Singular Value Decomposition–Based Collocation Spectral Method for Quasi-Two-Dimensional Laminar Water Hammer Problems

Abstract

The radial distributions of velocity components need to be resolved in quasi-two-dimensional laminar water hammer problems. In a collocation spectra method, the radial distributions are approximated with Chebyshev expansions and the equations are assumed valid at the collocation points. The traditional collocation method requires an equal number of equations and unknown expansion coefficients, which is sometimes difficult to implement. The proposed model adopts extra collocation points to provide extra equations for expansion coefficients to construct an overdetermined system. Singular value decomposition is used to solve the overdetermined system. In the new method, the boundary conditions can be naturally incorporated into the system. However, the accuracy of the boundary condition equation is not acceptable because of least-squares approximation. Large multipliers are introduced to enhance the accuracy of the boundary condition equations. Spatial variation in the axial direction and time advancement are treated using the method of characteristics.