mathbb{R}^+\) into \(\mathbb{R}\):

\[ {\partial u\over\partial t}+ a.\nabla u-\nu\Delta u= f, \] where \(\nu\) is the viscosity, \(f\) is the forcing term and \(a\) is some divergence-free vector field. The initial conditions are that \(u(x,0)= u_0(x)\) for all \(x\in ]0,2\pi[^d\) and \(u\) is \(2\pi\)-periodic in all space variables.
The spatial discretization uses the space \(S_M\) of all trigonometric polynomials of degree \(\leq M\) in each variable. A two-level decomposition is introduced by considering another parameter \(m\), \(0< m< M\) and writing \(S_M= S_M= S_m+ (I- P_m)S_M\) where \(P_m\) is the \(L^2\)-projection onto \(S_m\).
The numerical integration then relies on two different interpolation opeators and the approximate solution is obtained as the sum of a term in \(S_m\) and a term in \((I- P_m)S_M\). The two components are advanced in time using different time steps.
The authors investigate the stability fo this scheme and derive error estimates. These indicate that the high-frequency term can be integrated with a larger time-step.
The paper contains a few numerical tests for 1D-problems. It is found that the two-level method allows a significant gain in computing time with respect to the classical method. The accuracy is always better than the one of the classical method based on \(S_m\) and close to the one based on \(S_M\).

65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
65M55 Multigrid methods; domain decomposition 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
35K15 Initial value problems for second-order parabolic equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] Finite element methods for fluids, Masson, 1989. · Zbl 0712.76001
[2] Bercovier, Appl Math Model 7 pp 89– (1983)
[3] Douglas, SIAM J Numer Anal 19 pp 871– (1982)
[4] Suli, Numer Math 53 pp 459– (1988)
[5] and A high-order characteristics/finite element method for incompressible Navier-Stokes equations, Rapport de l’Université Pierre et Marie Curie, R 92032, 1992.
[6] and Stability of the Lagrange-Galerkin method with nonexact integration, M2 AN, 22 (1988), 625-653. · Zbl 0661.65114
[7] and Stability and convergence of the spectral Lagrange-Galerkin method for periodic and non-periodic convection-dominated diffusion problems, Oxford Univ Comp Lab Rpt, 92/19, 1993.
[8] Résolution numérique des équations de Navier-Stokes instationnaires par méthodes spectrales. Méthode de Galerkin non linéaire, Thèse, Université Paris-Sud, 1990.
[9] Simulation numérique d’écoulement homogènes et non-homogènes par des méthodes multi-résolution, Thèse, Université Paris-Sud, 1993.
[10] Debussche, Theo Comp Fluid Dynam 7 pp 279– (1995)
[11] Burie, SIAM J Numer Anal 34 pp 1574– (1997)
[12] Burie, Math Comp
[13] Marion, SIAM J Numer Anal 32 pp 1170– (1995)
[14] and Spectral methods in fluid dynamics, Springer-Verlag, Berlin, Heidelberg, 1987.
[15] and ?Approximations spectrales des problèmes aux limites elliptiques,? Mathématiques et Applications, Springer-Verlag, New York, 1992.
[16] Méthodes de caractéristiques multi-niveaux en espace et en temps pour une équation de convection-diffusion?Cas d’une approximation pseudo-spectrale, Thèse, Ecole Centrale de Lyon, 1998.
[17] Méthodes en temps d’ordre élevé par décomposition d’opérateurs. Applications aux équations de Navier Stokes, Thèse, Université Paris 6, 1993.
[18] Kreiss, SIAM J Numer Anal 16 pp 421– (1979)
[19] Suli, SIAM J Numer Anal 28 pp 423– (1991)
[20] Méthodes spectrales multi-niveaux en espace et en temps pour des problèmes paraboliques: estimations d’erreurs et algorithmes adaptatifs, Thèse, Ecole Centrale de Lyon 1996.
[21] Pironneau, Numer Math 38 pp 309– (1982)
[22] and The spectral Lagrange-Galerkin method for the Navier-Stokes equations: Convergence and non-linear stability, Oxford Univ Comp Lab Rpt, 89/10, 1989.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.