Interpolation Errors in Rectangular and Diamond Characteristic Grids

Abstract

Analytical predictions of numerical errors in method of characteristics analyses using time-line interpolation in a rectangular grid are obtained for (1) both time-line and space-line interpolation and (2) both rectangular and diamond grids. Amplitude and frequency errors are investigated for each of these four cases using purpose-developed polynomial transfer matrices. Both semiinfinite and finite pipes are investigated. The time-line analysis permits reach back in time and the space-line analysis permits reach out in space. A common definition is adopted for the Courant number in the four cases and it is shown why stability can be achieved in reach-out analyses with Courant numbers greater than 1. In contrast with most work on error estimation, the predicted errors are obtained analytically, not numerically. This is made possible by restricting the analysis to special, but important, cases such as liquid-filled pipes in which the waves may be assumed to propagate at constant speed. Furthermore, the development is restricted to inviscid flows, thereby enabling interpolation errors to be assessed in the absence of complicating influences of discretization errors. In contrast with the latter, it is found that interpolation errors are more sensitive to the shape of numerical grids (i.e., Courant number and rectangular versus diamond grid) than to the size of the numerical time step.