Orthogonal double-sweep method

A variant of the double-sweep method based on an orthogonal transformation of the unknowns. Let, for , a boundary value problem be examined for a pair of linear ordinary differential equations

(1)

(2)

with conditions of the form

(3)

(4)

Let the given functions , , be continuous on the segment . A solution of the boundary value problem (1)–(4) by the orthogonal double-sweep method is realized as follows.

I) The auxiliary Cauchy problem

(5)

(6)

(7)

is solved, where

(the direct double-sweep).

II) The condition is tested, and if it is fulfilled, the Cauchy problem

(8)

(9)

where

is solved in the direction from to (the inverse double-sweep).

III) The required functions are calculated using the formulas

If the solution of the boundary value problem (1)–(4) exists and is unique and stable with respect to small changes of the coefficients and the free terms defining the problem, then and the method in question is also stable (see [2]).

A system of linear algebraic equations

(10)

(11)

(12)

(13)

where , , , is solved according to the following rules.

1) Using the formulas

the quantities are calculated successively when (the direct double-sweep).

2) The condition is tested, and if it is fulfilled, then

and

are calculated, when (inverse double-sweep).

3) The values of the required solution of the system of equations (10)–(13) are calculated using the formulas

If a solution of the system of equations (10)–(13) exists and is unique and stable with respect to small changes of the coefficients and the free terms, then the orthogonal double-sweep method in question is also stable (see [2]).

Methods based on the use of a fundamental system of solutions of a homogeneous system of equations which aim at transferring the boundary conditions are sometimes called orthogonal double-sweep methods (see [1], [3]). However, these methods are really variants of the shooting method.

References

[1]	N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)
[2]	V.I. Krylov, V.V. Bobkov, P.I. Monastyrnyi, "Computing methods of higher mathematics" , 2 , Minsk (1975) (In Russian)
[3]	A.A. Samarskii, E.S. Nikolaev, "Numerical methods for grid equations" , 1–2 , Birkhäuser (1989) (Translated from Russian)

Comments

Care should be taken with some steps in the above algorithm, as severe cancellation may occur (e.g. in ). Also, the method requires solving non-linear equations, whereas the system is simple and linear. This aspect is also shared by the Riccati method (or invariant imbedding), which however uses one non-linear equation only. Other variants, with similar ideas as this "orthogonal double-sweep method" , are in [a2].

References

[a1]	U.M. Ascher, R.M.M. Mattheij, R.D. Russell, "Numerical solution for boundary value problems for ordinary differential equations" , Prentice-Hall (1988)
[a2]	G.M. Meyer, "Continuous orthonormalization for boundary value problems" J. Comput. Phys. , 62 (1986) pp. 248–262