Method to Solve 1D Unsteady Transport and Flow Equations

Abstract

A method of solution for a fixed and nonstaggered grid is proposed. This method is obtained by a modification of the standard method of integration applied in the Galerkin procedure. The proposed approach can be used to the linear basis functions. For approximation of any function in an element, the weighted averages of weighting parameter ω are used. This approach yields a six-point implicit scheme that can be used either for the transport equation with dominated advection or for the full flow equations and their simplifications. It ensures a uniform approach to the solution of both problems. The particular cases of this method are the various well-known difference schemes and the standard finite-element method. The Fourier analysis carried out for the linear equations showed that the proposed approach has advantageous numerical properties. It has been confirmed by calculations carried out for the different kinds of transport and flow.