Lie groups of point transformations and their corresponding symmetry algebras are determined for a general system of second order differential equations, special cases of which include the multigroup diffusion equations and the ''FLIP form'' of the P/sub L/ equations. It is shown how Lie symmetry algebras can be used to motivate, formulate and simplify double sweep algorithms for solving two-point boundary value problems that involve systems of second order differential equations. A matrix Riccati equation that appears in double sweep algorithms is solved exactly by regarding a set of first integrals of the second order system as a set of …
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Los Alamos National Lab., NM (USA)
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New Mexico
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Lie groups of point transformations and their corresponding symmetry algebras are determined for a general system of second order differential equations, special cases of which include the multigroup diffusion equations and the ''FLIP form'' of the P/sub L/ equations. It is shown how Lie symmetry algebras can be used to motivate, formulate and simplify double sweep algorithms for solving two-point boundary value problems that involve systems of second order differential equations. A matrix Riccati equation that appears in double sweep algorithms is solved exactly by regarding a set of first integrals of the second order system as a set of first order differential invariants of the group of point transformations that is admitted by the system. A second computational application of symmetry algebras is the determination of invariant difference schemes which are defined as difference schemes that admit the same groups of point transformations as those admitted by the differential equations that they simulate. Prolongations of symmetry algebra vector fields that are required to construct invariant difference equations are defined and found. Examples of invariant difference schemes are constructed from the basic difference equation invariance conditions and shown to be exact. 15 refs.
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