Gröbner bases and generation of difference schemes for partial differential equations

Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

Volumn 2, Issue , 2006, Pages

  Gerdt, Vladimir P ^a   Blinkov, Yuri A ^b   Mozzhilkin, Vladimir V ^b  

^a JOINT INSTITUTE FOR NUCLEAR RESEARCH   (Russian Federation)

^b SARATOV STATE UNIVERSITY   (Russian Federation)

Author keywords

Burgers equation; Computer algebra; Conservative difference schemes; Difference algebra; Falkowich Karman equation; Gr bner basis; Janet like basis; Linear difference ideal; Partial differential equations

Indexed keywords

EID: 84889236623     PISSN: None     EISSN: 18150659     Source Type: Journal    
DOI: 10.3842/SIGMA.2006.051     Document Type: Article

References (51)

1. Godunov S.K., Ryaben'kii V.S., Difference schemes. An introduction to the underlying theory, New York, Elsevier, 1987.
2. Strikwerda J.C., Finite difference schemes and partial differential equations, 2nd ed., Philadelphia, SIAM, 2004.
3. Ganzha V.G., Vorozhtsov E.V., Numerical solutions for partial differential equations: problem solving usingMathematica, Boca Raton, CRC Press, 1996.
4. Quarteroni A., Valli A., Numerical approximation of partial differential equations, 2nd ed., Berlin, Springer-Verlag, 1997.
5. Thomas J.W., Numerical partial differential equations: finite difference methods, 2nd ed., New York, Springer-Verlag, 1998.
6. Thomas J.W., Numerical partial differential equations: conservation laws and elliptic equations, New York, Springer-Verlag, 1999.
7. Samarskii A.A., The theory of difference schemes, New York, Marcel Dekker, 2001.
8. Morton K.W., Mayers D.F., Numerical solution of partial differential equations, 2nd ed., Cambridge, Cambridge University Press, 2005.
9. Ganzha V.G., Vorozhtsov E.V., Computer-aided analysis of difference schemes for partial differential equations, New York, Wiley-Interscience, 1996.
10. Liska R., Shashkov M.Yu., Algorithms for difference schemes construction on non-orthogonal logically rectangularmeshes, in Proceedings of ISSAC'91, New York, ACM Press, Addison Wesley, 1991, 419-426.
11. Liska R., Shashkov M.Y., Solovjov A.V., Support-operators method for PDE discretization: symbolic algorithmsand realization, Mathematics and Computers in Simulation, 1994, V.35, 173-183.
12. Fournié M., Symbolic derivation of different class of high-order schemes for partial differential equations, inComputer Algebra in Scientific Computing / CASC'99, Springer-Verlag, 1999, 93-100.
13. Köller R., Mohl K.D., Schramm H., Zeitz M., Kienle A., Mangold M., Stein E., Gilles E., Method of lineswithin the simulation environment DIVA for chemical processes, in Adaptive Method of Lines, Chapmannand Hall, 2001, 371-406.
14. Mozzhilkin V.V., Blinkov Yu.A., Methods of constructing difference schemes in gas dynamics, Transactionsof Saratov University, 2001, V.1, N 2, 145-156 (in Russian).
15. Buchberger B., An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomialideal, PhD Thesis, University of Innsbruck, Institute for Mathematics, 1965 (in German).
16. Buchberger B., Winkler F. (Editors), Gröbner bases and applications, Cambridge University Press, 1998.
17. http://www.maplesoft.com/
18. http://www.wolfram.com/
19. Greuel G.-M., Pfister G., Schönemann H., Singular 3.0. A computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserslautern, 2005, http://www.singular.uni-kl.de/
20. http://magma.maths.usyd.edu.au/magma/
21. Cohn R.M., Difference algebra, Tracts in Mathematics, Vol. 17, Interscience Publishers, 1965.
22. Kondratieva M.V., Levin A.B., Mikhalev A.V., Pankratiev E.V., Differential and difference dimensionpolynomials. Mathematics and its applications, Dordrecht, Kluwer, 1999.
23. Buchberger B., Gröbner bases: an algorithmic method in polynomial ideal theory, in Recent Trends inMultidimensional System Theory, Dordrecht, Reidel, 1985, 184-232.
