A positive splitting method for mixed hyperbolic-parabolic systems

Numerical Methods for Partial Differential Equations

Volumn 17, Issue 2, 2001, Pages 152-168

  Gerisch, Alf ^a   Griffiths, David F ^b   Weiner, Rüdiger ^a   Chaplain, Mark A J ^b  

^a MARTIN LUTHER UNIVERSITY HALLE WITTENBERG   (Germany)

^b UNIVERSITY OF DUNDEE   (United Kingdom)

Author keywords

Finite difference approximation; Method of lines; Mixed hyperbolic parabolic PDE system; Positive methods; Splitting methods

Indexed keywords

EID: 0000460082     PISSN: 0749159X     EISSN: None     Source Type: Journal    
DOI: 10.1002/1098-2426(200103)17:2<152::AID-NUM5>3.0.CO;2-A     Document Type: Article

References (26)

1. A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bull Math Biol 60 (1998), 857-899.
2. A. R. A. Anderson, M. A. J. Chaplain, E. L. Newman, R. J. C. Steele, and A. M. Thompson, Mathematical modeling of tumour invasion and metastasis, J Theor Med, to appear.
3. T. Höfer, J. A. Sherratt, and P. K. Maini, Cellular pattern formation during Dictyostelium aggregation, Physica D 85 (1995), 425-444.
4. M. A. J. Chaplain and A. M. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, IMA J Math Appl Med Biol 10 (1993), 149-168.
5. A. Gerisch, Finite difference methods for coupled nonlinear hyperbolic and parabolic partial differential equations in one and two dimensions, Master's thesis, U Dundee, 1997.
6. A. Gerisch, D. F. Griffiths, R. Weiner, and M. A. J. Chaplain, Splitting methods for mixed hyperbolic-parabolic systems, Tech Rept NA/186, U Dundee, 1998.
7. W. Hundsdorfer, B. Koren, M. van Loon, and J. G. Verwer, A positive finite-difference advection scheme, J Comp Phys 117 (1995), 35-46.
8. Zoltán Horváth, Positivity of Runge-Kutta and diagonally split Runge-Kutta methods, Appl Numer Math 28 (1998), 309-326.
9. W. H. Hundsdorfer, Numerical solution of advection-diffusion-reaction equations, Note NM-N9603, Dept Numer Math, CWI, 1996.
10. D. F. Griffiths, A. M. Stuart, and H. C. Yee, Numerical wave propagation in an advection equation with nonlinear source term, SIAM J Numer Anal 29 (1992), 1244-1260.
11. P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J Numer Anal 21 (1984), 995-1011.
12. B. Koren, A robust upwind discretization method for advection, diffusion and source terms, Numerical methods for advection diffusion problems, V 45 of Notes on numerical fluid mechanics, C. B. Vreugdenhill and B. Koren (Editors), Vieweg, Braunschweig, 1993, pp. 117-138.
13. U. M. Ascher, S. J. Ruuth, and B. T. R. Wetton, Implicit-explicit methods for time-dependent partial differential equations, SIAM J Numer Anal 32 (1995), 797-823.
14. U. M. Asher, S. J. Ruuth, and R. J. Spiteri, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl Numer Math 25 (1997), 151-167.
15. J. Frank, W. Hundsdorfer, and J. G. Verwer, On the stability of implicit-explicit linear multistep methods, Appl Numer Math 25 (1997), 193-205.
16. J. F. B. M. Kraaijevanger, Absolute monotonicity of polynomials occurring in the numerical solution of initial value problems, Numer Math 48 (1986), 303-322.
17. C. Bolley and M. Crouzeix, Conservation de la positivité lors de la discrétisation des problèmes d'évolution paraboliques, RAIRO Anal Numer 12 (1978), 237-245.
18. W. Hundsdorfer, Trapezoidal and midpoint splittings for initial-boundary value problems, Math Comp 67 (1998), 1047-1062.
19. C. Eichler-Liebenow, N. H. Cong, R. Weiner, and K. Strehmel, Linearly implicit splitting methods for higher space-dimensional parabolic differential equations, Appl Numer Math 28 (1998), 259-274.
20. K. Strehmel and R. Weiner, Numerik gewöhnlicher Differentialgleichungen, Teubner, Stuttgart, 1995.
21. E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations I, Springer, New York, 1987.
22. R. Weiner, B. A. Schmitt, and H. Podhaisky, ROWMAP-a ROW-code with Krylov techniques for large stiff ODEs, Appl Numer Math 25 (1997), 303-319.
23. G. D. Byrne, Pragmatic experiments with Krylov methods in the stiff ODE setting, Computational ordinary differential equations, J. Cash and I. Gladwell (Editors), Oxford Univ. Press, Oxford, 1992, pp. 323-356.
24. M. A. J. Chaplain, The mathematical modeling of tumour angiogenesis and invasion, Acta Biotheor 43 (1995), 387-402.
25. M. A. J. Chaplain, Avascular growth, angiogenesis and vascular growth in solid tumours: The mathematical modeling of the stages of tumour development, Math Comp Model 23 (1996), 47-87.
26. R. Weiner and B. A. Schmitt, Order results for Krylov-W-methods, Comp 61 (1998), 69-89.