The numerical Green's function technique for boundary value problems in networks

Abstract

The numerica Green's function technique was initially proposed for the solution of partial differential equations in particular problems of non-steady flow and water quality modelling in hydraulic networks. In this paper the method is presented in a general framework applicable to a variety of differential equations in networks. The basic idea behind the method lies in the unified manner of considering the boundary conditions at the common network nodes, and decomposing the large sparse system of equations produced by the numerical scheme in easy-to-solve systems for each reach of the network, and a smaller system of equations for the network nodes. The statement of the boundary value problems in ordinary differential equations in networks is presented first, followed by a general formulation of the proposed numerical method for the steady state (ordinary differential equation) case. In each network reach the sought solution is represented as a superposition of three numerically obtained auxiliary solutions: one homogeneous (zero boundary conditions) solution, and two Green's function solutions (one for each reach end) multiplied by the unknown values at the two ends of the reach. To obtain the numerical Green's function corresponding to one reach end, a value of one is imposed at that boundary and a value of zero at the other. The fluxes at the reach ends are expressed by the same values, and continuity balance relations at the network nodes are used to form a system of linear equations for the values of the unknown quantities at the network nodes Finally, the possibilities of application of the proposed technique to partial differential equations (non-steady problems) are briefly discussed.