Increase of accuracy of projective-difference schemes
In this paper with simple examples there is examined one of the improvement methods of approximate solution, which is derived with integral equalities for elliptic differential problems. The improvement method is to use some approximate systems having low order of accuracy and depending on the mesh size as the parameter. A linear combination of solutions of these problems is made, which has a given order of accuracy limited by only a degree of smoothness and the data of the differential problem.
An idea of this method is due to L.F.Richardson, but E.A.Volkov and some other mathematicians obtained a constructive proof for some problems in the 1950's.
We research the realization of this method for an ordinary differential equation (in detail, as as illustration), an elliptic differential equation in a rectangle and in the domain with a smooth boundary, and an evolutional equation with a bounded operator.
- 1.G.I.Marchuk. Methods of computing mathematics. Novosibirsk, "Nauka", 1973.Google Scholar
- 2.R.Bellman. Introduction in matrix theory. New York, Toronto, London, 1960.Google Scholar
- 3.S.L.Sobolev. Some applications of functional analysis in mathematical physics. Printed by Siberian Brunch of AS of the USSR, Novosibirsk, 1962.Google Scholar
- 4.E.A.Volkov. Solving Dirichlet's problem by: method of improvement by higher order differences "Differential equations", v. 1, No 7, 8, 1965.Google Scholar
- 5.O.A. Ladizenskaja, N.N. Uraltzeva. Linear and quasi-linear equations of elliptic type. Moscow, "Nauka", 1964.Google Scholar
- 6.J.L.Lions. Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris, 1969.Google Scholar
- 7.A.A. Samarski. Introduction in theory of difference schemes. Moscow, "Nauka", 1971.Google Scholar
- 8.H.Cartan. Calcul différentiel. Formes différentielles. Paris, 1967.Google Scholar