On monotonicity of difference schemes for computational physics

SIAM Journal on Scientific Computing

Volumn 25, Issue 5, 2004, Pages 1557-1584

  Borisov, V S ^a   Sorek, S ^a  

^a BEN GURION UNIVERSITY OF THE NEGEV   (Israel)

Author keywords

Boundary maximum principle; Difference schemes; Differential equations; Grid connectedness; Monotonicity; Stability; TVD schemes

Indexed keywords

COMPUTATIONAL FLUID DYNAMICS; DIFFERENCE EQUATIONS; HEAT TRANSFER; MATHEMATICAL MODELS; OSCILLATIONS; PROBLEM SOLVING; STABILITY;

DIFFERENCE SCHEMES; GRID CONNECTEDNESS; MONOTONICITY;

COMPUTATIONAL METHODS;

EID: 6344284739     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827502406695     Document Type: Article

References (36)

1. L. ABRAHAMSSON AND S. OSHER, Monotone difference schemes for singular perturbation problems, SIAM J. Numer. Anal., 19 (1982), pp. 979-992.
2. V. V. AKIMENKO, Monotone high-order accurate schemes for transport and parabolic equations, Comput. Math. Math. Phys., 39 (1999), pp. 590-600.
3. V. V. AKIMENKO, The maximum principle and nonlinear monotone schemes for parabolic equations, Comput. Math. Math. Phys., 39 (1999), pp. 805-816.
4. D. A. ANDERSON, J. C. TANNEHILL, AND R. H. PLETCHER, Computational Fluid Mechanics and Heat Transfer, Hemisphere Publishing Corporation, New York, 1984.
5. V. S. BORISOV, On discrete maximum principles for linear equation systems and monotonicity of difference schemes, SIAM J. Matrix Anal. Appl., 24 (2003), pp. 1110-1135.
6. P. G. CIARLET, Discrete maximum principle for finite-difference operators, Aequationes Math., 4 (1970), pp. 338-352.
7. S. R. CHAKRAVARTHY AND S. OSHER, High Resolution Applications of the Osher Upwind Scheme for the Euler Equations, AIAA paper 83-1943, in 6th Computational Fluid Dynamics Conference (Danvers, MA) Collection of Technical Papers, 1983.
8. M. ELIASSI AND R. J. GLASS, On the continuum-scale modeling of gravity-driven fingers in unsaturated porous media: The inadequacy of the Richards equation with standard monotonic constitutive relations and hysteretic equations of state, Water Resources Research, 37 (2001), pp. 2019-2035.
9. B. ENGQUIST AND S. OSHER, Stable and entropy satisfying approximations for transonic flow calculations, Math. Comp., 34 (1980), pp. 45-75.
10. P. A. FORSYTH AND M. C. KROPINSKI, Monotonicity considerations for saturated-unsaturated subsurface flow, SIAM J. Sci. Comput., 18 (1997), pp. 1328-1354.
11. V. G. GANZHA AND E. V. VOROZHTSOV, Numerical Solutions for Partial Differential Equations, CRC Press, New York, 1996.
12. S. K. GODUNOV, A finite difference method for the numerical computation and discontinuous solutions of fluid dynamics, Mat. Sb., 47 (1959), pp. 271-306 (in Russian).
13. A. HARTEN, J. M. HYMAN, AND P. D. LAX, WITH APPENDIX BY B. KEYFIZ, On finite-difference approximations and entropy conditions for shocks, Comm. Pure Appl. Math., 29 (1976), pp. 297-322.
14. A. HARTEN, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49 (1983), pp. 357-393.
15. R. HORVATH, On the monotonicity conservation in numerical solutions of the heat equation, Appl. Numer. Math., 42 (2002), pp. 189-199.
16. N. N. KALITKIN, Numerical Methods, Nauka, Moscow, 1978 (in Russian).
17. O. A. LADYŽENSKAJA, V.A. SOLONIKOV, AND N.N. URALCEVA, Linear and Quasi-Linear Equations of Parabolic Type, AMS, Providence, RI, 1968.
18. P. LANCASTER, Theory of Matrices, Academic Press, New York, 1969.
19. P. D. LAX AND R. D. RICHTMYER, Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math., 9 (1956), pp. 267-293.
20. L. LEBOUCHER, Monotone scheme and boundary conditions for finite volume simulation of magnetohydrodynamic internal flows at high Hartman number, J. Comput. Phys., 150 (1999), pp. 181-198.
21. L. MIRSKY, An Introduction to Linear Algebra, Dover, New York, 1990.
22. K. W. MORTON AND D. F. MAYERS, Numerical Solution of Partial Differential Equations, Cambridge University Press, Cambridge, UK, 1994.
23. K. W. MORTON, Numerical Solution of Convection-Diffusion Problems, Chapman & Hall, London, 1996.
24. E. S. ORAN AND J. P. BORIS, Numerical Simulation of Reactive Flow, Elsevier, New York, 1987.
25. J. M. ORTEGA AND W. C. RHEINBOLDT, Iterative Solutions of Nonlinear Equations in Several Variables, Academic Press, New York, London, 1970.
26. V. V. OSTAPENKO, On monotonicity of difference schemes, Siberian Math. J., 39 (1998), pp. 959-972.
27. V. V. OSTAPENKO, On the strong monotonicity of nonlinear difference schemes, Comput. Math. Math. Phys., 38 (1998), pp. 1119-1133.
28. M. REED AND B. SIMON, Methods of Modern Mathematical Physics, Academic Press, New York, 1972.
29. R. D. RICHTMYER AND K. W. MORTON, Difference Methods for Initial-Value Problems, 2nd ed., Wiley-Interscience, New York, 1967.
30. A. A. SAMARSKIY, On the monotone difference schemes for elliptical and parabolic equations in the case of not self-conjugate elliptical operator, Zh. Vychisl. Mat. Mat. Fiz., 5 (1965), pp. 548-551 (in Russian).
31. A. A. SAMARSKIY AND A. V. GULIN, Stability of Finite Difference Schemes, Nauka, Moscow, 1973 (in Russian).
32. A. A. SAMARSKIY, Theory of Finite Difference Schemes, Nauka, Moscow, 1977 (in Russian).
33. A. H. SCHATZ, A weak discrete maximum principle and stability of the finite element method in L∞ on plane polygonal domains. I, Math. Comp., 34 (1980), pp. 77-91.
34. V. V. SMELOV, Algebraic aspect of the discrete maximum principle, Russian J. Numer. Anal. Math. Modelling, 16 (2001), pp. 175-190.
35. G. STOYAN, On maximum principle for monotone matrices, Linear Algebra Appl., 78 (1986), pp. 147-161.
36. G. WINDISH, M-Matrices in Numerical Analysis, BSB Teubner, Leipzig, 1989.