24. Chyzak F., Salvy B., Non-commutative elimination in Ore algebras proves multivariate identities, J. Symbolic Computation, 1998, V.26, 187-227.
25. Chyzak F., Quadrat A., Robertz D., Ore Modules: a symbolic package for the study of multidimensionallinear systems, in Applications of Time-Delay Systems, Editors J. Chiasson and J.-J. Loiseau, Springer, toappear (see http://wwwb.math.rwth-aachen.de/OreModules).
26. Greuel G.-M., Levandovskyy V., Schönemann H., Plural. A subsystem of the computer algebra systemSingular 3.0 for computations with non-commutative polynomial algebras, Centre for Computer Algebra, University of Kaiserslautern, 2005, http://www.singular.uni-kl.de.
27. Gerdt V.P., Involutive algorithms for computing Gröbner bases, in Computational Commutative and Non-Commutative Algebraic Geometry, Amsterdam, IOS Press, 2005, 199-225, math.AC/0501111.
28. http://invo.jinr.ru.
29. Gerdt V.P., Robertz D., A Maple package for computing Gröbner bases for linear recurrence relations, cs.SC/0509070.
30. Janet M., Leçons sur les Systèmes d'Equations aux Dérivées Partielles, Cahiers Scientifiques, IV, Paris, Gauthier-Villars, 1929.
31. Thomas J., Differential systems, New York, American Mathematical Society, 1937.
32. Gerdt V.P., Completion of linear differential systems to involution, in Computer Algebra in Scientific Computing CASC'99, Editors V.G. Ganzha, E.W. Mayr and E.V. Vorozhtsov, Berlin, Springer, 1999, 115-137, math.AP/9909114.
33. Gerdt V.P., Blinkov Yu.A., Janet-like monomial division. Janet-like Gröbner bases, in Computer Algebrain Scientific Computing / CASC 2005, LNCS, Springer, 2005, 174-195.
34. Gerdt V.P., On computation of Gröbner bases for linear difference systems, math-ph/0509050.
35. von zur Gathen J., Gerhard J., Modern computer algebra, 2nd ed., Cambridge University Press, 2003.
36. Gerdt V.P., Blinkov Yu.A., Involutive bases of polynomial ideals. Minimal involutive bases, Mathematicsand Computers in Simulation, 1998, V.45, 519-560, math.AC/9912027, math.AC/9912029.
37. Apel J., Hemmecke R., Detecting unnecessary reductions in an involutive basis computation, RISC Linz Report Series 02-22, 2002.
38. http://www-sop.inria.fr/saga/POL.
39. http://www.math.uic.edu/_jan/demo.html.
40. Shokin Yu.I., Yanenko N.N., Method of differential approximation. Application to gas dynamics, Nauka, Siberian Division, 1985 (in Russian).
41. Richtmyer R.D., Morton K.W., Difference methods for initial-value problems, 2nd ed., New York, John Wiley & Sons, 1967 (reprinted Krieger Publishing Company, New York, 1994).
42. Godunov S.K., A finite difference method for numerical computation and discontinuous solutions of theequations of fluid dynamics, Math. Sb., 1959, V.47, 271-306 (in Russian).
43. Toro E.F., Riemann solvers and numerical methods for fluid dynamics, 2nd ed., Berlin, Springer-Verlag, 1997.
44. Czapor S.R., Solving algebraic equations: combining Buchberger's algorithm with multivariate factorization, J. Symbolic Computation, 1989, V.7, 49-53.
45. von Karman T., Collected works, von Karman Institute for Fluid Dynamics, 1975 (see also Butterworth Scientific Publ., London, 1956).
46. Bykov V., Kutmanov A., Lazman M., Elimination methods in polynomial computer algebra, Kluwer Academic Publishers, 1997.
47. Jameson A., Transonic flow calculations, von Karman Institute for Fluid Dynamics Lecture Series, Vol. 87, 1976.
48. Jameson A., Numerical methods in fluid dynamics, Hemisphere, 1978, 1-87.
49. Spanier J., Oldham K.B., The unit-step u(x - a) and related functions, Ch. 8, in An Atlas of Functions, Washington, DC, Hemisphere, 1987.
50. Yee H.C., A class of high-resolution explicit and implicit shock-capturing methods, Technical Report Lecture Series 1989-04, von Karman Institute for Fluid Dynamics, 1989.
51. Manizini M., Numerical methods for 1D compressible flows, an interactive book, see here.