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

The WR-HSS iteration method for a system of linear differential equations and its applications to the unsteady discrete elliptic problem