Automatic Finite Volume Discretizations Through Symbolic Computations

The finite volume method is a popular method for the solution of systems of partial differential equa-tions whenever local conservation properties are desired. However, engineering software usually does not reflect the underlying continuous mathematical formulation internally, but instead implements the discrete system directly, sacrificing a higher-level representation. As a consequence, the discretiza-tion code needs to be altered whenever the mathematical formulation changes. For modeling pur-poses, however, it is often desirable to only modify the mathematical formulation and let the software deal with the derivation of the discrete equations. In the context of computational fluid dynamics, OpenFOAM [1] has successfully demonstrated that such a goal can in principle be achieved. We present results obtained from coupling the symbolic computation library ViennaMath [2] to a finite volume code to solve systems of PDEs on structured and unstructured grids in one, two, and three dimensions. As an example, a fully automatic Scharfetter-Gummel-like stabilization [3] of a finite volume method for a drift-diffusion system in the context of semiconductor device simulation is discussed. Nonlinear couplings are automatically handled through Picard iterations or a globalized Newton-Raphson scheme. Moreover, we demonstrate that the mathematical formulation can be easil