Comparison of numerical and analytical Jacobians

Abstract

The results are presented of comparison between numerical and analytical approaches for forming the Jacobian matrices that arise from implicit discretizations of the governing equations The derivation of the exact analytical Jacobians using the MACSYMA symbolic manipulation expert system is discussed in detail. The numerical differentiation procedure used to develop the numerical Jacobians is described. The Jacobian entries obtained from the preceding approaches were tested by comparing the density residuals obtained for selected test cases. The test cases include supersonic flat plate and compression corner geometries. The results indicate that the two approaches are virtually identical, although the numerical Jacobian approach may have some implementation advantages over the exact analytical approach.