Exact finite-difference schemes for two-dimensional linear systems with constant coefficients

An exact finite-difference scheme for a system of two linear differential equations with constant coefficients, ( d / d t ) x ( t ) = A x ( t ) , is proposed. The scheme is different from what was proposed by Mickens [Nonstandard Finite Difference Models of Differential Equations, World Scientific, New Jersey, 1994, p. 147], in which the derivatives of the two equations are formed differently. Our exact scheme is in the form of ( 1 / φ ( h ) ) ( x k + 1 - x k ) = A [ θ x k + 1 + ( 1 - θ ) x k ] ; both derivatives are in the same form of ( x k + 1 - x k ) / φ ( h ) .