×

An adaptive finite element semi-Lagrangian implicit-explicit Runge-Kutta-Chebyshev method for convection dominated reaction-diffusion problems. (English) Zbl 1128.65073

Summary: We introduce in this paper an adaptive method that combines a semi-Lagrangian scheme with a second order implicit-explicit Runge-Kutta-Chebyshev (IMEX RKC) method to calculate the numerical solution of convection dominated reaction-diffusion problems in which the reaction terms are highly stiff. The convection terms are integrated via the semi-Lagrangian scheme, whereas the IMEX RKC treats the diffusion terms explicitly and the highly stiff reaction terms implicitly. The space adaptation is done in the framework of finite elements and the criterion for adaptation is derived from the information supplied by the semi-Lagrangian step; so that, this can be considered a heuristic approach to adaptivity that is somewhat similar to the so-called \(r\)-adaptivity strategy.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
80A25 Combustion

Software:

RKC; ALBERT; ALBERTA
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Allievi, A.; Bermejo, R., Finite element modified method of characteristics for Navier-Stokes equations, Int. J. Numer. Meth. Fluids, 32, 439-464 (2000) · Zbl 0955.76048
[2] Bangerth, W.; Rannacher, R., Adaptive Finite Element Methods for Differential Equations (2003), Birkhäuser: Birkhäuser Basel · Zbl 1020.65058
[3] Bermejo, R.; Staniforth, A., The conversion of semi-Lagrangian advection schemes to quasi-monotone schemes, Month Weath. Rev., 120, 2622-2632 (1992)
[4] Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · Zbl 0445.73043
[5] Dörfler, W., A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal., 33, 1106-1124 (1996) · Zbl 0854.65090
[6] Eriksson, K.; Estep, D.; Hansbo, P.; Johnson, C., Introduction to adaptive methods for differential equations, Acta Numerica, 1729-1749 (1995) · Zbl 0835.65116
[7] Eriksson, K.; Johnson, C., Adaptive finite element methods for parabolic problems IV: Non-linear problems, SIAM J. Numer. Anal., 32, 1729-1749 (1995) · Zbl 0835.65116
[8] Estep, D.; Larson, M.; Williams, R., Estimating the Error of Numerical Solutions of Systems of Reaction-Diffusion Equations, Memoirs of the American Mathematical Society, vol. 696 (2000), 109 pp · Zbl 0998.65096
[9] Fernandez-Tarrazo, E.; Vera, M.; Liñán, A., Lift-off and blow-off of a diffusion flame between parallel streams of fuel and air, Combust. Flame, 144, 261-276 (2006)
[10] Garrido-López, D.; Sarkar, S., Effects of imperfect premixing coupled with hydrodynamic instability on flame propagation, Proc. Combst. Instit., 30, 621-628 (2005)
[11] Schmidt, A.; Siebert, K. G., ALBERT: Designe of Adaptive Finite Element Software. The Finite Element Toolbox ALBERTA, Springer Lectures Notes in Computational Science and Engineering (2005), Springer: Springer Berlin · Zbl 1068.65138
[12] Verwer, J. G.; Sommeijer, B. P.; Hundsdorfer, W., RKC time-stepping for advection-diffusion-reaction problems, J. Comput. Phys., 201, 61-79 (2004) · Zbl 1059.65085
[13] Verwer, J. G.; Sommeijer, B. P., An implicit-explicit Runge-Kutta-Chebyshev scheme for diffusion-reaction equations, SIAM J. Sci. Comput., 25, 1824-1835 (2004) · Zbl 1061.65090
[14] Verwer, J. G.; Hundsdorfer, W. H.; Sommeijer, B. P., Convergence properties of the Runge-Kutta-Chebyshev method, Numer. Math., 57, 157-178 (1990) · Zbl 0697.65072
[15] Wathen, A. J., Realistic eigenvalue bounds for the Galerkin mass matrix, IMA J. Numer. Anal., 7, 449-457 (1989) · Zbl 0648.65076
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.