SEVERAL papers have dealt with the solution by the mesh method of the Sturm-Liouville problem ( u′ p ) − qu + λru = 0, u (0) = u (1) = 0, (0.1) 0 < c 1 ⩽ p(x) ⩽ c 2, 0 ⩽ q(x) ⩽ c 3, 0 < c 4 ⩽ r(x) ⩽ c 5 , (0.2) In the majority of these [1–4], difference schemes of a particular kind are considered, and reasonably smooth coefficients are assumed when examining convergence and accuracy, and the expansion of the error in powers of the mesh interval h. It was shown in [1] that a solution of problem (0.1) exists when p, q, r ϵ C (2), and that the solution of the difference problem is convergent. The problem u″ + λru = 0, u(0) = u(1) = 0 ( p = 1, q = 0, r ϵ C (4) was solved in [2], and bounds of O( h) 2 obtained for the convergence rate of the difference eigenfunctions and eigenvalues. A more exact two-sided bound for the error of the eigenvalues was obtained in [3], pp. 188–201, by means of the variational principle. The error of the solution was expanded in powers of h in [4], where 2nd or 4th order schemes were used for solving (0.1) when p, q, r ϵ C (2)( C (4)). Practical utilization of high-speed computers has shown that it is pointless to develop methods for specific problems. We need numerical algorithms suitable for solving classes of problems. In this connection, a family of homogeneous difference schemes was introduced and studied in [5], whereby problems can be solved in a unified manner, whether the coefficients are continuous or discontinuous These schemes were used in [6] for solving problem (0.1). The aim of the present paper is to develop and investigate, in the class of piecewise continuous functions p( x), q( x), r( x) ( p, q, r ϵ Q (0)) homogeneous three-point difference schemes of a high order of accuracy for the problem (0.1). Such schemes are obtained from the exact scheme by analogy with [7, 8], where a boundary value problem for the equation ( u′ p )′ − qu = − ƒ(p, q, ƒ ϵ Q (0)) was solved on a non-uniform mesh.