  We consider the numerical method for non-self-adjoint positive definite
linear differential equations, and its application to the unsteady discrete
elliptic problem, which is derived from spatial discretization of the unsteady
elliptic problem with Dirichlet boundary condition. Based on the idea of the
alternating direction implicit (ADI) iteration technique and the
Hermitian/skew-Hermitian splitting (HSS), we establish a waveform relaxation
(WR) iteration method for solving the non-self-adjoint positive definite linear
differential equations, called the WR-HSS method. We analyze the convergence
property of the WR-HSS method, and prove that the WR-HSS method is
unconditionally convergent to the solution of the system of linear differential
equations. In addition, we derive the upper bound of the contraction factor of
the WR-HSS method in each iteration which is only dependent on the Hermitian
part of the corresponding non-self-adjoint positive definite linear
differential operator. Finally, the applications of the WR-HSS method to the
unsteady discrete elliptic problem demonstrate its effectiveness and the
correctness of the theoretical results.
