Hybrid Symbolic-Numeric Methodology for Fluid Dynamics Simulations

Abstract

With all of the experience and knowledge gained from symbolic and numeric computing in the last 20 years along with advances in software engineering and the advances in terms of the availability of powerful computers, it seems that the time is at hand to revisit the original problem of solving partial differential equations using a combination of numerical calculation and symbolic manipulation. We believe that hardware technology today has become powerful enough to realize the original dream of implementing mathematics at a high level on machines in robust and efficient ways. We used computer aided symbolic computation in our previous paper 1 to explore the problem of solving partial differential equations in a more general way. Here we revisit some classic techniques in light of new technology and extend the new methodology emerging here. Continued development of the hybrid methodology is presented with the hope of inspiring new thought in the creation of hybrid symbolic-numeric algorithms for solving the equations of mathematical physics from a fundamental perspective.

Generalized Approaches for Constructing Compact Schemes

“…, a l . The path operator algebra was implemented using ExprLib [10][11][12][13][14][15][16] and this was used to perform all relevant calculations in this paper.…”

Mathematical Analysis and Computational Integration of Massive Heterogeneous Data from the Human Retina

“…In classical implementations of the active contour algorithms by level set method, the contours around objects approximated as a zero level of the function which is the numerical solution of equation (2)[8], [9]. It is possible to replace these numerical methods by deriving an analytic model of the active contour and obtain analytic expressions for their gradients using symbolic computation.…”