zbMATH — the first resource for mathematics

An adaptive Rothe method for nonlinear reaction-diffusion systems. (English) Zbl 0789.65075
The authors study a reaction-diffusion system of the form (1) \(u_ t - \nabla \cdot (a(x)\nabla u) = f(u,t)\) for \(x\in \Omega \subset \mathbb{R}^ 1\), \(t\in [0,T]\), \(u: \Omega \times [0,T] \to \mathbb{R}^ k\), where \(u\) satisfies some boundary conditions and takes an initial value \(u_ 0\) in \(L_ 2(\Omega)\). The problem (1) is reformulated as an abstract Cauchy problem as follows:(2) \(\dot u(t) + Au(t) = f(u(t),t)\), \(t\in [0,T]\), \(u(0) = u_ 0\), where \(u: [0,T] \to L_ 2(\Omega)\) and \(A\) denotes the weak representation of the diffusion operator in the original problem (1).
After adding on both sides of (2) a bounded linear operator representing (with the sign minus) an approximation of the Jacobian \(f_ u(u_ 0)\) one can write out explicitly, using the theory of semigroups, a formal solution for this modified problem (2). The numerical analogue of the abstract solution formula is obtained via an adaptive two-stage Runge- Kutta method. Using suitable stability functions one gets strongly \(A\)- stable or \(L\)-stable Euler methods. A way to achieve a prescribed tolerance of the local error is given.
Elliptic problems arising during the realization of one time step of the above adaptive Runge-Kutta method are reformulated in variational forms and then solved approximately by the finite element method applying quadratic trial functions. Both local and global error estimates of the finite element method are given. For the whole solution process some criteria for global error estimation are proposed and investigated.
Next, the full adaptive algorithm is written out in a pseudocode. Finally, the authors use their method for the numerical solution of two interesting nonlinear reaction-diffusion systems, one from population ecology and the second one from flame propagation.
Reviewer: S.Burys (Kraków)

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
35K57 Reaction-diffusion equations
Full Text: DOI
[1] Babuška, I.; Rheinboldt, W.C., Error estimates for adaptive finite element computations, SIAM J. numer. anal., 15, 736-754, (1978) · Zbl 0398.65069
[2] Bornemann, F.A., An adaptive multilevel approach to parabolic equations I: general theory and 1D-implementation, IMPACT comput. sci. engrg., 2, 279-317, (1990) · Zbl 0722.65055
[3] Bornemann, F.A., An adaptive multilevel approach to parabolic equations II: variable-order time discretization based on a multiplicative error correction, IMPACT comput. sci. engrg., 3, 93-122, (1991) · Zbl 0735.65066
[4] Deuflhard, P.; Bader, G., A semi-implicit mid-point rule for stiff systems of ordinary differential equations, () · Zbl 0522.65050
[5] Dwyer, H.A.; Kee, R.J.; Sanders, B.R., Adaptive grid method for problems in fluid mechanics and heat transfer, Aiaa j., 10, 1205-1212, (1980) · Zbl 0454.76074
[6] Golub, G.H.; Van Loan, C.F., Matrix computations, (1989), The John Hopkins University Press Baltimore, MD · Zbl 0733.65016
[7] Higham, D.J., Bounding the error in Gaussian elimination for tridiagonal systems, () · Zbl 0716.65025
[8] Lang, J.; Walter, A., A finite element method adaptive in space and time for nonlinear reaction-diffusion-systems, (), Preprint SC 92-5
[9] Murray, J.D., Lecture on nonlinear-differential-equation models in biology, (1977), Oxford University Press Oxford · Zbl 0379.92001
[10] Ostermann, A.; Kaps, P.; Bui, T.D., The solution of a combustion problem with rosenbrock methods, ACM trans. math. software, 12, 354-361, (1986) · Zbl 0619.76088
[11] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer Berlin · Zbl 0516.47023
[12] Rothe, E., Zweidimensionale parabolische randwertaufgaben als grenzfall eindimensionaler randwertaufgaben, Math. ann., 102, 650-670, (1930) · JFM 56.1076.02
[13] Strehmel, K.; Weiner, R., Linear-implizite Runge-Kutta-methoden und ihre anwendungen, (1992), Teubner Stuttgart · Zbl 0759.65047
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.